THE CALCULUS A SERIES OF MATHEMATICAL TEXTS EDITED BY EARLE RAYMOND HEDRICK Brenke. PLANE AND SOLID ANALYTIC GEOMETRY ""b, A..-™- ^-" -^ ^-" ^^"=' ::;„TH PLANE AND SPHERICAL TRIGONOMETRY WITH COMPLETE TABLES B, A„K.. Mo»«o. K.».o. and Lo.-s I»»o.p. PLANE AND SPHERICAL TRIGONOMETRY WITH BRIEF TABLES By A.rKED Mo.koe Kenvon and Louis I.oo.d. THE MACMILLAN TABLES ,, havmond Hedrick. Prepared under the direction of Earle Ra.mod ^^Tw:r:™i» .o.. a., c„..... a.,»...u». THE CALCULUS BY ELLERY WILLIAMS DAVIS H PKOFE880K OF MATHEMATICS, THE UNIVERSITY OF NEBRASKA ASSISTED BY WILLIAM CHARLES BRENKE ASSOCIATE PROFESSOR OF MATHEMATICS, THE UNIVERSITY OF NEBRASKA EDITED BY EARLE RAYMOND HEDRICK THE MACMILLAN COMPANY 1915 All rights reserved Copyright, 1912, By the MACMILLAN COMPANY. , Set up and electrotyped. Published September, 1912. Reprinted October, 191a ; May, iqt^; August, October, 1914; February, 1915. tti lilf- ^ V^-u— U-wt-^ ^M-^4 Ct^ Nortoooli ^tf«8 J. 8. Gushing Co. — Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE The significance of the Calculus, the possibility of applying it in other fields, its usefulness, ought to be kept constantly ind vividly before the student during his study of the subject, rather than be deferred to an uncertain future. Not only for students who intend to become engineers, but ilso for those planning a profound study of other sciences, the isef ulness of the Calculus is universally recognized by teachers ; t should be consciously realized by the student himself. It is )bvious that students interested primarily in mathematics, particularly if they expect to instruct others, should recognize ;he same fact. To all these, and even to the student who expects only gen- jral culture, the use of certain types of applications tends to nake the subject more real and tangible, and offers a basis for m interest that is not artificial. Such an interest is necessary ;o secure proper attention and to insure any real grasp of the essential ideas. For this reason, the attempt is made in this book to present IS many and as varied applications of the Calculus as it is possible to do without venturing into technical fields whose lubject matter is itself unknown and incomprehensible to the ;tudent, and Avithout abandoning an orderly presentation of fundamental principles. The same general tendency has led to the treatment of lOpics with a view toward bringing out their essential useful- less. Thus the treatment of the logarithmic derivative is vitalized by its presentation as the relative rate of change of a quantity; and it is fundamentally connected with the important ' compound interest law," which arises in any phenomenon in VI PREFACE which the relative rate of increase (logarithmic derivative) is constant. Another instance of the same tendency is the attempt, in the introduction of the precise concept of curvature, to explain the reason for the adoption of this, as opposed to other simpler but cruder measures of bending. These are only instances, of two typical kinds, of the way in which the effort to bring out the usefulness of the subject has influenced the j)resentation of even the traditional topics. Rigorous forms of demonstration are not insisted upon, es- pecially where the precisely rigorous proofs would be beyond the present grasp of the student. Rather the stress is laid upon the student's certain comprehension of that which is done, and his conviction that the results obtained are both reasonable and useful. At the same time, an effort has been made to avoid those grosser errors and actual misstatements of fact which have often offended the teacher in texts otherwise attractive and teachable. Thus a proof for the formula for differentiating a logarithm is given which lays stress on the very meaning of logarithms ; while it is not absolutely rigorous, it is at least just as rigorous as the more traditional proof which makes use of the limit of (1 -I- 1/?*)" as n becomes infinite, and it is far more convincing and instructive. The proof used for the derivative of the sine of an angle is quite as sound as the more traditional proof (which is also indicated), and makes use of fundamentally use- ful concrete concepts connected with circular motion. These two proofs again illustrate the tendency to make the subject vivid, tangible, and convincing to the student; this tendency will be found to dominate, in so far as it was found possible, every phase of every topic. Many traditional theorems are omitted or reduced in impor- tance. In many cases, such theorems are reproduced in exer- cises, with a sufficient hint to enable the student to master them. Thus Taylor's Theorem in several variables, for which PREFACE vil wide applications are not apparent until further study of nuithe- niatics and science, is presented in this manner. On the other hand, many theorems of importance, both from mathematical and scientific grounds, which have been omitted traditionally, are included. Examples of this sort are the brief treatment of simple harmonic motion, the wide application of Cavalieri's theorem and the prismoid formula, other api)roxi- mation formulas, the theory of least squares (under the head of exercises in maxima and minima), and many other topics. The Exercises throughout are colored by the views expressed above, to bring out the usefulness of the subject and to give tangible concrete meaning to the concepts involved. Yet formal exercises are not at all avoided, nor is this necessary if the student's interest has been secured through conviction of the usefulness of the topics considered. Far more exercises are stated than should be attempted by any one student. This will lend variety, and will make possible the assignment of different problems to different students and to classes in successive years. It is urged that care be taken in selecting from the exercises, since the lists are graded so that certain groups of exercises prepare the student for other groups which follow ; but it is unnecessary that all of any group be assigned, and it is urged that in general less than half be used for any one stu- dent. Exercises that involve practical applications and others that involve bits of theory to be worked out by the student are of frequent occurrence. These should not be avoided, for they are in tune with the spirit of the whole book ; great care has vbeen taken to select these exercises to avoid technical concepts strange to the student or proofs that are too difficult. An effort is made to remove many technical difficulties by the intelligent use of tables. Tables of Integrals and many other useful tables are appended ; it is hoped that these will be found usable and helpful. Parts of the book may be omitted without destroying the essential unity of the whole. Thus the rather complete treat- Viu PREFACE ment of Differential Equations (of the more elementary types) can be omitted. Even the chapter on Functions of Several Variables can be omitted, at least except for a few paragraphs, without vital harm ; and the same may be said of the chapter on Approximations. The omission of entire chapters, of course, would only be contemplated where the pressure of time is un- usual ; but many paragraphs may be omitted at the discretion of the teacher. Although care has been exercised to secure a consistent order of topics, some teachers may desire to alter it ; for example, an earlier introduction of transcendental functions and of por- tions of the chapter on Approximations may be desired, and is entirely feasible. But it is urged that the comparatively early introduction of Integration as a summation process be retained, since this further impresses the usefulness of the subject, and accustoms the student to the ideas of derivative and integral before his attention is diverted by a variety of formal rules. Purely destructive criticism and abandonment of coherent arrangement are just as dangerous as ultra-conservatism. This book attempts to preserve the essential features of the Calculus, to give the student a thorough training in mathematical rea- soning, to create in him a sure mathematical imagination, and to meet fairly the reasonable demand for enlivening and en- riching the subject through applications at the expense of purely formal work that contains no essential principle. E. W.DAVIS, W. C. BRENKE, E. R. HEDRICK, Editor. June, 1912. CONTENTS f Pajre numbers in Roman type refer to the body of the book ; those in italic type refer to pages of the Tables.] PAGES CHAPTER I FUNCTIONS 1-6 § 1. Dependence 1 § 2. Variables. Constants. Functions . . . ,1-2 Exercises I. Functions and Graphs ..... 2-3 § 3. The Function Notation 3 Exercises II. Substitution. Function Notation . . , 3-5 CHAPTER II RATES-^ LIMITS DERIVATIVES . 6-27 § 4. Rate of Increase. Slope 6-8 § 5. General Rules 8 § 6. Slope Negative or Zero. [Maxima and Minima.] . 8-11 Exercises III. Slopes of Curves 11-12 § 7. Speed 12-1-4 § 8. Component Speeds 14 § 9. Continuous Functions 14-15 Exercises IV. Speed 15-16 § 10. Limits. Infinitesimals 16-17 § 11. Properties of Limits 17-18 § 12. Ratio of an Arc to its Chord 18-19 § 13. Ratio of the Sine of an Angle to the Angle . . .19 § 14. Infinity 19-20 Exercises V. Limits and Infinitesimals .... 20-22 § 15. Derivatives 22-23 § 16. Formula for Derivatives 23-24 § 17. Rule for Differentiation 24-26 Exercises VI. Formal Differentiation 26-27 CHAPTER III DIFFERENTIATION OF ALGEBRAIC FUNCTIONS 28-57 Part I Explicit Functions . . . 28-43 § 18. Classification of Functions 28-29 § 19. Differentiation of Polynomials 29-31 iz CONTENTS Exercises VII. Differentiation of Polynomials § 20. Differentiation of Rational Functions. [Quotient.] Exercises VIII. Differentiation of Rational Functions § 21. Derivative of a Product § 22. Derivative of a Function of a Function Exercises IX. Short Methods. Rational Functions § 23. Differentiation of Irrational Functions . § 24. Collection of Formulas § 25. Illustrative Examples of Irrational Functions Exercises X. Algebraic Functions PAGES 31-32 32-33 34-35 35-36 36-37 37-38 38-40 40-41 41-42 42-43 Part II Equations not in Explicit Form Differentials 44-57 § 26. Solution of Equations 44-45 § 27. Explicit and Implicit Functions 45-46 § 28. Inverse Functions 46-47 § 29. Parameter Forms 47 Exercises XI. Functions not in Explicit Form . . . 47-49 § 30. Rates 49-50 § 31. The Differential Notation 50-52 § 32. Differential Formulas 52-53 Exercises XII. Differentials 54-57 CHAPTER IV FIRST APPLICATIONS OF DIFFEREN- TIATION 58-90 Part I Application to Curves Extremes . 58-70 § 33. Tangents and Normals 58-59 § 34. Tangents and Normals for Curves not in Explicit Form 59 § 35. Secondary Quantities 60 § 36. Illustrative Examples 60-61 Exercises XIII. Tangents and Normals .... 62-63 § 37. Extremes. [Maxima and Minima ] . . . .63 § 38. Critical Values 63-64 § 39. Fundamental Theorem 64 § 40. Final Tests 64-65 § 41. Illustrative Examples in Extremes .... 65-67 Exercises XIV, Extremes Part II Rates § 42. Time Rates § 43. Speed . 67-70 70-90 70 71 CONTENTS XI PAGEk § 44. Tangential Acceleration 71 § 45. Second Derivative. Flexion 71-72 Exercises XV. Second Derivatives. Acceleration . . 73-75 § 46. Concavity. Points of Inflexion .... 75 § 47. Second Test for Extremes 75-76 § 48. Illustrative Examples 76-77 § 49. Derived Curves 77-79 Exercises XVI. Flexion. Derived Curves . . . 79-81 § 50. Angular Speed 81 § 51. Angular Acceleration 81-82 § 52. Momentum. Force 82-83 Exercises XVII. Time Rates 83-85 § 53. Related Rates 85-87 Exercises XVIII. Related Rates 88-90 CHAPTER V REVERSAL OF RATES INTEGRATION SUMMATION 91-129 Part I Integrals by Reversal of Rates . 91-109 § 64. Reversal of Rates . . .... 91-92 § 65. Principle Involved m § 54 . . . . .92 § 56. Illustrative Examples 92-94 Exercises XIX. Reversal of Rates 94-95 § 67. Integral Notation 96-97 Exercises XX. Notation. Indefinite Integrals . . . 97-98 § 58. Fundamental Theorem 99 § 59. Definite Integrals 100-101 Exercises XXI. Definite Integrals 102-103 § 60. Area under a Curve 103-104 Exercises XXII. Area 105 § 61. Lengths of Curves 106-107 § 62. Motion on a Curve. Parameter Forms . . . 107 § 63. Illustrative Examples 108-109 Exercises XXIII. Length. Total Speed .... 109 Part II Integrals as Limits of Sums . 110-120 § 64. Step-by-Step Process 110-111 § 65. Approximate Summation 111-112 Exercises XXIV. Step-by-Step Summation. Approximate Results 112-113 xu CONTENTS § 66. Exact Results. Summation Formula § 67. Integrals as Limits of Sums § 68. Water Pressure .... Exercises XXV. Integrals as Limits of Sums §69. Volumes § 70. Volume of Any Frustum . Exercises XXVI. Volumes of Solids. Frusta § 71. Cavalieri's Theorem. The Prismoid Formula Exercises XXVII. General Exercises PAGES 114-116 116-117 117-119 119-120 120-121 121-123 124-125 125-127 128-129 CHAPTER VI TRANSCENDENTAL FUNCTIONS 130-173 Pakt I Logarithms Exponential Functions 130-149 § 72. Necessity of Operations on Transcendental Functions § 73. Properties of Logarithms § 74. Graphical Representation Exercises XXVIII. Logarithms and Exponentials § 75. Slope of y = logio x at x = 0. [Modulus M] § 76. Differentiation of logio x §77. Differentiation of loga X. [Napierian Base] § 78. Illustrative Examples Exercises XXIX. Logarithms § 79. Differentiation of Exponentials § 80. Illustrative Examples Exercises XXX. Exponentials §81. Compound Interest Law Example 1. Work in Expanding Gas . (142) Example 2. Cooling in a Moving Fluid . (142-143) Example 3. Bacterial Growth . . . (143) Example 4. Atmospheric Pressure . . (143) §82. Percentage Rate of Increase. [Relative Rates] Exercises XXXI. Compound Interest Law § 83. Logarithmic Differentiation. Relative Increase § 84. Logarithmic Methods .... Exercises XXXII. Logarithmic Differentiation Part II Trigonometric Functions § 85. Introduction of Trigonometric Functions . S 86. Differentiation of Sines and Cosines . 130 130-131 131-132 132-133 133-134 134 135-136 136-137 137-138 138-139 139-140 140-141 141-143 144 144-146 146-147 147-148 149 150-173 150 150-153 CONTENTS xiu PAGES § 87. Illustrative Examples 152-153 Exercises XXXIII. Trigonometric Functions . . . 153-155 § 88. Simple Harmonic Motion 155-156 § 89. Relative Acceleration 156 § 90. Vibration 157 §91. Waves 157-158 Exercises XXXIV. Simple Harmonic Motion. Vibrations 158-100 § 92. Damped Vibrations 160-162 Exercises XXXV. Damped Vibrations .... 162-163 § 93. Inverse Trigonometric Functions .... 163-164 § 94. Integrals of Irrational Functions .... 164 § 95. Illustrative Examples 164-165 Exercises XXXVI. Inverse Trigonometric Functions . 165-166 § 96. Polar Coordinates 166-168 Exercises XXXVII. Polar Coordinates .... 168-169 §97. Curvature 169-171 Exercises XXXVIII. Curvature 171-172 § 98. Collection of Formulas 173 CHAPTER VII TECHNIQUE TABLES SUCCES- SIVE INTEGRATION 174-226 Part I Technique of Integration . . 174-200 § 99. Question of Technique. Collection of Formulas . 174-175 § 100. Polynomials. Other Simple Forms . . .176 § 101. Substitution. [Algebraic and Trigonometric] . . 176-177 § 102. Substitution in Definite Integrals . . . .177 Exercises XXXIX. Elementary Integration. Substitution 178-181 § 103. Integration by Parts. [Algebraic and Transcen- dental] 181 Exercises XL. Integration by Parts 182-183 § 104. Rational Functions. [Partial Fractions] . . 184-186 Exercises XLI. Rational Functions 186-188 § 105. Rationalization of Linear Radicals .... 188-189 § 106. Quadratic Irrationals 189-190 Exercises XLII. Integrals involving Radicals. [Trigono- metric Substitutions] 191-194 § 107. Elliptic and Other Integrals 195 § 108. Binomial Differentials 195-196 XIV CONTENTS § 109. General Remarks. [Tables.] . Exercises XLIII. General Integration. [All Methods.] PAGES 196 197-200 Part II Improper and Multiple Integrals 201-226 § 110. Limits Infinite. Horizontal Asymptote . § 111. Integrand Infinite. Vertical Asymptotes § 112. Precautions Exercises XLIV. Improper Integrals § 113. Repeated Integration § 114. Successive Integration in Two Letters Exercises XLV. Successive Integration § 115. Double Integrals .... § 116. Illustrative Examples [A] Volumes by Double Integration . [B] Area in Polar Coordinates . [C] Moment of Inertia of a Thin Plate [D] Moment of Inertia in Polar Coordinates Exercises XLVI. Double Integrals . § 117. Triple and Multiple Integrals . Exercises XLVII. Multiple Integrals §118. Other Applications. Averages. Centers of Gravity Exercises XLVIII. General Problems in Integration (212) (212-213) (21.3-214) (214) 201 202-203 203 204-205 206 206-207 208-209 210-211 211-214 214-217 217-218 218-219 219-220 221-226 CHAPTER VIII METHODS OF APPROXIMATION . 227-280 Part I Empirical Curves Increments . 227-250 § 119. Empirical Curves 227 § 120. Polynomial Approximations 227 § 121. Review of Elementary Methods .... 227-229 § 122. Logarithmic Plotting . . . . - . . 229-230 § 123. Semi-Logarithmic Plotting 230 Exercises XLIX. Empirical Curves. Elementary Methods 230-233 § 124. Method of Increments 233-236 Exercises L. Empirical Curves by Increments . . . 236-238 § 125. Approximate Integration 239-240 § 126. Integration from Empirical Fornmlas . . . 240-241 § 127. Derived and Integral Curves 241-242 Exercises LI. Approximate Evaluation of Integrals . 242-243 5128. Integrating Devices. [Planimeter. Integraph] . 243-246 CONTENTS XV PAGES § 129. Tabulated Integral Values ..... 246-247 Exercises LII. Integrating Devices. Numerical Tables . 248-250 Part II Polynomial Approximations Series Taylor's Theorem . . . 250-280 § 130. Rolle's Theorem 250 § 131. The Law of the Mean. [Finite Differences] . . 251 § 132. Increments. [Small Errors] 252-253 Exercises LIII. Increments. Law of the Mean . . 253-254 § 133. Limit of Error 251-257 § 134. Extended Law of the Mean. Taylor's Theorem . 257-259 Exercises LIV. Extended Law of the Mean . . . 259-260 § 135. Application of Taylor's Theorem to Extremes . 260-261 Exercises LV. Extremes 261-263 § 136. Indeterminate Forms. [Form h- 0] . . . 263-265 §137 Infinitesimals of Higher Order .... 265-266 Exercises LVI. Indeterminate Forms. Infinitesimals . 266-267 § 138. Double Law of the Mean 267-268 § 139. The Indeterminate Form (» h- co. Vertical Asymptotes 268-269 § 140. Other Indeterminate Forms 269-270 Exercises LVII. Secondary Indeterminate Forms . . 270-271 § 141. Infinite Series 271-273 § 142. Taylor Series. General Convergence Test . , 273-275 Exercises LVIII. Taylor Series 275-276 § 143. Precautions about Infinite Series .... 276-279 Exercises LIX. Infinite Series 279-280 CHAPTER IX SEVERAL VARIABLES PARTIAL DERIVATIVES APPLICATIONS GEOMETRY 281-344 Part I Partial Differentiation Elementary Applications .... 281-297 § 144. Partial Derivatives 281-282 § 145. Technique 282 § 146. Higher Paitial Derivatives 282-283 Exercises LX. Technique of Partial Differentiation . . 283-284 § 147. Geometric Interpretation 284-285 § 148. Total Derivative 285-287 CONTENTS § 149. Elementary Use § 150. Small Errors. Partial Differentials Exercises LXI. Total Derivatives and Differentials . § 151. Significance of Partial and Total Derivatives . Example 1. Isothermal Expansion . . (292-293) Example 2. Adiabatic Expansion . . (293-294) Example 3. Implicit Equations. Contour Lines (294) Example 4. Flow of Heat in a Metal Plate (294-295) Example 5. Flow of Water in Pipes . . (295-296) Exercises LXII. Applications of Total Derivatives . Part II Applicatbons to Plane Geometry . Evolute Evolutes § 152. Envelopes § 153. Envelope of >Jormals. Exercises LXIII. Envelopes. § 154. Properties of Evolutes .... § 155. Center of Curvature .... § 156. Rate of Change of ^ § 157. Illustrative Examples .... Exercises LXIV. Properties of Evolutes . § 158. Singular Points § 159. Illustrative Examples .... § 160. Asymptotes § 161. Curve Tracing Exercises LXV. Singular Points. Asymptotes. Tracing Curve PAG! 287-2 ■ 288-2" 290-2 292-2 - 296-297 298-315 298-300 300-303 303 304-305 305 305-306 306-307 308-309 309-310 310-311 311-313 313 313-314 Part III Geometry of Space Extremes . 315-344 162. R^sum^ of Formulas (a) Distance between Two Points (&) Distance from Origin to {x, y, z) (c) Direction Cosines (d) Angle between Two Directions (e) The Plane .... (/) The Straight Line (f/) Quadric Surfaces . § 163. Loci of One or More Equations Exercises LXVI. R6sum6 of Solid Geometry (315) (315) (315) (315) (316-317) (317 (318) 315-318 318-319 319-320 CONTENTS XVU PAOBS § 164. Tangent Plane to a Surface 321-322 § 165. Extremes on a Surface. [Least Squares] . . 322-325 §166. Final Tests 325-326 Exercises LXVII. Tangent Planes. Extremes . . 327-329 § 167. Tangent Planes. Implicit Forms .... 329-330 § 168. Line Normal to a Surface 330 § 169. Parametric Forms of Equations .... 330-332 § 170. Tangent Planes and Normals. Parameter Forms . 332-333 Exercises LXVIII. Equations not in Explicit Form . . 333-334 § 171. Area of a Curved Surface 334-335 Exercises LXIX. Area of a Surface 336 § 172. Tangent to a Space Curve 337 § 173. Length of a Space Curve 338 Exercises LXX. Tangents to Curves. Lengths . . 338 Exercises LXXI. General Review. Several Variables . 339-344 CHAPTER X DIFFERENTIAL EQUATIONS 345-383 Part I Ordinary Differential Equations of THE First Order . . . . § 174. Reversal of Rates 345 § 175. Other Reversed Problems 345-346 § 176. Determination of the Arbitrary Constants . . 346-347 § 177. Vital Character of Inverse Problems . . . 347-348 §178. Elementary Definitions. Ordinary Differential Equa- tions 348 § 179. Elimination of Constants 348-350 § 180. Integral Curves 350-351 Exercises LXXII. Elimination. Integral Curves . . 351-352 § 181. General Statement 352-353 § 182. Type I. Separation of Variables . . . .353 § 183. Type II. Homogeneous Equations .... 354-355 Exercises LXXIII. Separation of Variables . . . 355-^356 § 184. Tjije III. Linear Equations 356-358 § 185. Extended Linear Equations 358 Exercises LXXIV. Linear Equations .... 359 § 186. Other Methods. Non-linear Equations . . . 359-360 Exercises LXXV. Miscellaneous Exercises . . . 360-362 XVlll CONTENTS Part II Ordinary Differential Equations of THE Second Order .... Order Part III Generalizations § 192. Ordinary Equations of Higher Order § 193. Linear Homogeneous Type § 194. Non-homogeneous Type . Exercises LXXX. Linear Equations of Higher § 195. Systems of Differential Equations § 196. Linear Systems of the First Order §197. dx/P=dy/Q = dz/R Exercises LXXXI. Systems of Equations § 198. Partial Differential Equations § 199. Relation to Systems of Ordinary Equations Exercises LXXXII. Partial Differential Equations 363-374 § 187. Special Types § 188. Type I : d-s/df- = ± k'^s Exercises LXXVI. Type I §189. Type II. Homogeneous Linear — Constant Coeffi- cients Exercises L XXVII. Type II. Linear Homogeneous § 190. Type III. Non-homogeneous Equations . Exercises LXXVIII. Non-homogeneous. Type III . §191. Type IV. One of the quantities a;, j/,?/' absent (a) Type IV (a) : (y") = . . . (371) (6) Type IV (6): (.7;, 2/',?/") = . . (371-373) (c) Type IV (c) : 4> (V, V' , V") =0 . • (373-374) Exercises LXXIX. Type IV 374 363-365 368-369 369-370 371 371-374 375-383 375 375-377 377-378 378-379 379 379 379-381 381-382 TABLES [Note page numbers in italic numerals] TABLE I SIGNS AND ABBREVIATIONS . 1-3 TABLE II STANDARD FORMULAS. . . 3-16 A. Exponents and Logarithms 3 B. Factors 4 C. Solution of Equations. Determinants 4-5 CONTENTS xix PAGES D. Applications of Algebra 5-6 E. Series. [Special. Theorems of Taylor and Fourier] . . 7-8 F. Geometric Magnitudes. Mensuration S-11 G. Trigonometric Relations 12-13 H. Hyperbolic Functions 13-14 I. Analytic Geometry _. 14-15 J. Differential Formulas * . 15-16 Table III Standard Curves . . . 17-32 A. Curves y = .r». [Chart of Entire Family] .... 17- IS B. Logarithmic Paper. Curves y = x", y ~ k.r" . . . .IS C. Trigonometric Functions 19 D. Logarithms and Exponentials. [Bases 10 and r] . . 19 E. Exponential and Hyperbolic Functions 20 F. Harmonic Curves. [Simple and Compound] . . . 31-22 G. The Roulettes. [Cycloid, Trochoids, etc.] .... 22-24 H. The Tractrix 24 I. Cubic and Quartic Curves. Contour Lines .... 25-27 J. Error or Probability Curves 28 K. Polynomial Approximations. [Taylor and Lagrange] . 28-29 L. Trigonometric Approximations. [Fourier] . . . . 29 M. Spirals 30-31 N. Quadrlc Surfaces 31-32 Table IV Standard Integrals . . 33-48 A. Fundamental General Formulas 33 B. Integrand Rational Algebraic 34-36 C. Integrand Irrational 37-39 (a) Linear radicals r = Vox+T . . . {37) (b) Quadratic radicals (3T-39) D. Binomial Differentials — Reduction Formulas . . .39 E. Integrand Transcendental 39-44 (a) Trigonometric (,39-42) (6) Trigonometric — Algebraic . (42-43) (c) luverse Trigonometric .... {43) {(i) Exponential and Logarithmic . . . (43-44) F. Some Important Definite Integrals 44-4^ G. Approximation Formulas 45-46 H. Standard Applications of Integration 4(>-48 XX CONTENTS PA6KB Table V Numerical Tables . . . 49-58 A. Trigonometric Functions. [Values and Logarithms] . . 49 B. Common Logarithms 50-51 C. Exponential and Hyperbolic Functions. Natural Logarithms 52 D. Elliptic Integral of the First Kind 53 E. Elliptic Integral of the Second Kind 53 F. Values of n(p) = r(p + 1). Gamma Function . . .54 G. Values of the Probability Integral 54 H. Values of the Integral | (e*/a;)dx 54 I. Eeciprocals ; Squares ; Cubes 55 J. Square Roots , . . 56 K. Radians to Degrees 56 L. Important Constants 57 M. Degrees to Radians 57 N. Short Conversion Table and Other Data . , . .68 IXDEX 59 Be not the first by whom the new is tried, Nor yet the last to lay the old aside." — POPK. THE CALCULUS CHAPTER I FUNCTIONS 1. Dependence. There are countless instances in -which one quantity depends upon another. The speed of a body falling from rest depends upon the time it has fallen. One's income from a given investment depends upon the amount invested and the rate of interest realized. The crops depend upon rain- fall, soil fertility and proper cultivation. In mathematics we usually deal with quantities that are definitely and completely determined by certain others. Thus the area ^ of a square is determined precisely when the length s of its side is given : A = s^; the volume of a sphere is 4 tti^/S ; the force of attraction between two bodies is k ■ m • m' /d^, where m and m' are their masses, d the distance between them, and k a certain number given by experiment. The Calculus is the stvidy of the relations between such interdependent quantities, with special reference to their rates of change. 2. Variables. Constants. Functions. A quantity which may change is called a variable. The quantities mentioned in § 1, except k and tt, are examples of variables. A quantity which has a fixed value is called a constant. Ex- amples of constants are ordinary numbers : 1, V2, — 7, 2/3, tt, 30°, log 5, and the number k in § 1. If one variable y depends on another variable x, so that ?/ is determined when x is known, y is said to be a function of x. B 1 FUNCTIONS [I, §2 The variable x, thus thought of as determining the other, is called the independent variable ; the other variable y is called the dependent variable. Thus, in § 1, the area ^ of a square is a function, A = s^, of the side s. In Algebra we learn how to express such relations by means of equations. In Analytic Geometry such relations are represented graphically. For example, if the principal at simple interest is a fixed sum p and if the interest rate r also is fixed, then the amount a, of principal and interest, varies solely with (is a function of) the time t that the principal has been at interest. In fact, Up = 100 and r = 6%, - - ^ ' : - ?> - -K ie - 1 1 1 . < (Years) Fig. 1. a=p + 2:)tr = 100 + 6 t. This is represented graphically in Fig. 1. parts of a day are neglected. The relation ^ = s* of § 1 is repre- sented in Fig. 2. EXERCISES I. — FUNCTIONS AND GRAPHS Represent graphically the following : — 1. a = 100 + 3 «, a = 300 + 4 «, a = 150 + 7«. 2. The number of feet / in terms of the number of yards ?/ in a given length is given by the equation f = 3y. In practice fractional / A \ (8 3.fe et)J 1 \ \ \ 1 4=s 2 k 1 L > s (fe et) 12 3 4 Fig. 2. 3. The temperature in degrees Fahrenheit, F, is 32 more than 9/5 the temperature in degrees Centigrade, C. 4. The distance s that a body falls from rest in a time t is given by s = 16 t'^. (Measure t horizontally and s vertically dovirnward.) I, § 3] GR.IPHS NOTATION 5. (a) y = «2 + 3x+l. (4) y = 2x'-6x. (c) y = oi?+2. w,=^;^. «^=l^- Wy.-±^ 6. The volume v of a fixed quantity of gas at a constant temperature varies inverse!}' as tlie pressure p upon the gas. 7. The amount of 6 1.00 at compound interest at 10 % per annum for t years is a =(1 + 1/10)*. 8. The area A of an equilateral triangle is a function of its side s. Determine this function, and represent the relation graphically. Express the side in terms of the area. 9. Determine the area a of a circle in terms of its radius r. Deter- mine the- radius in terms of the area. 10. The radius, surface, and volume of a sphere are functionally re- lated. Find the equations connecting each pair. Also express each of the three as a function of the circumference of a great circle of the sphere. 11. The area A bounded by the straight line y = ax + b, the ordinate y, and the axes, is a function of x. Determine it ; and also express y as a function of the area. 3. The Function Notation. A very useful abbreviation for functions consists in writing /(^) (read /of x) in place of the given expression. Thus iff{x) = x^ + 3x + l,we may write /(2) = 2^ + 3 • 2 + 1 = 11, that is, the value of u^ + Sx + l when x = 2 is 11. Likewise /(3) = 19, /(- 1) = - 1, /(O) = 1, and so on. /(«) = a^ + 3 a + 1. fill + v) = (u-\- vf + 3(w + v) + 1. Other letters than / are often used, to avoid confusion, but /is used most often, because it is the initial of the word func- tion. Other letters than x are often used for the variable. In any case, given f{x), to find /(a), simply substitute a for x in the given expression. EXERCISES II. — SUBSTITUTION FUNCTION NOTATION 1. If /(x) = x2 - 5 X + 2 find /(I), /(2), /(3), /(4), /(O), /(- 1), /(-2). From these values (and others, if needed) draw the graph of 4 FUNCTIONS [I, § 3 the curve y =/(.r). Mark its lowest point, and estimate the values of jt and y there. 2. Proceed as in Ex. 1 for each of the following functions, using the function notation in calculating values ; mark the highest and lowest points if any exist, and estimate the values of x and y at these points. (a)x3-2x + 4. (6)3x2_2x + l. (c)^^. (d)-^+-^. 2x — 3 x + 1 X — 1 (e) y = sin x, taking x =ir/Q, tt/^, t/2, 3 7r/4, tt, 0, - 7r/2. (/) j/ = logiox, taking x = 1, 2, 10, 1/10, 1/100. 3. If fix) = X* - 6 x3 + 3 x2 - 2 X + 3, calculate /(I), /(4), /(5). Hence show that one solution of the equation /(x) = is x = 1 ; and that another solution lies between 4 and 5. (This work is simplified by using the theorem that /(a) is equal to the remainder obtained by dividing f{x) by [x—a); and by using synthetic division.) 4. If /(x) = 2x2-3x + 5, show that /(a) = 2a2_3rt4-5,/(m + M) = 2 (m + 7i)2 _ 3 (m + n) + 5 ; find f{a - b), f{a + 2b), fia/b). 5. If /(x) = x2 4- 3 and (x) = 3 x + 1, show that f(l) =

0(3). Draw y =/(x) and y = 0(x). 6. In Ex. 5, draw the curve y =f{x) —

(x), 2/ =/(x) — (x). Mark the points where /(x) = (x) and estimate the values of x and y there. 11. Taking /(x) = x^, compare the graph of y=fix) with that of y -fix) + 1, and with that of y =fix + 1). I, §3] GRAPHS NOTATION 5 12. Taking any two curves y=f(x), y = (.<) ? Draw y^x^ +l/x. 13. How can you most easily draw y =f{x) + 5? y =f{x + 5) ? as- suming that y =/(x-) is drawn. 14. Draw y = x- and show how to deduce from it the graph of y = 2x- ; the graph oi y = — a;-. Assuming that y =/(x) is drawn, show how to draw the graph of 2/ = 2/(.r); that of y=-f{x). 15. From the graph of y = x"^, show how to draw the graph of y = (2 a;)-; that of y = x^ + 2 ; that of y =(a;+2)2 ; that of y={2x-SyK 16. What change is made in a curve if x, in the equation, is replaced by - .r ? if y by - 2/ ? if both things are done ? Compare the graphs of y=fix), y=f{-x), -y=f{x); y=.2f(x); y=/(x) + 2. 17. What change is made in a curve if x is replaced by 2 x, 3 x, x/2 ? Compare the graphs of y=f{x), y=f{2x), t/=/(3x), y=f{x/2); y=f{x + 2). 18. What is the effect upon a curve if, in the equation, x and y are interchanged ? Compare the graphs of y =/(x), x =/(?/). 19. Plot the following curves : (a) y+2 = sin (3x + 2), (6) y=x+s\nx, (c) j/ = 2* — sinx, (d) 2/=2='cosx, (e) 3x + 4 y = 4 sin (4 x — 3?/), (/) ?/= (cosx)/(2x + 3), (gr) sin 2/ = cos 2 X, W y = logo (x^ + 1). 20. In polar coordinates (r, 0), what change is made in a curve if, in the equation, d is replaced by 2 ^, if r is replaced by 2 r ? 21. What change in 6 is equivalent to a change in the sense of r. 22. From the graph of r=f{d) derive those of («) r=f(2d), {b)r = 2f{9), (c) r=f{-e), (d) r=-f{e), (e) r + l=f{d), Lnr=fie+l), (r/)r+l=/(5 + 2). Take, for example, /((?)= 1, f{d) = e, /(^) = sin(?, f{e) = 2d, f{e) = arc tan d, and draw the variations from the original gi-apli. 23. Plot the following : (a) r = 2 + 3 cos ^, (6) r = 3 + 2 cos ^, (c) r = 2 + 2costf, (d) r = 29, (e) r'^ = ad, (/) = 2% (g) e^ = ar, {h)e = smr, {i)e = cosr, (j)^ = tanr, (A) r = sec (tf-o), (I) e=secr. 24. Show how to obtain the graph of y = ^ sin (at + b) by suitable modification of the simple sine-curve y = sin t. 25. Draw the graphs from the following equations : (a) 2 s = e« + e-', (b) 2s = e*-e-«, (c) s = {e* + e-')/(et - e-'), (d) s = sin ( + sin 2 «, (e) s = smt + e-« sin 2 t. Take e = 2.7, and use logarithms in com- putations. CHAPTER II RATES LIMITS DERIVATIVES 4. Rate of Increase. Slope. In the study of any quantity, its rate of increase (or decrease), when some related quantity changes, is very important for auy complete understanding. Thus, the rate of increase of the speed of a boat when the power applied is increased is a fundamental consideration. Graphically, the rate of increase of y with respect to x is shown by the rate of increase of the height of a curve. If the curve is very flat, there is a small rate of increase ; if steep, a large rate. The steepness, or slope, of a curve shows the rate at which the dependent variable is increasing with respect to the independent variable. When we speak of the slope of a curve at any point P we mean the slope of its tan- gent at that point. To find this, we must start, as in Analytic Geometry, with a secant through P. Let the equation of the curve. Fig. 3, be y = X-, and let the point P at which the slope is to be found, be the point (2, 4). Let Q be any other point on the curve, and let Ax represent the difference of the values of x at the two points P and Q* * Ax may be regarded as an abbreviation of the phrase, " difference of the a;'s." The quotient of two such differences is called a difference quotient. Notice particularly that Ax does not mean A X x. Instead of " difference of the x's," the phrases "change in x " and " increment of x " are often used. 1 T iT clUt n i y=x^l'^ IS i A '• \ W ; ^ yikt^,B 0V$ 3 i II, § 4] RATES 7 Then, in the figure, 0/1 = 2, AB = ^x, and OJ5 = 2 + Aar. Moreover, since y = x- at every point, the value of ?/ at Q is BQ=(2 + Axy. The slope S of the secant PQ is the quotient of the differ- ences A?/ and Aa; : Us- \ ^M^^-^- ^ = tanZ3/PQ = ^ = M=(2 + Aa;)--4^ Ax PJi Aa; The slope ?ii of the tangent at P, that is tan Z MPT, is the limit of the slope of the secant as Q approaches P. The slope of the secant is the average slope of the curve between the points P and Q. The slope of the curve at the single point P is the limit of this average slope as Q approaches P. But, since *S = 4 + A.r, it is clear that the limit of aS as Q ap- proaches P is 4, since A.c approaches zero when Q approaches P; hence the slope m of the curve is 4 at the point P. At any other point the argument would be similar. If the coordinates of P are (a, a-), those of Q would be [(a + Aa;), (a + Aa;)2] ; and the slope of the secant would be the difference quotient A?/ -=- Aa; : ^^Ay^(a + Axf-a^^2aAa; + A^^2a + Aa;. Aa; Aa; Aa; Hence the slope of the curve at the point (a, a-) is * m = lim S = lim Ay /Ax = lim (2 a + Aa;) = 2 a. Ax=0 Ai=0 Ax=0 On the curve y = a?, the slope at any point is numerically twice the value of x. When the slope can be found, as above, the equation of the tangent at P can be written down at once, by Analytic Geome- try, since the slope m and a point (a, h) on a line determine its equation : (y — 6)= m(a; — a). Hence, in the preceding example, at the point (2, 4), where we found m = 4, the equation of the tangent is * Read " Ax = " " as Ax approaches zero." A detailed discussion of limita is given in § 10, p. 16. 8 DERIVATIVES [II, § 5 (2/ — 4) = 4 (a; — 2), or 4 a; — ?/ = 4. At the point (a, oF) on the curve y = x^, we found m = 2 a ; hence the equation of the tangent there is (?/ — a?) = 2 a{x —a), or 2ax — y = a^. 5. General Rules. A part of the preceding work holds true for any curve, and all of the work is at least similar. Thus, for any curve, the slope is m = lim S = lim(Ai//A.i;) ; Ai=0 Ai=0 that is, the slope m of the curve is the limit of the differenee quo- tient Ay/Ax. The changes in various examples arise in the calculation of the difference quotient, Ay -7- Ax, or S. This difference quotient is alvmys obtained, as above, by find- ing the value ofyat Q from the value of x at Q, from the equa- tion of the curve, then finding Ay by subtracting from this the value of y at P, and finally forming the difference quotient by dividing Ay by Ax. 6. Slope Negative or Zero. If the slope of the curve is negative, the rate of increase in its height is negative, that is, the height is really decreasing with respect to the independent variable.* If the slope is zero, the tangent to the curve is horizontal. This is what happens ordinarily at a highest point (maximum) or at a lowest point (minimum) on a curve.t Example 1 . Thus the curve y = x^, as we have just seen, has, at any point a: = a, a slope m = 2 a. Since m is positive when a is positive, the * Increase or decrease in the height is always measured as we go toward the right, i.e. as the independent variable increases. t A maxinmm need not be the highest point on the entire curve, but merely the highest point in a small arc of the curve about that point. See § 37. p. 63. Horizontal tangents sometimes occur without any maximum or any minimum. See § 38, p. 63. 11, § 6] SLOPES OF CURVES 9 curve is vising on the right of the origin ; since m is negative when a is negative, the curve is f;illing (tliat is, its height y decreases as x increases) on the left of the origin. At the origin m = ; the origin is the lowest point (a minimum) on the curve, because the curve falls as we come toward the origin and rises afterwards. Example 2. Find the slope of the curve (1) 2/ = x2 + 3a:-5 at the point where x = — 2 ; also in general at a point x = a. Use these values to find the equation of the tangent at a; = 2 : tangent at any point ; 1/= X- *■ 6 r - j i \ \ / 1 \ \ / \ ^ /l ^U \ 1 1 ^ / ^ \ Az M \ T the the maximum or minimum points if any exist. When X = — 2, we find y = - 7, (P in Fig. 4) ; taking any second point Q, (— 2 + Ax, — 7+Ay), its coordi- nates must satisfy the given equation, therefore (2) -7 + A2/ = (-2 + Aa;)2 + 3(-2 + A.r)-5, or (3) Ay = - 4 Ax +Ax^ + 3 Ax = — Ax + Ax^, where Ax^ means the square of Ax. Hence the slope of the secant PQ is ^^^- *• (4) S = Ay /Ax = - 1 + Ax. The slope m of the curve is the limit of S as Ax approaches zero ; i.e. (5) m = lim S= lim ^^= lim (_ 1 + Ax) = - 1. Ai=0 Ax=0 Ax Ax = It follows that the equation of the tangent at (— 2, — 7) is (<>) (2/ + 7) = - ](x + 2), or x + 2/ + 9 = 0. Likewise, if we take the point P (a, 5) in any position on the curve whatsoever, the equation (1) gives (7) ft = a2 + 3(j_5. Any second point Q has coordinates (a + Ax, b + Ay) where Ax and 10 DERIVATIVES [II, § 6 Ay are the differences in x and in ?/, respectively, between Pand §. Since § also lies on the curve, these coordinates satisfy (1) : (8) !> + A2/=(a + Aa:)2 + 3(a + Ax)-5. Subtracting the equation (7) from (8), Ay = 2 aAx + Ax"^ + 3 Ax, vrhence S = Ay /Ax = (2 a + 3) + Ax, and (9) m = lim S= lim ^ = lim [(2 a + 3) + Ax] = 2 a + 3. Ai=0 Aj^MjAX Aj^ = Therefore the tangent at («, 6) is (10) 2/-(a2 + 3a-5) = (2a + 3)(x-a), or (2a + 3)x-?/ = a2 + 5. From (9) we observe that to=0, when 2 a + 3 = 0, i.e. when a = — 3/2. For all values greater than —3/2, m=(2a + 3) is positive; for all values less than — 3/2, m is negative. Hence the curve has a minimum at (—3/2, —29/4) in Fig. 4, since the curve falls as we come toward this point and rises afterwards. Example 3. Consider the curve y = x^ — 12x + 7. If the value of x at any point P is a, the value of ?/ is a^ — 12 a + 7. If the value of x at Qis a + Ax, the value of ?/ at (^ is (a + Ax)^ _ 12 (a + Ax) + 7. Hence c,_Ay _ [(a + Ax)« - 12 (g + Ax) + 7] - [g^ _ 12 g + 7] Ax Ax IfBiPilH :|::::;:::::::..|^', \': i|ii \\\y {\\\ m^^\f4b ^^dmmM ^:::::a.LH^- .in- ;io,'^i\4, the curve is rising when x<— 2 and when x>+ 2. At X = ± 2, the slope is zero. At x = + 2 there is a minimum (see Fig. II, §6] SLOPES OF CURVES 11 5), since the curve is falling before this point and rising afterwards. At x= — 2 there is a maximum. At x=+ 2, y =(2)8 — 12 ■ 2 + 7 =— 0, which is the lowest value of y near that point. At x = — 2, y = 23, the highest value near it. This information is quite useful in drawing an accurate figure. We know also that the curve rises faster and faster to the right of x = 2. Draw an accurate figure of your own on a large scale. EXERCISES III. — SLOPES OF CURVES 1. Find the slope of the curve y = x'^ + 2 at the point where x = 1. Find the equation of the tangent at that point. Verify the fact that the equation obtained is a straight line, that it has the correct slope, and that it passes through the point (1,3). 2. Draw the curve y = x- + 2 on a large scale. Tlirough the point (1, 3) draw secants which make Ax = 1, ^, 0.1, 0.01, respectively. Calcu- late the slope of each of these secants and show that the values are ap- proaching the value of the slope of the curve at (1, 3). 3. Find the slope of the curve and the equation of the tangent to each of the following curves at the point mentioned. Verify each answer as in Ex. 1. (ffl) j/ = 3x2; (1,3). ((?) 2/=x2-t-4x-5; (1,0) (6) 2/ = 2x2- 5; (2,3). (e) y = x^ + x^ ; (1,2). (c) y = x^; (1, 1). (/) y = x^-3x + 4:; (2,6). 4. Find the slope of the curve ?/ = x2 — 3x+l at any point x = a ; from this find the highest (maximum) or lowest (minimum) point (if any), and show in what portions the curve is rising or falling. 5. Draw the following curves, using for greater accuracy the precise values of x and y at the highest (maximum) and the lowest (minimum) points, and the knowledge of the values of x for which the curve rises or falls. The slope of the curve at the point where x = is also useful in (6), (c), (e), (g). (ffl) y = x2 -I- 5 X + 2. {(I) y = x*. (g) y = 2 .r^ - 8 x. (6) y = xK (e) 2/ = -x2 + 3x. (h) y = x^ - C> x + 5. (c) 2/ = x3-3x4-4. (/) y = 3 + 12x-x3. (0 y = x'^ + x-. 6. Show that the slope of the graph of y = ax + b is always m = a, (1) geometrically, (2) by the methods of § 6. 7. Show that the lowest point on y = x- + px + q is the point where z = — p/2, (1) by Analytic Geometry, (2) by the methods of § 0. 8. The normal to a curve at a point is defined in Analytic Geometry 12 DERIVATIVES [II, § 6 to be the perpendicular to the tangent at that point. Its slope n is shown to be the negative reciprocal of the slope m of the tangent -. n =— \/m. Find the slope of the normal, and the equation of the normal in Ex. 1 •, in each of the equations under Ex. 3. 9. The slope w of the curve y = x^ at any point where x = a is m=2 a. Show that the slope is +1 at the point where a = 1/2. Find the points where the slope has the value — 1, 2, 10. Note that if the curve is drawn by taking different scales on the two axes, the slope no longer means the tangent of the angle made with the horizontal axis. 10. Find the points on the following curves where the slope has the values assigned to it ; (a) ?/=x2-3x + 6; (m = l, -1,2). (6) y=r?; (m = 0, + 1, + 6). (c) 2/ = a;3 - 3 X + 4 ; (in = 9, 1) . 11. Show that the curve y = x^ — 0.03x + 2 has a minimum at (0.1, 1.998) and a maximum at (— 0.1, 2.002). Draw the curve near the point (0, 2) on a very large scale. 12. Draw each of the following curves on an appropriate scale ; in each case show that the peculiar twist of the curve through its maximum and minimum would have been overlooked in ordinary plotting by points : (ffl) 2/ =48x3-x + l• [HINT. Use a very small vertical scale and a rather large horizontal scale. The slope at x = is also useful.] (6) 2/ = x3^30x2 + 297x. [Hint. Use an exceedingly small vertical scale and a moderate hori- zontal scale. The slope at x = 10 is also useful.] 7. Speed. An important case of rate of change of a quan- tity is the rate at which a body moves, — its speed. Consider the motion of a body falling from rest under the influence of gravity. During the first second it passes over 16 ft., during the next it passes over 48 ft., during the third over 80 ft. In general, if t is the number of seconds, and s the entire distance it has fallen, s = 16 «^ if the gravitational constant g be taken as 32. The graph of this equation (see Fig. 6) is a parabola with its vertex at the origin. 'm^' II, § 7] SPEED 13 The speed, that is the rate of increase of the space passed over, is the slope of this curve, i.e. liin As/ At. At=0 This may be seen directly in another way. The average speed for an interval of time At is found by dividing the dif- ference between the space passed over at the beginning and at the end of that interval of time by the difference in time : i.e. the average speed is the difference quotient As -i- At. By the speed at a given instant we mean the limit of the average speed over an interval A^ beginning or ending at that instant as that interval approaches zero, i.e. speed = lim As/ At. Taking the equation s = 16 t-, if < = 1/2, s After a lapse of time At, the new values are t = 1/2 -|- At, and s = 16(l/2 + A02(«inFig. 6). Then 4 (see point P in Fig. 6). As 16(1/2 + A02- 4 16 At + 16 M^, As/ At = 16 + 16 At. Whence speed — lim — A(^ A« = lim (16 + 16A0=16; Af=0 that is, the speed at the end of the first half second is 16 ft. per second. Likewise, for any value of t, SAy t= T, s= 16 r- ; while for t ^T + At, s = 16 (r+ At)-; hence , As 16(r- average speed = — = — ^^ — 4- ^ /- 4^ t V -1 X 4- \ I J 64 C \ S: = 16/-i/ V J^ A t t U^ \ M 1 >^Jjr A/ [ , 1 iw ■' Aty 32 r + 16 At and speed ■■ : lim — = 32 T. Al=^ At 14 DERIVATIVES [II, § 8 Thus, at the end of two seconds, T = 2, and the speed is 32 ■ 2 = 64, in feet per second. 8. Component Speeds. Any curve may be regarded as the path of a moving point. If a point P does move along a curve, both X and y are fixed when the time t is fixed. To specify the motion completely, we need equations which give the values of X and ?/ in terms of t. The horizontal speed is the rate of increase of x with respect to the time. This may be thought of as the speed of the pro- jection iW of P on the a;-axis. As shown ih § 7, this speed is the limit of the difference quotient A.t; -i- A^ as A^ = 0. Likewise the vertical speed is the limit of the difference quo- tient Ay -7- At as M=0. /' y Since the slope m of the curve P is the limit of Ay -H Ax as Ax = ; and since Ay _ Ay . Ax Ax~~At ' Ai' it follows that m = (vertical speed) -¥■ (horizontal spieed) ; that is, the slope of the curve is the ratio of the rate of increase of y to the rate of increase of x. Fig. 9. Continuous Functions. In §§ 4-8, we have supposed that the curves used were smooth. The functions which we have had have all been representable by smooth curves ; except perhaps at isolated points, to a small change in the value of one coordinate, there has been a correspondingly small change in the value of the other coordinate. Throughout this text, unless the contrary is expressly stated, the functions dealt with will be of the same sort. Such functions are called continuous. (See § 10, p. 17.) II, § 9] SPEED 15 The curve y=l/x is continuous except at the point a; = ; y = tan x is continuous except at the points x = ± 7r/2, ± 3 ir/2, etc. Such excep- tional points occur frequently ; we do not discard a curve because of them, but it is understood that any of our results may fail at such points. EXERCISES IV. — SPEED 1. From the formula s = 16 t-, calculate the values of s when f = 1, 2, 1.1, 1.01, 1.001. From these values calculate the average speed between t = \ and t = 2; between t — 1 and t—l.l; between t = \ and t = 1.01 ; ■ between t = 1 and t = 1.001. Show that these average speeds are succes- sively nearer to the speed at the instant < = 1. 2. Calculate as in Ex. 1 the average speed for smaller and smaller in- tervals of time after t =2; and show that these approach the .speed at the instant t = 2. 3. A body thrown vertically downwards from any height with an original velocity of 100 ft. per second, passes over in time t (in seconds) a distance s (in feet) given by the equation s — 100 t + \Q (^ (if gr = 32, as in § 7). Find the speed v at the time t = \ ; at the time t — 2; at the time t = i; at the time t = T. 4. In Ex. 3 calculate the average speeds for smaller and smaller in- tervals of time after ( = ; and show that they approach the original speed vo = 100. Repeat the calculations for intervals beginning with t=2. 5. Calculate the speed of a body at the times indicated in the follow- ing possible relations between s and t : (a) s = «2; « = 1, 2, 10, T. (c) s = -16 «2 + 160 < ; < = 0, 2, 5. (6) s = 16f^-100t; t = 0,2, T. (d) s = «3-3« + 4; « = 0, 1/2, 1. 6. The relation (c) in Ex. 5 holds (approximately, since gr = 32 ap- proximately) for a body thrown upward with an initial speed of 160 ft. per second, where s means the distance from the starting point counted positive upwards. Draw a graph which represents this relation between the values of s and t. In this graph mark the greatest value of s. What is the value of v at that point ? Find exact values of s and t for this point. 7. A body thrown horizontally with an original speed of 4 ft. per second falls in a vertical plane curved path so that the values of its hori- zontal and its vertical distances from its original position are respectively, X = 4 «, 1/ = 16 (-, where y is measured downwards. Show that the vertical speed is 32 T, and that the horizontal .speed is 4, at the instant t = T. Eliminate t to show that the path is the curve y = z*. 16 DERIVATIVES [II, § 10 8. Show by Ex. 7 and § 8 that the slope of the curve ?/ = a;^ at the point where t = l, i.e. (-i, 10), is 32 ~ 4, or 8. Write the equation of the tangent at that point. 9. Show that the slope of the curve y = x^ (Ex. 7) at the point (a, a^), i.e. t = a/4, is 2 a, from Ex. 7 and § 8 ; and also directly by means of § 6. 10. If a body moves so that its horizontal and its vertical distances from the starting point are, respectively, x = 16 t^, y = 4 t, show that its path is the curve y'^ = x ; that its horizontal speed and its vertical speed are, respectively, 32 T and 4, at the instant t = T. 11. From Ex. 10 and § 8 show that the slope of the curve y^ ^x&t the point (16, 4), i.e. when f = 1, is 4 h- 32 = 1/8. Write the equation of the tangent at that point. 12. From Ex. 10 and § 8 show that the slope of the curve y"^ = x aX the point where t= T is 4 -- (32 T) = 1/(8 T) = 1/(2 A:), where k is the value of y at the point. Compare this result with that of Ex. 8. 10. Limits. Infinitesimals. We have been led in what pre- cedes to make use of limits. Thus the tangent to a curve at the point P is defined by saying that its slope is the limit of the slope of a variable secant through P; the speed at a given instant is the limit of the average speed ; the difference of the two values of x, Ax, was thought of as approacldng zero ; and so on. To make these concepts clear, the following precise statements are necessary and desirable. When the difference betiveen a variable x and a constant a he- comes and remains less, in absolute value,* than ani/ preassigyied positive quantity, however small, then a is the limit of the vari- able x. We also use the expression "x approaches a as a limit," or, more simply, " x approaches o." The symbol for limit is Urn ; the symbol for approaches is = : thus we may write lim x==a, or x = a, or lim (a — x) = 0, or a — x=0. When the limit of a variable is zei'o, the variable is called * When dealing with real numbers, absolute value is the value without regard to signs so that the absolute value of — 2 is 2. A convenient symbol for it is two vertical lines ; thus |3 — 7 1= 4. II, § 11] LIMITS 17 an infinitesimal. Thus a — x above is an infinitesimal. The difference between any variable and its limit is always an in- finitesimal. When a variable x approaches a limit a, an;/ con- tinuous function J\x) approaches the limit /(a): thus, if y=f{x) and b =f{a), we may write lim y = b, or lim/(a;) =f(a). This condition is the precise definition of continuity at the point x = a. (See § 0, p. 14.) 11. Properties of Limits. The following properties of limits will be assumed as self-evident; some of them have already been used in the articles noted below. Theorem A. The limit of the siim. of two variables is the sum of the limits of the ttco variables. This is easily extended to the case of more than two variables. (Used in §§4, 6, and 7.) Theorem B. The limit of the product of two variables is the product of the limits of the variables. (Used in §§4, 6, and 7.) Theorem C. The limit of the quotient of one variable divided by another is the quotient of the limits of the variables, provided the limit of the divisor is not zero. (Used in § 8.) The exceptional case in Theorem C is really the most in- teresting and important case of all. The exception arises because when zero occui-s as a denominator, the division can- not be performed. In finding the slope of a curve, we consider lim (Ay/A-r) as A.r approaches zero; notice that this is pre- cisely the case ruled out in Theorem C. Again, the speed is lim.(As/A^) as M approaches zero. The limit of any such difference quotient is one of these exceptional cases. Now it is clear that the slope of a curve (or the speed of an object) may have a great variety of values in different cases : no one answer is sufficient for all examples, in the case of the limit of a quotient when the denominator approaches zero. 18 DERIVATIVES [II, § 12 Theorem D. Tlie limit of the ratio of two infinitesimals de- pends upon the law connecting them; otherwise it is quite inde- terminate. Of this the student will see many instances ; for the Differential Calculus consists of the consideration of just such limits. In fact, the very reason for the existence of the Diifer- ential Calculus is that the exceptional case of Theorem C is important, and cannot be settled in an offhand manner. The thing to be noted here is, that, no matter how small two quantities may be, their ratio may be either small or large ; and that, if the two quantities are variables whose limit is zero, the limit of their ratio may be either finite, zero, or non-existent. In our work with such forms we shall try to substitute an equivalent form whose limit can be found. Obviously, to say that two variables are vanishing implies nothing about the limit of their ratio. 12. Ratio of an Arc to its Chord. Another important illus- tration of a ratio of infinitesimals is the ratio of the chord of a curve to its subtended arc : , , „ „ p _ chord PQ ~ arcPQ If Q approaches P, both the arc and the chord approach zero. At any stage of the process the arc is greater than the chord ; but as Q approaches P this difference di- minishes very rapidly, and the ratio R approaches 1 : V „ T chord PQ ^ hm R = lim --^ = 1. -mf ■ ,, - ,. X" (0, 71 >1, (h) lim-= - ' ^ ' x-o X ( 1, n = 1. Show that lira 2x2 + 3 = 2. x=aoa;2 + 43; -I- 5 [Hint. Divide numerator and denominator by x"^ ; then such terms as S/x* approach zero as x becomes infinite.] 7. Evaluate : 2x + 1 3x + 2' (a) lim !x2-4 (d) lim VI ax+b mx + n Vax- + bx 4- c Vx2 - 1 8. Let O be the center of a circle of radius r — OB, and let a = Z COB be an angle at the center. Let BT be perpen- dicular to OB, and let BF be perpendicular to OC. Show that OF approaches OC as a ap- proaches zero ; likewise arc CB = 0, arc DB=: 0, and FC = 0, as a = 0. 9. In the figure of Ex. 8, show that the ob- vious geometric inequality FB <_ arc CB and therefore the derivative is (5) 4?^ = lim fl^ ^ lim /(•^ + A.r)-/(^) ^ ^j ^^ p^ This formula is often convenient; we shall apply it at once. 17. Rule for Differentiation. The process of finding a de- rivative is called differentiation. To apply formula (5) of § 16 : (A) Find {y + Ay) by substitzUing (x + Ax) for x in the given function or equation; this gives y + Ay =f(x -f Ax*). > (B) Subtract y froin y-\- Ay ; this gives Ay =/(£c-fAa;)—/(a;). (C) Divide Ay by Ax to find the difference quotient Ay /Ax; simplify this result. (D) Find the limit of Ay /Ax as Ax approaches zero ; this I result is the derivative, dy/dx. * Instead of slope, read speed in case the problem deals with a motion, as in § 7. In general, Ajz/As; is the average rate of increase, and dy/dx is the i instantaneous rate. II, § 17] FORMAL DIFFERENTIATION 25 Example 1. Given y =f(x)=x'^, to find dy/dx. (A) f{x + Ax) = {x+Axy^. (B) Ay =f{x + Ax)-/(x) = (x + AxY - x^ = 2xAx + A^'. (C) Ay/Ax = (2 xAx + Ax") H- Ax = 2 X + Aa:. (Z>) dy/dx= lim Ay/Ax= lim (2x + Ax) = 2x. Ax = Ai = Compare this work and the answer with the work of § 4, p. 6. Example 2. Given y =/(x) =x3 — 12x + 7, to find dy/dx. {A) /(x + Ax) = (x+ Ax)3-12(x + Ax)+7. {B) Ai/=/(x + Ax)-/(x) = 3x2Ax + 3xA? + Ax^- 12Ax. ( C) Ay /Ax = 3 x2 + 3 xAx + Ax- - 12. (Z>) dy/dx = lim Ay/ Ax = lim (3 x'- + 3 xAx + Ax" - 12) = 3 x2 - 12. Ai = Ai = Compare this work and the answer with the work of Example 3, § 6. Example 3. Given y =/(x) = l/x^, to find dy/dx. {A) /(x+Ax)= 1 {B) Ay=/(x + Ax)-/(x)= 1 1 ^ _ 2xAx + A^ (C) Ay/ Ax ^ ^' x2(a; + Ax)2 (2)) d2,/dx= lim^= lim T - -1^+-4^1 = - 2^ = - 1 . Ax=oAx Ai=oL x2(x + Ax)2J X* X* Example 4. Given y =/(x)= Vx, to find dy/dx, or df{x)/dx. (X + Ax)2 1 1 (X + Ax)2 x2 2x + Ax (^4) /(x + Ax) = Vx + Ax. 1^ (B) Ay = /(x + Ax) - /(x) = Vx + Ax - Vx. (C) Ay _ Vx + Ax — Vx _ Vx + Ax - Vx ^ Vx + Ax + Vg Ax Ax ~ Ax VxTAx + Vx 1 Vx + Ax + Vx (m ^= lim^= lim- 1 = -i-. '' dx Ax=oAx Ax=oVx + Ax + Vx 2Vx (Compare Ex. 11, p. 16.) 26 RATES [II. § 17 Example 5. Given y = /(x) = .r^ to find df{x)/dx. _^ (^) /(x + Ax) = {X + Ax)^ = x^ + 7 x^Ax + (terms with a factor Ax ) . (5) Ay = /(x + Ax) - /(x) = 7 x6Ax + (terms with a factor Ax'). (C) Ay/Ax=: 7x6+ (terms with a factor Ax). (D) dy/dx = lira A2//Ax= lira [7 x^ + (terms with a factor Ax)] = 7 x^. EXERCISES VI. -FORMAL DIFFERENTIATION 1 Find the derivative of y = x^ with respect to x. [Corapare Ex 3 (c), p. 11.] Write the equation of the tangent at the pomt (2, 8) to the curve y = x*. 2. Find the derivatives of the following functions with respect to x : (a)x2-3x + 4. (6)x3-6x + 7. (c) x^ + 5. (d) x^ + 3x^-2. ie) x3 + 2x;^-4. (/) x^-3x3+5x. (.) h.- ^'^ x-TT- ^"^ ^^' 2x+3 :-2 3. Find the equation of the tangent and the equation of the normal to the curve y = 1/x at the point where x = 2. (See Ex. 8, p. 11. ) 4 Find the values of x for which the curve y = x^-\bx + 1 rises and those for which it falls ; find the highest point (maximum) and the lowest point (minimum). Draw the graph accurately. 5 Draw accurate graphs for the following curves : (a) 2/ = x3-18x + 3. (c) y = x*-32x. (ft)y^x3 + 3x^. (d)y = x4-18x2. 6. Determine the speed of a body which moves so that s= 16«2 + 10« + 5. N FA body thrown down from a height with initial speed 10 ft. per sec- ond moves in this way approximately, if . is measured downward from a mark 5 ft. above the starting point.] 7 If a body moves so that its horizontal and its vertical distances from a point are, respectively, . =. 10 ^ y = - 16 i^ + 10 e, find xts hori- zontal speed and its vertical speed. Show that the path is 2/ = - 16 xVlOO + X' and that the slope of this path is the ratio of the vertical speed to th« II, § 17] FORMAL DIFFERENTIATION 27 horizontal speed. [These equations represent, approximately, the motion of an object thrown upward at an angle of 45° with a speed 10 V2.] 8. A stone is dropped into still water. The circumference c of the growing circular waves thus made, as a function of the radius ?•, isc = 2 irr. Show that dcldr = 2 ir, i.e. that the circumference changes 2 ir times as fast as the radius. Let ^1 be the area of the circle. Show that dAjdr ^'l-irr \ i.e. the rate at which the area is changing compared to the radius is numerically equal to the circumference. 9. Determine the rates of change of the following variables : (a) The surface of a sphere compared with its radius, as the sphere expands. (6) The volume of a cube compared with its edge, as the cube enlarges. (c) The volume of a right circular cone compared with the radius of its base (the height being fixed) , as the base spreads out. 10. If a man G ft. tall is at a distance x from the base of an arc light 10 ft. high, and if the length of his shadow is s, show that s/6 = x/4, or s = 3x/2. Find the rate (ds/dx) at which the length s of his shadow increases as compared with his distance x from the lamp base. 11. The specific heat of a substance (e.g. water) is the amount of heat required to raise the temperature of a unit volume of that substance 1° (Centigrade). This amount is known to change for the same substance for different temperatures. The average specific heat between two tem- peratures is the ratio of the quantity of heat AH consumed in raising the temperature divided by the change At in the temperature ; show that the actual specific heat at a given temperature is dH/dt. 12. The coefficient of expansion of a solid substance is the amount a bar of that substance 1 ft. long will expand when the temperature changes 1°. Express the average coefficient of expansion, and show that tlie coeGB- cient of expansion at any given temperature is dl/dt, if the bar is precisely 1 ft. long at that temperature. (See also Ex. 12, p. 145.) CHAPTER III DIFFERENTIATION OF ALGEBRAIC FUNCTIONS PART I. EXPLICIT FUNCTIONS 18. Classification of Functions. For convenience it is usual to classify functions into certain groups. A function which can be expressed directly in terms of the independent variable x by means of the three elementary operations of multiplication, addition, and subtraction is called a poljmomial in x. Thus, ic^(= cc • ic), 2 x^ + 4 a^ — 7 aj + 3, ar^ — 4 x- + 6, etc., are polynomials. The most general polynomial is ayX" + aiX""^ + ••• + a„_iX + a„, where the coefficients Uq, Uj, •••, a„ are constants, and the exponents are positive integers. Notice that raising a quantity to a positive integral power can be regarded as a succession of multiplications, A function which can be expressed directly in terms of the independent variable x by means of the four elementary opera- tions of multiplication, division, addition, and subtraction, is called a rational function of x. Thus, 1/x, (ar* — 3 x)/{2 x + 7), etc., are rational. The most general rational function is the quotient of two polynomials, since more than one division can be reduced to a single division by the rules for the combination of fractions. All polynomials are also rational functions. If, besides the four elementary operations, a f miction re- quires for its direct expression in the independent variable x at most the extraction of integral roots, it is called a simple algebraic function * of x. Thus, Vx, ( Vx-^ + 1 — 2)/(3 — \/x), ♦Since the expression "algebraic function" is used in tlie broader sense of § 27 in advanced matliematics, we sliall call these simple algebraic functions. 28 Ill, § 19] CLASSIFICATION OF FUNCTIONS 29 etc., are simple algebraic functions. All rational functions are also simple algebraic functions. Simple algebraic functions which are not rational are called irrational functions. A function which is not an algebraic function is called a transcendental function. Thus, sin x, log x, e^, j^+ tan-^ (1 -f- x), etc., are transcendental. In this chapter we shall deal only with algebraic functions. 19. Differentiation of Polynomials. We have differentiated a number of polynomials in Chapter II. To simplify the work to a mere matter of routine, we need four rules : Tlie derivative of a constant is zero : [I] 1^ = 0. dx The derivative of a constant times a function is equal to the constant times the derivative of the function : [II] d{c-u) ^^ du dx dx The derivative of the sum of two functions is equal to the sum of their derivatives : rjjj^ d(u + v) _ du . dv ^ dx dx dx The derivative of a power, x^, loith respect to x is na;""^* [IV] ^ = nx"^-^. dx [We shall prove this at once in the case when n is a positive integer ; later we shall prove that it is true also for negative and fractional values of ?i.] Each of these rules was illustrated in Chapter II, § 17. To prove them we use the rule of § 17. Proof of [I]. If ?/ = c, a change in x produces no change in y ; hence A?/ = 0. Therefore dy/dx = lim Ay/ Ax = lim = as Ax approaches zero. Geometrically, the slope of the curve y = c (a horizontal straight line) is everywhere zero. 30 ALGEBRAIC FUNCTIONS [III, § 19 Proof of [II]. liy = c -u where m is a function of x, a change Art' in x produces a change Ati in u and a change Ay in y; following the rule of § 17 we find : (A) y + Ay = C'(u-\- Au). (B) A?/ = c • Au. (C) Ay/Ax = c • (Au/Ax). (D) dy/clx = lim [c • (Au/Ax)! = c • Urn Au/Ax = c(du/dx). Ar=0 Ai=0 Thus d(7 x^)/dx = 7 • d{x')/dx = 7 ■2x = Ux. (See § § 4, 17. ) Proof of [III]- If y=u-\-v, where ?« and v are functions of x, a change A;c in x produces changes Ay, Au, Av in y, u, v, respec- tively, hence {A) y + Ay={u + A^i) + {o + Av); (B) Ay = Au + Av ; (O) Ay /Ax = An /Ax -\- Ay/Ar ; (D) dy/dx = lim (Au/Ax) + lim (Av/Aa;) = dxi/dx + dv/dx, Ax=0 Ai=0 Thus d(a^-12a;4-7) ^ d(x^) d(12x) ^ d(7)^^^, ^^ ^ ^^ dx dx dx dx by applying the preceding rules and noticing that dx^/dx = 3 x^. [See Ex. 1 of Exercises VI and compare Example 3, p. 10, and Example 2, p. 25]. Proof of [IV]. If y = x", we jjroceed as in Example 5, p. 26: (A) y + Ay — (x + Ax)" = x" -^ nx'^-'^Ax -\- (terms which have a^ common factor Ax ). (B) Ay = nx^'^Ax -f (terms with a common factor Ax ). (C) Ay / Ax = nx''-'^ + (terms which have a factor Ax). (D) dy/dx = lim (Ay /Ax) = nx"-^. Ax = This proof holds good only for positive integral values of n. For negative and fractional values of n, see §§ 20, 23. Ill, § 19] POLYXOMIALS 31 Example 1. (l{x^)/dx = 9 x?. (This would be serious without the rule. ) Example 2. dx/dx = 1 . a;"^ = 1, since a;'' = 1. This is also evident directly: dx/tte = lim Ax/Ax = 1. Notice how- ever that no new rule is necessary. Example 3. — (x* - 7 x- + 3 x - 5) = 4 x=^ - 14 a; + 3, dx Example 4. — (^x"» + -B.':" + C) = mAz"'-'^ + nBx''-\ dx EXERCISES VII. — DIFFERENTIATION OF POLYNOMIALS Calculate the derivative of each of the following expressions with re- spect to the independent variable it contains (x or r or s or « or ?/ or m). In this list, the first letters of the alphabet, down to n, inclusive, represent constants. 1. (a) 2/ = 5x3. (d) y = 5(x3 + l). (gr) ?/ =-10xW + 10. (6) y = x*/4. (e) y = (x* - 2)/4. (h) y = 8 x^ + 6 x*. (c) 2/ = 5 .<-3 + 1. (/) 2/ r= - 10 xio. (t) 2/ = 7 x6 - 6 x^ + 5. 2. (rt) 2/ = «->-^- (c) 2/ = (« + 'j) ■^•^. (e) 1/ = ^^-^ - kx^ + I- (6) 2/ = - c-^x9. (d) 2/ = (rt- - h'^) xK (/) 2/ = ^ + Bx + Cx2. 3. (a) s = i-«-2. (ft) s = (2(3^ + ^2). (c) s = c (a<3 + 6«4), 4. (a) g=s(s2-n. (c) g = (1 - «3) (2 + gS). (6) g = s^{a -bs + cs"^). (d) q = as(b + cs) + d. 5. {a) z = (y + a){y-b). (c) z = {y''-> + 2)(y^o _ s). (6) z = ay\y- + by^). (d) z = (3 y'- + 2)2. 6. (a) v={hti*-ku^ + l)u\ (b) v = a (rfi + u + l){ic^ - u + I). 7. (a) y = kx'* + /.<;"•. (ft) i/ = a^e'-" - &•»""• 8. (a) 2/ = a;2»+'" + x''+2'" + i. (ft) ?/ = a + ftx"'. 9. Determine the slope of the curve y = x2 — 2 x at the origin. Where is the slope 2 ? Where is the tangent horizontal ? Draw the graph. 10. Locate the vertex of the parabola y = x"^ + 8 x + 19 by finding the point at which the tangent is horizontal. 11. Proceed for each of the following curves as in Ex. 10 : (a) 2/ = x2 - 2 X + 2. (6) J, = _ x2 + 2 X - 10. (c) y = ax^ + bx + c 32 ALGEBRAIC FUNCTIONS [III, § 19 12. Where on the parabola y = x^ is the slope 1 ? Where is the slope 1 on the curve y = x^? Ou ?/ = x* '? On y = x" ? Where is the slope on each of these curves ? 13. What is the slope of the curve ?/ = 2 x^ — 3 x^ + 4 at x = 0, ±2, ±4? Where is the slope 9/2? —3/2? Where is the tangent hori- zontal ; are these points highest or lowest points, or neither ? Dravf the graph. 14. What is the slope of the curve y = xV4 — 2 x^ + 4 x^ at x = 0, 1, — 1, — 2 ? Where is the tangent horizontal ; are these points maxima or minima ? Where is the slope equal to eight times the value of x. 15. Show that the function x^ + Sx^ + Sx + l always increases with X. Where is the tangent horizontal ? Show that there is no maximum or minimum at this point. 16. Locate the maxima and minima (if any exist) on each of the fol- lowing curves and draw their graphs accurately : (a) y = x^-27x+ 15. (d) y = 4x^- 11 x^ - 70x + 20. (6) y = 2 x3 - 9 x2 + 12 X - 10. (e) y = 3 x* - 4 x^ + 5. (c) y = x3 - 9x2 -I- 27 X - 15. (y) y = 3x^- 80x3 + iqoo. 17. At what angle does the line y = 2x meet the parabola y = x^ + 4x + l? 18. Find the angle between the curve y = x^ and the straight line 2/ = 9 X at each of their points of intersection. 19. At what angles does the curve y ={x — l){x — 2){x — 3) cut the X-axis ? 20. If a sphere expands — as when a rubber balloon is distended, or when an orange is growing — the volume and the radius both increase. Find the rate of growth of the volume with respect to the radius. 21. In an expanding sphere, find the rate of growth of the surface with respect to the radius. 22. Find the rate of change of the total surface of a right circular cylinder with respect to the radius, the altitude being fixed ; with respect to the altitude when the radius is fixed. Do the same for a right circular cone. 20. Differentiation of Rational Functions. In order to dif- ferentiate all rational functions, we need only one more rule, — that for differentiating a fraction. Ill, § 20] RATIONAL FUNCTIONS 33 Tlie derivative of a quotient N/D of two functions N and D is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square oj the denominator : '-*-' dx ~ D' To prove this rule, let y = iVyZ), where N and D are func- tions of x; then a change Ax in x produces changes A?/, AiV, AD in y, N, and D, respectively ; hence, by the rule of § 17 : JSr+AN (A) y + Ay D + AD /„N A N+AN N D-AN-NAD {G) D + AD D D{D + AD) Ay Ax Ax Ai~ D{D + AD) ' j)dJ[_^dD (D) 'll=lim^ = -J^—^. dx Ax^y) Ax D - (3 X - 7) — (x2 + 3) - (.v:2 + 3) — (3 .r _ 7) E^,nplel. ilf±^W- "^ ,' -,. "• dx\3x-7j (3a:- 0" ^ (3x-7)-2x-(x^+S)-S ^ 3 3-2-14X-9 (3a;-7)2 (3a;-7)2 * TT 7 o ^ / 1 \ dx dx - 2 X 2 Example 2. —( — )=: — — = - — tL±= — ±. dx\xy (x2)2 X* a;3 (Compare Example 3, § 17, p. 25.) Example 3. ^ (a^*) =^f lU ^ziiL^ = ^= -kx-'-\ dx ' dxXx") x2* a-*+i Note that formula IV holds also when n is a negative integer, for if n = — A;, formula IV gives the result we have just proved. 34 • ALGEBRAIC FUNCTIONS [III, § 20 EXERCISES VIII. DIFFERENTIATION OF RATIONAL FUNCTIONS Calculate the derivative of each of the following : 1. (a) 2/ = ^- (e) y^'^-^^. x + 4. w.=^ + !- X— 1 <")-!f^- (d) 2/=-^- 1 + x 2. (-)y=~ '''y=^^- (c), = x-( = l) (d) y^^x'K (e) y = ^x-\ 3. ia)v--\-\. u^ — 1 (^)-'-^- ^^^-^.■ id) v=-l-^. u -- u 4. (»)« = <,+5_£. <')-^ri7- (c) s = ht^-kt-^. id)p-r^ '•:+; if)y-^'^' (9) y a;2- 1 X2 + x-3 x2 - 2 X + 6 (fif) y = 2x-3-3x-2. (A) y = 4x-6 + 4- X* (»■) ?/ = 8.r-w + ^-15. (j) y = ax-"* + bx-". (e) tJ u- — tl + 1 (/)s ] 1 t^ + 2t-S f' + t + 6 {g) s 2t+ 1 5 «3 - 1 fi^t+l (h) s = a+ ^ . '^\ (e) s ^t-^{t^- 2<5 + l). {/), ■,^ g^/4 + 2g3_ X ■ ^ ^ ^ 2 ff^/7 3 xV2 ^ ^2 + 2 . 5. («)2/ = (c)p= ^^^^^ Ill, § 21] PRODUCT 35 6. Draw the following curves ; obtain the equation of the tangent at the point indicated, and also at any point (xq, yo) ; determine the hori- zontal tangents if any exist, and show whether these points correspond to maxima or minima or neither. 1 + X X 7. Compare the slopes of the curves y = x, y = x-^ at the points at which they intersect. What is the angle between them ? 8. Compare the slopes of the family of curves y = x", where n = 0, + 1, + 2, etc., —1,-2, etc., at the common point (1, 1). What is the angle between ?/ = x- and y — x~^ ? See Tables, III, A. 21. Derivative of a Product. — The following rule is often useful in simplifying differentiations : TJie derivative of the product of two functions is equal to the first factor times the derivative of the second plus the second factor times the derivative of the first : dx dx dx li y = u • V where u and v are functions of x, a change Aa: in X produces changes Ay, Aw, Ay in y, u, and v, respectively: {A) y + Ay = {u + A «) (v + Ay) ; (B) Ay =(u + A.u)(y + Av)— m - v = uAv -}-v A/t + Au Au ; ( C) A.y/ A.'c = u (Av/Ax) + v(A?//Aa;) + A?< — ; (D) dy/dx = lim (Ay /Ax) = n (dv/dx) + v(du/dx). Aa:=0 Example 1. To find the derivative of y = (x^ + 3) (.r^ + 4). Method 1. We may perform the indicated multiplication and write: ^ = A [ (x2 + 3)(x3 + 4) ] = i^ [x5 + 3 x8 4- 4 x2 + ] 2] = 5 X* + 9 x2 + 8 a;. ax ax dx 36 ALGEBRAIC FUNCTIONS [III, § 21 Method 2. Using the new rule, we write : ^ =(x2 + S)4-(x^ + 4) + (x3 + 4) I- (x2 + 3) dx dx dx = [y? + 3)3 x2 + (x3 + 4)2 x=5 X* + 9 x2 + 8 x. In other examples which we shall soon meet, the saving in labor due to the new rule is even greater than in this example. 22. The Derivative of a Function of a Function. Another convenient rule is the following : The derivative of a function of a variable ii, ivhich itself is a function of another variable x, is found by multij)lying the deriva- tive of the original function loith respect to u by the derivative of ?* loith respect to x. [VH] dy ^dy du dx du doc If y is a function of u, and m is a function of x, a change Acb in X produces a change A?/- in u; that in turn produces a change A?/ in ?/ ; hence : Aj/_ A// ^ Au ^ Ax Aii Aa; Taking limits on both sides, we find : dy _dy du^ dx du. dx This is really the same as the rule used in § 8, p. 14 ; for, if we divide both sides by du/dx, we find [Vila] rly^rJM^^* du dx dx which is the same as the rule of § 8, except that different letters are used. Example 1. To find the derivative of j/ =(x2 + 2)^ Method 1. We may expand the cube and write : ^ = A [ (x2 + 2)31 = -^ (x6 + 6 X* + 12 x2 + 8) = 6 x5 + 24 x3 + 24 x. dx dx dx Ill, § 22] FUNCTION OF A FUNCTION 37 Method 2. Using tlie new rule, we may simplify this work : let w = x2 4- 2, then y = (a;^ + 2)8 = u^ ; rule [VI] gives dx du dx du dx = 3(x2 + 2)2. (2a;) = 3(a:« + 4x2 + 4) • (2 x) = 6x5 + 24 x8 + 24 x. Example 2. liy = f^ + 2 and x = 3 < + 4, to find dy/dx. Method 1. We may solve the equation x = 3 f + 4 for < and substitute this value of t in the first equation : ^ \, 3 ; ^ 9 9^9 Method 2. Using the new rule (with letters as used in § 8, p. 14) we write : dy_di^dx^ d(t^ + 2) ^ d{S t + 4) ^ 2t ^S = -t dx dt ' dt dt ' dt ■ ' 3 ■ EXERCISES IX. — SHORT METHODS. RATIONAL FUNCTIONS Calculate the derivative of each of the following : 1. (a) 2/=3x(x2+ 1). (d) y = (2x+l)(l-x + x2). (^) ?/ = x3(x2 + 3). (e) 2/ = (x2-4)(l +x3). (c) ?/ = (3x + 2)(2x-3). (/) 2/ = (x3 + 3x-2)(x2-2x). 2. (a) y= (x2 + 1)2. - (c) y = (1 - x2)2. (g) y = {a +6x)». (6) 2/ = (x2 - 1)3. (d) y = (1 - x2)3. (/) y={a + bx)K 3. (ffl) y = (1 + 2 X -. 3 x2)2. {d) 2/ = (« + 6x + cx2)3. (i>) y = (x-i + 3x + 7)3. (e) s = (3«2 + 2«-4)*. (c) s = («3 _ f _ 4)2. (/) 2/ = (a + 6x + cx2)5. 4. (a) y= ^ (d) y = (2 +3x2)-2r= ?— t-:1- *" ' ^ (l+2x- 3x2)2 ^ ^ ^ ^ ^ ^ L (2 + 3x2)2j (b) s= (e) » = (a + 6x)-8. (x2 + 2)3 (a _ 6s - cs2)3 5. (a) 2/= (1 -5x2)(3-4xS)(l-x). (c) 2/ = (a;2 + 2)3(3x - 5)^ (6) 2/ = a:(x2 + 3)(x3 + 4). (d) s = (<3 _ 2)2(2 «- l)". 38 ALGEBRAIC FUNCTIONS [III, § 22 6. Determine dyjUx in each of the following pairs of equations : ^^ |M = 3a;-4. ^ ^ 1 m = x2 _ 1/2. r 6;s-4^ [2^-4 2=2-4x. 4x8 4x 7. Draw each of the curves represented by the following pairs of parameter equations and determine dy/dx : , . fx = «2, ... fx = 2f + 3«2, («) \y=3t + 2. (^) [y = 2t+i. What is the slope in each case when t = 1? Show this in your graphs. Find the value of the slope in each case at a point where the parameter has the value 2. 8. Draw the graph of the function y = (2x - l)2(3x + 4)2. De- termine its horizontal tangents. 9. Proceed as in Ex. '8, for the function?/ =(2 x - 1)2-- (3 x + 4)2. 10. Show that if y =(x - 1)2(2 x + 3)2, the derivative dy/dx has a factor (x — 1) and a factor (2 x + 3) ; hence show that the given equation represents a curve tangent to the x-axis at x = 1 and at x = — 3/2. 11. Show that if 2/ = (x - 2)3(x3 + 4x — 7), the derivative dy/dx has a factor (x — 2)2. Show that the given curve is tangent to the x-axis at X = 2, but has no minimum or maximum there. 12. Apply the same reasoning which was used in Ex. 11 to the equa- tion y = {x — aY{x — by. 13. Show that the curve j/ = x^ + «x2 + 6x + c is tangent to the x-axis at x = k if (x — A:)2 is a factor of the right-hand side. 14. Show that y = P (x) where P (x) is any polynomial, is tangent to the X-axis at x = A; if (x — k)^ is a factor of P(x). 23. Differentiation of Irrational Functions. In order to differentiate irrational expressions, we proceed to prove that the formula for the derivative of a power (Rule [IVJ) holds true for all fractional powers : nv n <*a^" Hi . P Ill, §23] IRRATIONAL FUNCTIONS 39 First iiroof. Let y = a;p/«, where x> and q are positive integers ; then, raising both sides to the power q, (1) y^^x^. If {x, ?/) and {x + A.r, y + A?/) are pairs of related vahies of X and y, each pair must satisfy equation (1) ; hence (1) holds for X and y, and also (2) (2/ + Ay)« = (.K + AaO'', (3) y^ + q • y'^^y + (several terms) a/ = a;" +p.i"'~^Ax' -I- (several terms) A.r . Subtracting (1) from (3), and dividing both sides of the result- ing equation by Ax : [g?/«~^ + (several terms) A?/] -^ =2)x''^^ + (several terms) Ax, Ax Ay _ j)xp~^ + (several terms) Ax Ax Qi/9~^ + (several terms) Ay' whence ^= lim (^\ = P^^ P^"-' =Pxv/a-y, dx A===o\^Axy gy«-^ qixp^'^y-'^ q This is the same as formula [IV] with ?« = -; hence [IV] holds for any positive fractional exponent. That [IV] also holds for negative fractional exponents is now proved by means of Ex. 3, p. 33; hence [IV] holds for any positive or negative fractional exponent. Second proof. Another proof will seem simpler to some students : if we set (1) X = t", then y = t^, which together are equivalent io y = x^''', and apply formula [Vila] with suitable changes of letters, we find: dy^dy^dx^ , ^ , ^ 22 f,-, dx dt dt Q 40 ALGEBRAIC FUNCTIONS [III, § 23 but since t = x^^^, substitution for t gives dx q q This proves [IV] for positive fractional values of n ; the proof for negative fractional exponents is as given in Ex. 3, p. 33. The rule also holds when n is incommensurable ; for example, given y=x'^^, it is true that dy/dx = ^2x'^^~'^; we shall post- pone the proof of this until § 84, p. 147. 24. Collection of Formulas. Any formula may be combined with [VIIJ, for in any example, any convenient part may be denoted by a new letter, as in § 22. For example, Rule [IV] may be written ^" = ^ . ^, by [VII], =nn'^-' . ^, by [IV]. dx du dx' -^ •- j» dx ^ ^ -^ The formulas we have proved are collected here for easy reference : [11] dx dec [HI] d(u + v) ^^_^^ Holds for subtraction also. doc doc doc [IV] ^ = nt."-if^. doc doc a(¥.] D^-N^ d'- -cf rv'T V-D/ doc doc o • 1 ^* "^ ryi] d{u v)^^dv,^dM^ doc doc doc Special case: — = 1. (y = x.) (I 19) dx Ill, § 25] IRRATIONAL FUNCTIONS 41 These formulas enable us to differentiate any simple alge. braic function. 25. Illustrative Examples of Irrational Functions. In this article the preceding formulas are applied to examples. Examplel. ^ = ^'= ia:V2-i = L-^/z =^ . (See Ex. 4, p. 25.) dx dx 2 2 2v'x Example 2. Given y = VS x- + 4, to find dy/dx. Method 1. Set M = 3 a;2 + 4, then y=^u; by Rule [VII], dy_dy^du _ 1 g _ 6 a; _ 3 x v^3 xi^ + 4 dx du dx 2\/m 2\/3x2 + 4 3x^ + 4 Method 2. Square both sides, and take the derivative of each side of the resulting equation with respect to x : dCy^)^d(3x^jM)^6x. dx dx But by Rule [IV], d(y^^d(y^ . % = 2y^; dx dy dx dx' hence, 2y^ = 6x, or^^3x^ _3x ^ SxVS^^+l dx ' dx y V3 x2 + 4 3 x2 -1- 4 This method, vs-hich is excellent when it can be applied, can be used to give a third proof of the Rule [IV] for fractional powers. The next example is one in which this method cannot be applied directly. Example 3. Given y = a:^ _ 2 V3 x^ + 4, to find dy/dx. dy^d(x^) 2'^ V3^n~i=3x' 6xV3x-^ + 4 dx dx dx 3x2 + 4 Example 4. Given y = (x^ - 2) VS x^ + 4, to find dy/dx. ^ = V3x-i + 4-!^(x8i-2) + (x3-2)A(V3x2 + 4) [by Rule VI] dx dx dx ^ V3 X- + 4 . 3 x2 + (x8 - 2) 3 a: V3 x2 4- 4 .j^^ Example 3x*+ 4 = V3^H4r3x2 + (x3-2).-^^1 = V3^M^ . 12£i±12x^z:6^, L ^ ^ 3x2-|-4j 3x2 + 4 Example 5. Given ?/ = ^ a; + I-V x^ ^^ ^^^^ di//dx. .Vx + 1 + Vx 42 ALGEBRAIC FUNCTIONS [HI, § 25 First reduce y to its simplest form : Vx+l-Vx Vx+l— Vx 2a;+l-2Vx^ '"viTl "+Vx ViC + 1- Vx Then f C2X + 1)- dx -2- where m = X2 + X ; hence dy. dx = 2 2 1 2 vT dx (x+l)-x 2x + l-2Vx2+x. dx du dx (2x + l). V X^ + X This example may be done also by first applying the rule for the deriva- tive of a fraction [Rule V] ; but the work is usually simpler, as in this example, if the given expression is first simplified. EXERCISES X.— ALGEBRAIC FUNCTIONS Calculate the derivatives of 1. (a) y = x4/3. {d) y = Vx^. {h) s = 10^5/2, (e) s = 2V¥\ (c) y = x^'K if) v=i^u\ (/i) 2/ = 6x-2/3. 2. {a)y = x- '_ '''^ (b)v-. Vx« Vx5 (C) S = <3(2«2/3 + 3^-2/3). (^a) y _ 6 _ 2 = 2v/x(xi/3 + a;6/3). U 23^r3J 3. 2/ = f f X \/x5 - if x2 Vx^ + i^ x3 y/x. 4. (a) ?/= V2 + 3X. (Sr) 2/ = ^1 + xK (6)s = V3f-4. (A) 2/= V2x2 + 4x. (c) i; = M\/2 + 3 m. (0 ?/ = x^-v/Sx — 4. (d)s=v/«2-l. (j) 2/ = (5 + 3x)V6x-4. (e) s = V<^ - 3 1. (k) V = Vl - X + xK (/) s = - A+-3j- (Z) s =. V6 t - 5. Ve ( - 5 5 + 3 « 5. (a) 2/ = V 1 + Vx. (ft) s = \y-^- (<=) " Ill, § 25] ALGEBRAIC FUNCTIONS 43 (a) y^{9-6x + 5x2)V{l + x*y. (b) s = (I + r-) y/T^~ti, (b) r = f A + 1\ V(3 - 5w2)6. — ;z^i~- (First rationalize the denominator.) Vl+x+Vl-x (,^^^ V1 + X^ + X . (c), = ^^«^^. Vl + x2 — a; Va^ + a;^ V(20 — 3x»)"-' 10. Draw the graphs of the equations below, and determine the tan- gent at the point mentioned in each case. (ffl) 2/ = VT^, (X = I). {d) y= V(r+x)(2 + 3x),(x = 2). (6) y= Vl+x2, (x = |). (0 2/ = xVrT^, (x = l). (c) 2/ = Vx, (a: = 2). (/) ?/ = xV^ - xV3, (x = 1). 11. Find the angle between the curves y = x^/^ and y = x^ at each of their points of intersection. 12. Find the angle between the curves y = .r^/s and y = x^/^ at (1, 1). 13. Find the angle between the curves y = xp/« and y = x«/p at (1, 1). 14. In compressing air, if no heat escapes, the pressure and volume of the gas are connected by the relation pv^-*^^ = const. Find the rate of change of the pressure with respect to the volume, dp/dv. 15. In compressing air, if the temperature of the air is constant, the pressure and the volume are connected by the relation jiv = const. Find dp/dv, and compare this result with that of Ex. 14. 16. Find dy/dx for y = x- ; for y = x--^ ; for y = x^-s ; for y = x^-* ; for y = x^. Show that the value of dy/dx increases steadily in each case as X increases, and that the magnitudes of the derivatives are in the order of the exponents at x = 1 and for all larger values of x. 17. Draw a graph to show the values of the derivatives for each of the curves of Ex. 16 ; find graphically the values of x for which the derivative of each of them is the same as that of y — x-. 44 ALGEBRAIC FUNCTIONS [III, § 26 PART II. EQUATIONS NOT IN EXPLICIT FORM DIFFERENTIALS 26. Solution of Equations. An equation in two variables x and y is often given in unsolved form ; i.e' neither variable is expressed directly in terms of the other. Thus the equation (1) a^-^y' = l represents a definite relation between x and y ; graphically, it represents a circle of unit radius about the origin. Such an equation often can be solved for one variable in terms of the other ; thus (1) gives (2) y = Vl — X-, or y = — y/1 — x\ The first solution represents the upper half of the circle, the second the lower half. From this solution, we can find dy/dx as in § 25 : (3) dy^_ -^x_ ^ ^^ dy^ +x ^ dx Vl— ar^ ^^' Vl-ar' where the first holds true on the upper half, the second on the lower half, of the circle. By Rule [VII] such a derivative may be found directly without solving the equation. From (1) dx dx but ±(x^ + f)=^(^ + 'Mjn=2x + '^(fl .^, by VII; dx ^ dx dx dy dx '' hence (4) 2a.' + 22/g = 0, or (5) ^ = _^. ^ ^ dx y This result agrees with (3), since y = ± Vl — x^. This method is the same as that used in the first proof of [IVa] in § 23, p. 39, and also in the second solution of Ex. 2, Ill, § 27] IMPLICIT FUNCTIONS 45 p. 41. It may be used whenever the given equation really has any solution, without actually getting that solution. Such a formula as (4) is much more convenient than (3), since it is more compact, and is stated in one formula instead of in two. But the student must never use (5) for values of x and y without substituting those values in (1) to make sure that the point (x, y) actually lies on the curve; and he must never use (5) when (5) does not give a definite value for dy/dx* Thus it would be very unwise to use (4) at the point x = l, y = 2, for that point does not lie on the curve (1) ; it would be equally unwise to try to substitute x = l, y = 0, since that would lead to a division by zero, which is impossible. 27. Explicit and Implicit Functions. If one variable y is expressed directly in terms of another variable x, we say that y is an explicit function of x. If, as in § 26, the two variables are related to each other by means of an equation which is not solved explicitly for y, then y is called an implicit function of x. Thus, (1) in § 26 gives y as an implicit function of x ; but either part of (2) gives y as an explicit function of x. If an equation in x and y is given, so that y is an implicit function of x, we may either solve that equation for y, as in the first part of § 26, and then differentiate as we have done up to this point ; or we may proceed to find the derivative with- out solving, by means of Rule [VII], as in § 26. The latter method is especially fortunate when the given equation is difficult to solve. Definition. If the original equation is a simple polynomial in X and y equated to zero, any explicit function of x obtained by solving it for y is called an algebraic function. See § 18. * These precautions, which are quite easy to remember, are really suflScient to avoid all errors for all curves mentioued in this book, at least provided the equation like (4) [not (5)] is used in its original form, before any cancellation has been performed. 46 ALGEBRAIC FUNCTIONS [III, § 27 Example 1. x' + j/^— 3xi/=0. (Folium of Descartes : Tables, 111, h.) This equation is difficult to solve directly for y. Hence, as in § 26, we find dy/dx by Rule [VII] ; differentiating both sides with respect to x, we find : 3x2 + 32/2 ^^-32/-3x^ = 0; dx dx whence dy ^y_-^ dx 2/2 — X At the point (2/3, 4/3), for example, dy/dx = 4/5 ; hence the equation of the tangent at (2/3, 4/3) is (2/-4/3) = (4/5) (x-2/3) or 4 x-5 »/+4=0. Verify the fact that the point (2/3, 4/3) really lies on the curve. Note that this formula is useless at the point (0, 0) although that point lies on the curve. 28. Inverse Functions. If y is given as an explicit function of X, (1) y=fi^), and if this equation can be solved for x in terms of y, (2) X = cl>(y), then <})(y) is called the inverse function of f(x). If this solu- tion is substituted in the original equation (1), that equation must be satisfied: (3) y=fWy)}' Thus, it y = x^; we find x = y^'^\ substituting ?/^^' for x in the original equation gives y = {y^'^Y, which is an identity. Since it follows that ^ = 1 _=. ^, Ax ' Ay [VII 6] ^ = 1 -^ ^, doc ' dy unless dx/dy — 0.* This rule is really a special case of Rule [VII] ; for if, in Rule [VII], y = x, we get * The precautions to be observed are exactly the same as those of § 26. Ill, §29] IMPLICIT FUNCTIONS 47 dx du _ ^ du dx which agrees with [VII &] except that different letters are used. Thus if n = x^, x = ii^'^ ; du/dx = 3 x^ dx/du = 1/(3 m^/s) ; then (du/dx) • (dx/du) = (3 x'^) • [1/(3 m^/^)] = 1 since x = i(V3. 29. Parameter Forms. If both x and y are given as explicit functions of a third variable t : (1) x=f{t),y = (t), we call t a parameter, and the equations (1) parameter equa- tions. If we can eliminate t, we obtain an equation connecting a; and y directly : (2) F(x, y) = 0. From (2) we might find dy/dx as in § 26 ; but it is usually- easier to proceed as in § 8, p. 14, and § 22, p. 36, using the formula [Vila], in the letters x, y, t: rVTT 1 dy_dy^dx ^ "^ dx~ dt ' dt'. Thus in Example 2, p. 37, we found dy/dx by thi.s formula from equa- tions like (1) ; first, by eliminating t ; second, by using [Vila]- EXERCISES XL — FUNCTIONS NOT IN EXPLICIT FORM In each of these exercises the student should take some point on the curve, and find the equation of the tangent there. 1. From the equation xy = \ find dij/dx by the two methods of § 26, first solving for y, then without solving for y. Write the result in terms of X and y ; and also in terms of x alone, when possible. 2. Find dy/dx in the following examples by the two methods of § 20 : (a) x-^y = 10. ' (/) x2 - 2x2/ + 2x - 3?/ + 4 = 0. (b) x2 + xi/ - 5 = 0. (g) x3 - x^y -4 = 0. (c) x2 - yi = 1. (h) x3 - 2/2 = 0. (d) xy + x + y = 0. (0 x^ + y^ = a^. (e) 4 x2 - 2/2 = 16. 0') a:8 - j/3 = a\ 48 ALGEBRAIC FUNCTIONS [III, § 29 3. Find dyjdx in the following examples without solving for y ; check the answers when possible by the other method of § 26 : (a) x2 + 3 xt/ + 2/2 = 2. (c) ay? + 2 hxy + hy- = A;. (6) JcV _|. 2 x, = MQ, ^ = tan fi, ^^=tan«. Ax dx MK= dy is the change that would have taken place in y, for the change AB = dx in x, if dy/dx, the instantaneous rate of change (or the slope at P), had been maintained. The quan- 52 ALGEBRAIC FUNCTIONS [III, § 31 titles dx(= Ax),* dy(=: m Ax), Ay, Ay — dy (= KQ), are infini- tesimal when Ax approaches zero, i.e. they approach zero as Ax approaches zero. 32. Differential Formulas. For any given function y =/(x), dy can be computed in terms of dx{=Ax), by computing the derivative and multiplying it by dx. Thus, if y = x\ m = dy/dx = 2 X, and dy = mdx = 2xdx; again, it y = x'^—12 x+7, m = 3 x'^ — 12 and dy = m dx = (3 x^ — 12) rfx. Every formula for differentiation can therefore be written as a differential formula ; the first six in the list in § 24, p. 40, be- come after multiplication by dx : [I] dc = 0. (The differential of a constant is zero.) [II] d{G -u) —c • du. [III] d(u -\- v) = du + dv. [IV] d{u-) = nu^-Mu. NdD [VI] d(u ■ v) = u dv + V du. Rules [VII], [VII J, of § 24, p. 40, and [VII^], of § 28, p. 46, appear as identities, since the derivatives may actually be used as quotients of the differentials. From the point of view of the differential notation Rule [VII] merely shows that we may use algebraic cancellation in products or quotients which contain differentials. Rules [I]-[VI] are sufficient to express all differentials of simple algebraic functions. A great advantage occurs in the case of equations not in explicit form, since all applications of Rule [VII] reduce to algebraic cancellation of differentials. * This equation does not assign any particular value to dx but only makes it coincide with the value of Ax chosen above. While we usually think of an infinitesimal as small, because at last it always becomes small, any partic- ular value of an infinitesimal is a fixed finite quantity and may be chosen at pleasure. Ill, § 32] DIFFERENTIALS 53 Example 1. Given y = x^ — 12 x + 7, to find dy and m. dy = d(x^ - 12 a; + 7) = d(3fi) - d(l2 x) + d{l) = 3 x:^dx - 12 dx, whence m = dy ^ dx = 3 x:^ — 12 as in Example 3, p. 10. Example 2. Given y = ^^ — , to find dy (Example 1, p. 33). dv = ^-^ ^ ~ '^)^^'^^" + 3) - (a;" + 3) d(3 X- 7) (3x-7)2 ^ (3x-7)-2a;-(x--' + 3)-3 ^^^ (3x-7)2 Example 3. Given y = (x^ + 2)3, to find dy (Example 1, p. 36). dy = dl{x^ + 2)8 j =S(x-^ + 2)2 d(x^ + 2) = 3(x2 + 2)2 . 2 X • (Zx. Example 4. Given y = xP — 2y/Sx- + -4, to find dy (Example 3, p. 41). dy = J(x3) - 2 dv'3x2 + 4 = 3 x2dx - 2 ^ . d(3 x2 + 4) 2V3x2 + 4 = ^3x2 6 X ] dx. \ V3 x^ + 4 / Example 5. Given x- + y"^ = 1, to find dy in terms of dx (§ 26, p. 44). d(x2 + 2/2) = d(l) = ; but d(x2 + 2/2) = d(x2) + d(j/2) = 2 xdx + 2 ydy ; hence 2 x dx + 2 y dy = 0, or dy = — (x/y) dx, orm =dy/dx — —x/y. Example 6. To find dy and m when x^ + yS _ 3 xy = (Example 1, p. 46). d(x3) + d(2/3) - 3 d(xy) = 0, or 3 x2 dx + 3 2/2 dy - 3 X dy - 3 2/ dx = 0, or (x2 -y)dx+ (y2 - x) dy = 0, whence dy = i^-=^ die, or ?n = ^ = ?^fl^. y2 — X dx y2 — x Example 7. To find dy in terms of dx when x — 3 t + 4, y — (- -^ 2 (Example 2, p. 37). We find dx = d(3 « + 4) = 3 d< ; dy = d (t- + 2) =2tdt] hence m = dy ^ dx = (2/3) «, or dy = (2/3) « dx ; but since f = (x — 4)/3, this may be written : dy= (2/9) (x-4)dx, or m = ^' = (2/n)(x - 4). dx 54 ALGEBRAIC FUNCTIONS [III, § 32 EXERCISES XII. — DIFFERENTIALS [These exercises may be used for further drill in differenti- ation, and for reviews. It is scarcely advisable that all of them should be solved on first reading.] Calculate the differentials of the following expressions : 1. {a) y = a + 2bx + cx"-^. (6) y ={a-\- x^)^. • (c) y={a- bz^y. (d) y={a-\-bx- cx^)^. 2. w''=(''+^)'- ^'^ ^ (a-bxy 3. (a) y = x2(a - x)K (c) 2/ = x* (a - 2 x3)2. 4. (a) y = v'2 X + x'^. (c) J/ = v^l - X*. 5. (a) s = tVT+1. (c) 2/ - ' V'2 X + x'^ (ft) 2/ = a + 6x (d) y-- 1 (« + te + CX2)2 (b) y-- = (l-2x)(l+3x). (d) (a - 6x)2(c + dx^). (6) s = = «a/1 - <-. (.d) y = = (a + x) Va — X. (6) s. = t'y/a - t. (d) s. 1 -« l + t (6) 9 = l-2r2 9 ^2 1 (^) g= ,2 + 2r-3 ' ^^ ^ ^8 + 3-^ 7. Ca^ v=- - — . (ft) 2/ = -7 ^(2-^3)4- ^(tf-a)3 8. (a) 2/=(a + fta-»)p. (ft) y=-^a + bxn. (c) 2/ = ?-— -. (d) 2/ = ^ (a + 6x»)p ^a + ton 9. (a). = l + :^. (ft) .=V^- l-V?/ ^a-by (a + 6x)3/2 Ill, § 32] DIFFERENTIALS 55 10. (a)r=(-V^4-»^)V(5 + 2^. (,) , ^ ^^ _ ^^^'^^7|y . 11. (a) ^^^'i^. (i) g^«-^ + a«^ + a« + l. « - 1 a 4- 1 (C) S = «(a2 + «2) Va2 _ a2. ((JT) e = „3(a2 _ „2)3/2 12. (a) y = (« + 6x)-i. (6) ?/ = (a + 6x)-2. (c) y =(a + bx^)-K (d) y ={a + bx^)-^. X3.(„).=(ii|i:y. (^)"(f^:)-- (c) z =(Ay-3 + ^y-5)2. (rf) z ={Ay-^ + By-^y\ (e) y = A{a + bxy° + B{a - ftx)-i". 14. Determine dy in terms of dx from the equations below : (a) Ax + By + C -0. (g) y/x + Vy = c. X(6) xy + y = l. (h) (l-ax)(x^ + y^) = i. \,^c) x^-2xy-3y^ = 0. (i) x^ + y^ = {ax + by. x2 _ y2 U; y.2 a-bx (e) j^2(x - ffl) = (x + af. (k) y^ - 5 axy + x^ = 0. (/) y*-2y^x-l = 0. (Z) (X + y)3/2 + (X - 2/)3/2 = a3/2. 15. Obtain the equation of the tangent at (2,-1) to the curve 4 x' - 2 xj/ - 5 y- _ 6 X - 4 ?/ - 7 =0. 16. Obtain the equation of the tangent at (2, 1) to the curve / x3 - 7 x^j/ - 5 2/3 + 4 x2 - 10 xy + 8 X — 5 2/ + 18 = 0. 17. Obtain the equation of the tangent at (xo, j/o) to each of the fol- lovfing curves : Curve Tangent (a) y- =Aax; yyo = 2 a(x + Xo). (6) x2 + 2/2 r= a2 ; xxo + yyo = cfi- <=)S-S-^ a^o I yyo _ I C2 62 (d) (X + 2/)2 = 1 ; (X + y){xo + J^o) = 1- 56 ALGEBRAIC FUNCTIONS [III, § 32 18. Find the derivative dyjdx for the curves defined by each of the pairs of parameter equations given below : l + « I 3^^^' at 1 + « 1^2' i ^ a2«2 1 4 7rr-' (d) (« [ a; = 4 7rr-2, (e) i. = |... e-2. r 3 47rr3 19. If a particle moves so that its coordinates (a;, y) at any time t are "^ ~ 1 + <2' ^ 1 + i2' show on the same diagram the values of x and of y in terms of time ; what are the extreme values of x and of ?/, and when are they attained ? From the diagram construct another showing the (x, y) curve followed by the particle. 20. Calculate the x and y components of the speed {v^ and Vy) at any time «, and the resultant speed Vi'x- + Vy^^ along the path, in the motion of Ex. 19. Show that Uj, -=- Vx = dy/dx. See Ex. 7, p. 49. 21. If a particle moves so that its coordinates in terms of the time are X = 1 - « + <^ y = 1 +t + t-, show that its path is a parabola. Show that from the moment « = its speed steadily increases. 22. A point moves on a straight line so that its distance s from a fixed point on the line at any time t is as given below. Describe the motion from « = 0, giving the times when the speed is positive, negative, zero. Draw the (s, t) diagrams and the {v, t) diagrams. (a) s = «2-4< + 3. (&) s = «3 - 15 «2 + 68 « - 4. (c) s = 3<* - 40«3 + 54 «2 _ 10; ^ 23. If the volume of a sphere increases at the rate of 2 cu. ft. a second, calculate the rate of change per second of the radius and of the surface. What are these rates when the volume is 100 cu. ft. ? 24. If B denotes the radius of a sphere, S the surface, and V the volume, calculate the differential of each of these in terms of each of the othei-s. Ill, § 32] DIFFERENTIALS 57 25. If the radius of a cylinder expands at the rate of 1/2 in. a second, starting with a value 5 in., and if the height remains fixed at 10 in., at what rate per second is the volume changing at any time t ? When t — 10 sec. ? The same for the total surface ? ^ , 26. When you walk straight away from a street lamp with uniform speed, does the end of your shadow also move with uniform speed '.' Supposing that your height is 70 in., show how fast the shadow tip moves if you walk 6 ft. per second away from a lamp 10 "ft. above ground. 27. The electrical resistance of a platinum wire varies with the tem- perature, according to the equation B = Eo(l-ae + bd-)-^; calculate dB in terms of d0. What is the meaning of dR/dd ? 28. Van der Waal's equation giving the relation between the pressure and volume of a gas at constant temperature is Draw the graph when a = .0087, b = .0023, c = 1.1. Express dv in terms of dp. What is the meaning of dv/dp ? 29. The crushing strength of a hollow cast iron column of length I, inner diameter d, and outer diameter D, is r=46. Vo(^^ Calculate the rate of change of T with respect to D, d, and I, when each of these alone varies. 30. Show that the curve y = (x— ay + b has no maxima or minima. 31. Proceed as in Ex. 30 for y = (x — ay + b. 32. Show (see Exs. 10-14, p. 38) that the curve y = P(x) is tan- gent to the X-axis at points where the polynomial P(x) has a double root. 33. Show that if P{x) is a polynomial, its double roots are also roots of the polynomial P'{x) = dP(x)/dx. Hence the H. C. D. of P(x) and P'(x) contains as a factor x— k, where k is the double root. 34. Assuming the principle of Ex. 33, find the double roots of each of the following equations : (a) x3 + a:2 - 5x + 3 = 0. (f) x* + 2 x^ - 11 x2-12 x -|- 30 = 0. (6) x'^ + 3 x2 - 4 = 0. (J) X* -h 2 x3 - 2 X - 1 = 0. CHAPTER IV FIRST APPLICATIONS OF DIFFERENTIATION PART I. APPLICATIONS TO CURVES — EXTREMi:S 33. Tangents and Normals. We have seen in § 4, p. 6, that if the equation of a curve C is given in explicit form : (1) y=f{^), the derivative at any point P on C represents the rate of rise, or slope, of (7 at P: ^^^ [S ati-^t^'^^^' ofC]^,j. = slo2?e ofPT=tan a=[m] ^tP, where a is the angle XIIT, counted from the positive direction of the X-axis to the tangent PT, and where m^ denotes the slope of C at P. Hence (§ 4, p. 7) the equation of the tangent is where the subscript P indicates that the quantity affected is taken with the value which it has at P. If the slope nip is positive, the curve is risi7ig at P ; if lUp is negative, the curve is falling ; if mp is zero, the tangent is hori- zontal (§ 6, p. 8). Points where the slope has any desired value can be found by setting the derivative equal to the given number, and solving the resulting equation for x. Y \ V ^T .^ ^ X ^U AN\ IV, § 34] TANGENTS AND NORMALS 59 Since, by analytic geometry, the slope n of the normal PN is the negative reciprocal of the slope of the tangent, we have, (4) n, = slope ofPN=-^=- ) , as in Ex. 8, p. 11, hence the equation of the normal is : 34. Tangents and Normals for Curves not in Explicit Form. The equation of the curve may not be given in the explicit form (1); instead, it may not be solved for either letter (1) F(x,y)=0, as in §§ 26-27, pp. 44-45; or it may be solved for x: (2) x = (y), as in § 28, p. 46 ; or the equations in parameter form may be given : (3) x=f((), y = {t), as in § 29, p. 47. In any of these cases, chj/dx can be found by the methods of the articles just cited, and this value may be used in the formulas of § 33. No new formulas are necessary. In the particular case of the parameter form (3), however, a special formula is sometimes useful. Since by § 29, the equation of the tangent becomes, after simplification, ^ LdtAp ^ LdtSp' y-Vp z-Zp and the equation of the normal is A special formula for equations in the implicit form (1) will be given later (§ 164) ; just now it would actually be inconvenient. 60 APPLICATIONS OF DIFFERENTIATION [IV, § 35 35. Secondary Quantities, in Fig. 13, § 33, since tan « (= tiip = [rfy/dx]p), and AP( = yp), are supposed to be known, the right triangles HAP and PAN' can both be solved by the rules of Trigonometry, and the lengths HA, AN", HP, PiV can be found in terms of mp and yp : [Subtangentjp =^J. =AP-i- tan a = yp-^mp= [y/m]p. [Subnormal]/. = AN = AP • tan « = [?/. m]/>, since a = Z APX. [Length of tangent]/. = HP = ^AP'^ + HI^ = Vy% + [2//m]^ = [y Vl + (l/m)^]p [Length of normal]/. = P.V = ^AP^ + AN' = \/y], + (y • m)2, = ly Vi + TO-^]p. It is usual to give these lengths the names indicated above ; and to cal- culate the numerical magnitudes of these quantities without regard to signs, unless the contrary is explicitly stated. 36. Illustrative Examples. In this article, a few typical examples are solved. Example 1. Given the curve y = x^—12x + 7 (Ex. 2, p. 25) , we have m = dy/dx = 3x^—12. (1) The tangent (T) and the normal (N) at a point where x = a are (T) y- (a3-12a + 7) = (3a2-12) (a; - a), (,V) y_(a8_12a + 7)=^-=^(x-a); thus, at a; = 3, the tangent and normal are (T) y + 2 = 15(x-3), (iV) y + 2 = -j-V(x-3). (2) The tangent has a given slope k at points where 3x2-12 = A; -•- "- ' -'^' + 12 there are always two points where the slope is the same, if i > — 12 ; thus if A;=0, x=±2; il k = —9, x=±l ; if A; =-12, x=0 ; if ^•< — 12, no real value for x exists (see Fig. 17, p. 77). (3) The secondary quantities of § 35 may be calculated without using the formulas of § 35. Thus, at the point where x = 3, the tangent (T) cuts the X-axis where x = 47/15 ; the normal (N) cuts the x-axis where X = — 27. If the student will draw a figure showing these points and IV, § 36] TANGENTS AND NORMALS 61 lines, he will observe directly that the subtangent is 2/15, the subnormal 30, the length of the tangent y/W+lY/Wf, the length of the normal VSO- + 22. These values agree with those given by § 35. (4) The given curve cuts the curve y = a;^ — 5 at a point given by solv- ing the two etiuations simultaneously ; this gives a; = l,y = — 4; at this point the slopes of the two curves are m'l = — 9, jjio = + 3 ; hence, by Analytic Geometry, the acute angle between them is given by the formula tan^=:"'^-»^-^^--^^^L2^1, 1 + miTO.2 1-27 26 13 from which can be found by use of a trigonometric table ( Tables, V, A). From a larger table, we find 6 = 24^ 47'. Example 2. Given the circle x"^ -\- y- — \, we have m = dy/dx = — x/y [see § 26]. (1) The tangent (T") and normal (iV) at a point (xo, yo) are IT) (y-y,) = -^(x-Xo), (X)(y-yo)^y^(x-xo); yo xo or, since Xq^ + j/q^ =: 1, {T) xxo + 2/1/0 = 1, (^") y^o = yox ; thus, at the point (3/5, 4/5), which lies on the circle, we have (r)3a; + 4y = 5, (N)3y = 4x. (2) The tangent has a given slope k at points where -^ = k, i.e. xo + ktjo = 0. 2/0 The coordinates (xo, 2/o) can be found by solving this equation simul- taneously with the equation of the circle, or by actually drawing the line Xo + kyo=0. Thus the points where the slope is + 1 lie on the straight line a; 4- 2/ = ; hence, solving x + 2/ = and x- + y'^ = 1, the coordinates are found to be X = ± 1/ v'2, y = ^ 1/ V2 ; but these points are most readily located in a figure by actually drawing the line z + y = 0. (3) The given circle cuts the parabola Qy — 20x2 at the points (± 3/5, 4/6) ; at the point (3/5, 4/6) the slopes of the two curves are mi = — 3/4, ma = 40 x/9 = 8/3 ; hence the acute angle 6 between the two curves at that point is tan e = "'^ ~ "^- =~^ 3.4167, whence 6 = 73° 41' 10". 1 + ??li??l2 12 62 APPLICATIONS OF DIFFERENTIATION [IV, § 36 EXERCISES Xm. — TANGENTS AND NORMALS 1. Find the equations of the tangent and that of the normal, and find the four quantities defined in § 35, for each of the following curves at the point indicated : (a) 7j = x^-12x + '!; (1, - 4). (e) x = y^ - 3y'^ + 5 ; (3, 1). fb) y= ^^~^ ■ (-1, 3). (/) l*^*^^"^^'^! ; (1, 1). (c) 9x2 + 2/2 = 25; (1,4). , . \x = t-^+4.t-\\ . .^ ^ j. (d) X2/ + ?/2_2x = 5; (-4, 1). ^^^ j?/ = «' - 3 « + 5 j ^ ^' 2. Find the angle between the curves y = x^ and y^ = zat each of their common points. See Tables, III, A. 3. Find the points (if any) at which each of the curves in Ex. 1 has the slope zero ; the slope + 1. 4. Determine the values of x for which the slope, in each of the curves in Ex. 1, is positive ; and those for which it is negative. 5. In Ex. 1, the curves (a) and (c) pass through the point (1, — 4); at what angle do they cross ? 6. The curves y = x, y = x^, y = x^, ■•., y = x~^, y = x~^, •••, y — x^'^, y = x^/^, ••• all pass through the point (1, 1). Determine the angle which each of these curves makes with the first one of them at that point. 7. Determine the angle between the curves y = x^ and y = x™ at the point (1, 1) where m and n have any values whatever ; at the point (0, 0) (only if both n and m are positive). (Special case: n=p/q, m = q/p, where p and q are integers.) 8. Determine the equation of the tangent and that of the normal to the ellipse b^x^ + a'^y^ = a%'^ at any point (xo, yo) on it. [Solution : 2 b'^xdx+ 2 a^y dy = 0, hence dy/d3P=— b'^x/a-y ; the tan- gent and normal are, respectively, (T) (y-yo) = -^(x-xo),(^) (y-yo)=^'(x-x,), a-yo 0% or (T) b"xxQ + a^yyo = a^ft^, (iv") b-x^y — a-xy^, = xoyo^b'^ — a-), since 6%2 + (^a^^a _ (^252.] 9. Determine the equation of the tangent and that of the normal to each of the following curves at any point (Xq, j/q) on it : (a) y = A:x2. (e) b^x^ - aY = a^b\ (b) y2 = 2 px. (/) ax2 + 2bxy + cy^ = f. (c) x^ + y^ = a'\ (g) ax^ + 2bxy + cy"^ + 2dz + 2 ey +f = i Id) y = kx\ Qi) y = (ax + b)/{cx + d). IV, § 38] EXTREMES 63 10. The curve whose equations in parameter form are (1) x = 3t + 4, y = t^ + 2, gives (Example 2, p. 37): dt dt 3 hence this curve has a slope 1 when t = 3/2, i.e. when x = 17/2, y = 17/4. Its slope is when t = 0, i.e. at (4, 2). Verify these facts by drawing an accurate figure ; also by eliminating t in (1) and finding the derivative from the explicit equation. 11. Show that the slope of the curve z^ + y^ — Sxy = (Example 1, p. 46) is + 1 at points where it cuts the circle x- + y- — x — y = 0. Show that its slope is zero (tangent horizontal) where it cuts the parabola y = a;2 ; that the tangent is vertical (1/m = 0) where it cuts the parabola y2 =x. 12. Draw the curve of Ex. 11 by using its equations in parameter form (Ex. 6 d, p. 48) : 1 + «3' ^1+^3' and show that dy/dx =(2t — ?*)/(! — 2 «'), found from these equations, agrees with the value found from the implicit equation. 37. Extremes. In § 6, p. 8, and in numerous examples, we have found maxima and minima of functions by first finding the points at which the tangent is horizontal, and then testing these values. The value of /(x) at a point where x=a is /(a). This value is r maximum 1 , -^ -^ ■ f greater than ] ^, , a ^ . . \ value it it is K ^, any other value [ minimum J [ less than j *' of /(x") for values of x sufficiently near to re = a. A maximum or a minimum is called an extreme value, or an extreme oif(x). 38. Critical Values. We have seen that a horizontal tangent (i.e. slope zero) does not always give rise to an extreme. Thus, the curve y = x^ (Ex. 5(6), p. 11) has a horizontal tangent at the origin ; but the origin is neither a highest nor a lowest point. 64 APPLICATIONS OF DIFFERENTIATION [IV, § 38 On the other hand, extremes may also occur at points where the derivative has no meaning, or at points where the function becomes meaningless. Thus, the curve y = x-V3 gives m = 2/(3 x^s) ; hence m is meaningless when X = ; in fact, the curve has a vertical tangent at that point. It is easy to see that this is, however, the lowest point on the curve. Again, if a duplicating apparatus costs $ 150, and if the running expenses are 1 (* per sheet, the total Fig. 14. ^^^^ ^^ printing n sheets is « = 150 + .01 n. This equation represents a straight line ; geometri- cally there are no extreme values of t ; but practically t is a minimum when n = 0, since negative values of n are meaningless. Such cases are usually easy to observe. A value of x of any one of the types just mentioned, at which f(x) may have an extreme, is called a critical value; the cor- responding point on the curve y=f{^) is called a critical point. 39. Fundamental Theorem. We proceed to show that a func- tion f{x) cannot have an extreme except at a critical point : that is, assuming that f{x) and its derivative have definite meanings at x = a and everywhere near x = a,no extreme can occur if the derivative is not zero at x = a. We are supposing that all our functions are continuous ; if, then, the derivative m is positive at a; = a, it cannot suddenly ^ become negative or zero. Hence m is positive on both sides of x = a, and there can be no extreme there. Likewise if m is negative, the curve is falling near x = a on both sides of a; = a; there can be no extreme. 40. Final Tests. It is not certain that f(x) has an extreme value at a critical point. To decide the matter, we proceed to determine whether the curve rises or falls to the left and to IV, § 41] EXTREMES 65 the right of the critical point: it rises if m>0; it falls if m < 0. Near a maximum, the curve vises on the left and falls on the right. Near a minimum, the curve falls on the left and rises on the right. If the curve rises on both sides, or falls on both sides, of the critical point, there is no extreme at that point. 41. Illustrative Examples. Example 1. To find the extremes of the function y=f(x) = r'^— 12 x-f-7. (See § 6, p. 10.) {A) To find the Critical Values. Set the derivative equal to zero and solve for x : ^ _ f'.v . dx : 3 a:2 - 12 ; 3 x^ - 12 = ; x = 2 or x (JB) Precautions. Notice that/(x) and its derivative each has a mean- ing for every value of x ; hence x = + 2 and X =— 2 are the only critical values. (C) FinalTests. m=3x2-12=3(x2-4) is positive if x is greater than 2, nega- tive if X is slightly less than 2 ; hence the curve rises on the right and falls on the left of X = 2, therefore /(2) = — 9 is a minimum of /(x). The student may show that /(— 2) =23 is a maximum of /(x). (See Fig. 5, p. 10.) Example 2. To find the extremes of the function 2/=/(x)=3x*-12x8 + 50. {A) Critical Values. Setting f?2//f?x = 0, and .solving, we find : ^=12x8. dx 12 x8 - 36 x2 X = 0, or X = 3. {B) Precautions, y and dy/dx have a meaning everywhere ; the only critical values are and 3. 1 \ \ '\ \ ] 1 2/ = X - IL z|f+5(jj LL 1 2 i 1 I i Fig. 15. 66 APPLICATIONS OF DIFFERENTIATION [IV, § 41 (C) Final Tests. Near x = 0, m = 12x-(x — 3) is negative on both sides ; hence there is no extreme there, though the tangent is horizontal. Near x = 3, m = 12 x^{x — 3) is positive on the right, negative on the left ; hence / (3) = — 31 is a minimum. The information given above is of great assistance in accurate drawing. Example 3. Tvfo railroad tracks cross at right angles ; on one of them an eastbound train going 15 mi. per hour clears the crossing one minute before the engine of a southbound train running at 20 mi. per hour reaches the crossing. Find vyhen the trains were closest together. Let X and y be the distance in miles of the rear end of the first train and the en- gine of the second one from the crossing, respectively, at a time t measured in min- utes beginning with the instant the first train clears the crossing ; then 60 60^ " 16/25, m>0; hence D^ is diminishing before t = 16/25 and increasing afterwards. It follows that Z) is a mini- mum when t = 16/25. Substituting this value for t, we find 25 25 25 hence the minimum distance between the trains is 1/5 of a mile, and this occurs 16/25 of a minute after the first train clears the crossing. Example 4. To find the most economical shape for a pan with a square bottom and vertical sides, if it is to hold 4 cu. ft. Let X be the length of one side of the base, and let h be the height. Let V be the volume and A the total area. Then V = hx? = 4, whence h = 4/x2; and -.n, A = x'^ + 4:hx = x^ + — ; X IV, § 41] EXTREMES 67 whence we find dx x:^ x? When a; < 2, m = 2{x? — ^)lx^ is negative; when x>2, m is positive; hence A is decreasing when x is increasing toward 2, and A is increasing as X is increasing past 2 ; therefore x — 2 gives the minimum total area yl = 12. Notice that the height is A = 4/x2 = 1. The correct dimensions are x = 2, /i = 1 (in feet) . Example 5. The pan of the preceding example is to sit under a refrigerator which clears the floor by 8 in. How should it be made ? Since h cannot now exceed 8 in. = 2/3 ft., it is clear that the mini- mum of .4 found in Ex. 4 does not apply. The function A = x'^ -\- IG/x is meaningless if A > 2/3, i.e. if 4/x'^>2/3, or x-/4<3/2, orx< \/6 = 2.45 (in feet). Since A is increasing as x increases, x should be made as small as pos- sible ; practically, we ought to chose, say x = 2.5 ft. = 30 in. ; then h = 16/25 ft. =7.68 in., — we ought to take h about 7 3/4 in., which gives 1/4 in. clearance. This gives F= 6975 cu. in., in place of 6912 required, but this difference is on the safe side, and is practically negligible, because it corresponds to a difference in height of much less than 1/8 in, EXERCISES XIV.— EXTREMES 1. Determine the maximum and minimum values of the following functions and draw the graphs, choosing suitable scales : (a) 2/ = x3 - 3 x2 + 1. (ft) s = 2 i'i - 3 <2 - 36 « + 20. (c) p = g^ + 6 ^2 _ 15 q^ (f^) y = x3 - 2 ax^ + a2x. (e) X = ?/* - 8 >/2 + 2. (/) r = w4 _ 4 m3 + 4 „2 + 3. {g) m = 7i5 _ 10 ,^4 + 20 ?i3 + 32. {h) ^ = j-6-6r* + 4 r"'+9r2-12r+4. (0 s = ((-l)(2-0^- U) V=h(h-1)^. (k) r = (s2 - l)(s2 _ 4). (0 X = (2/ - 2)3 (y + 3)8. (.r + 2)-^ (0) y- "^ x2 + 2 X + 4 X3 - X ^'^' X^-X^ + l (s) Z» = rV2-j-2. («) V = dp) K: h ah- + bh + c 00 Q = -.k + Vl- k. (0 B = : -^.7+15 - X. 68 APPLICATIONS OF DIFFERENTIATION [IV, § 41 2. What is the largest rectangular area that can be inclosed by a line 100 feet long ? 3. What must be the ratio of the sides of a right triangle to make its area a maximum, if the hypothenuse is constant ? 4. Determine two possible numbers whose product is a maximum if the sum of their squares is 50. Is there any minimum ? 5. Determine two numbers whose product is 100 and such that the sum of their squares is a minimum. Is there any maximum ? Did you account for negative possible values of the two numbers ? 6. What are the most economical proportions for a cylindrical can ? Is there any most extravagant type ? Mention other considerations which affect the actual design of a tomato can. Is an ordinary flour barrel this shape ? What different considerations enter in making a barrel ? 7. What are the most economical proportions for a cylindrical pint cup ? (1 pint = 28| cu. in.) Mention considerations of practical design. 8. Determine the best proportions for a square tank with vertical sides, without a top. Is there any most extravagant shape ? 9. The strength of a rectangular beam varies as the product of the breadth by the square of the depth. What is the form of the strongest beam that can be cut from a given circular log ? Mention some other practical considerations which affect actual sawing of timber. 10. The stiffness of a rectangular beam varies as the product of the breadth by the cube of the depth. What are the dimensions of the stiffest beam that can be cut from a circular log ? 11. Is a beam of the commercial size 3" x 8" stronger (or stiffer) than the size 2" x 12" (1) when on edge, (2) when lying flat? [Commercial sizes of lumber are always a little short.] 12. What line through the point (3, 4) will form the smallest triangle with the coordinate axes ? Is there any other minimum ? Any maximum ? 13. Determine the shortest distance from the point (0, 3) to a point on the hyperbola x"^ — y^ = 16. Show that it is measured on the normal. [Hint. Use the square of the distance.] 14. The distance D from the point (2, 0) to the circle x^ -\-y^ = \ is given by the equation D'^ = 5 — 4 x. Discover the maximum and mini- mum values of D-, and show why the ordinary rule fails. 15. Show that the maximum and minimum on the cubic y = x? — ax + 6 are at equal distances from the y axis. Compute y at these points. IV, § 41] EXTREMES 69 16. Show that the cubic x^ - ax + h = has three real roots if the ex- treme values of the left-hand side (Ex. 15) have different signs. Express this condition algebraically by an inequality which states that the product of the two extreme values is negative. [Any cubic cau be reduced to this form by the substitution a: = 3;' + A ; heuce this test may be applied to any cubic] 17. Show that if the equation x^— ax + b =0 has two real roots, the derivative of the left-hand side (i.e. 3 x^ - «) must vanish somewhere between the two roots. Show that the converse is not true. 18. The line y = mx passes through the origin for any value of m. The points (1, 2.4), (3, 7.6), (10, 2.5) do not lie on any one such line : the values of y found from the equation y = mx at x = 1, S, 10 are m, 3 m, 10 m ; the differences between these and the given values of y are (m — 2.4), (3 m — 7.6), (10 vi —25). It is usual to assume that that line for which the su7n of the squares of these differences S=(m- 2.4)2 + (3 m- 7.6)2 + (lO m - 25)2 is least is the best compromise. Show that this would give m = 2.50 (nearly). Draw the figure. 19. In an experiment on an iron rod the amount of stretching s (in thousandths of an inch) and the pull p (in hundreds of pounds) were found to be (p = 5, s = 4), (p = 10, s = 8), {p = 20, s r= 17). Find the best compromise value for m in the equation s = m • p, under the assump- tion of Ex. 18. A71S. About 5/6. 20. A city's bids for laying cement sidewalks of uniform width and specifications are as follows: Job No. 1: length = 260 ft., cost, $110; Job No. 2: length, 600 ft., cost, §250; Job No. 3: 1500 ft., cost, §630. Find the price per foot for such walks, under the assumption of Ex. 18. How much does this differ from the arithmetic average of the price per foot in the three separate jobs ? 21. The amount of water in a .standpipe reaches 2000 gal. in 250 sec, 5000 gal. in 610 sec. From this information (which may be slightly faulty) find the rate at which water was flowing into the tank, under assumption of Ex. 18. 22. The values 1 in. = 2.5 cm., 1 ft. = 30.5 cm. are frequently quoted, but they do not agree precisely. The number of centimeters c, and the number of inches i, in a given length are surely connected by an equation of the form c = ki. Show that the assumptions of Ex. 18 give k = 2.641. Is this the same as the average of the values in the two cases ? Which result is more accurate ? 70 APPLICATIONS OF DIFFERENTIATION [IV, § 41 23. In experiments on the velocity of sound, it was found that sound travels 1 mi. in 5 sec, 3 mi. in 14.5 sec. These measurements do not agree precisely. Show that the compromise of Ex. 18 gives the velocity of sound 1084 ft. per second. How does this compare with the average of the two velocities found in the separate experiments ? 24. A quantity of water which at 0° C. occupies a volume vo, at 6° C. occupies a volume V = Uo(l - 10-* X .5758 d + 10-5 X .756 ff^ - lO"' x .351 6^). Show that the volume is least (density greatest), at 4° C. (nearly). 25. Determine the rectangle of greatest perimeter that can be in- scribed in a given circle. Is there any minimum ? 26. What is the largest rectangle that can be inscribed in an isosceles triangle ? Is there any minimum ? 27. Find the area of the largest rectangle that can be inscribed in a segment of the parabola ?/- = 4 ax cut off by the line x = h. 28. Determine the cylinder of greatest volume that can be inscribed in a given sphere. Is there also a minimum ? 29. Determine the cylinder of greatest convex surface that can be inscribed in a sphere. Is there a minimum ? 30. Determine the cylinder of greatest total surface (including the area of the bases) that can be inscribed in a given sphere. 31. What is the volume of the largest cone that can be inscribed in a given sphere ? 32. What is the area of the maximum rectangle that can be inscribed in the ellipse xVa- + y-Zb"^ = 1? PART II. RATES 42. Time-rates. All the applications- of derivatives are rates of increase (or decrease) of some quantity with respect to some other quantity which is taken as the standard of com- parison, or independent variable. Among all rates, those which occur most frequently are time-rates, that is, rate of change of a quantity with respect to the time. IV, § 45] RATES 71 43. Speed. Thus the speed of a moving body is the time- rate of increase of the distance it has traveled : (1) V = sjyeed * = lim — = — , as in § 7, p. 12, and in numerous examples. 44. Tangential Acceleration. The specfZ itself may change; the time-rate of change of speed is called the acceleration along the path, or the tangential acceleration.f (2) jr = tangential acceleration f = lim — • = — • A/ = oA^ dt Thus for a body falling from rest, if g represents the gravitational constant, s=lgt^; hence ^ = 1-^' and ^^-|- = it follows that the tangential acceleration of a body falling from rest is constant ; that constant is precisely the gravitational constant g. J In obtaining the tangential acceleration, we actually differ- entiate the distance s twice, once to get v, and again to get dv/dt or _/„ hence the tangential acceleration is also said to be the second derivative of the distance s passed over. 45. Second Derivatives, Flexion. It often happens, as in § 44, that we wish to differentiate a function twice. In any * The speed v is distinguished from the velocity v by the fact that the speed does not depend on the direction ; when we speak of velocity we shall always denote it by v (in black-faced type) and we shall specify the direction. t The general acceleration J is also a directed quantity ; when we speak of the acceleration J (not tangential acceleration j j) we shall denote it by J, and give its direction. As in the case of speed, the letter J, iu italic type, denotes the value oij without its direction. (See Ex. 17, p. 74.) t The value of c/ is approximately 32.2 ft. per second per second = 981 cm per second per second. 72 APPLICATIONS OF DIFFERENTIATION [IV, § 45 case, given y = f(x), the slope of the graph is m= -^= lim — ^« dx AcB=o^aj The slope itself may change (and it always does except on a straight line) ; the rate of change of the slope with respect to x will be called the flexion * of the curve: 0= flexion =■ — = hm , dx Aa!=oAa; and will be denoted by h, the initial letter of the word hend. Thus for y = a;2, m=2x, & = 2 1 ; iox y = oi?, m = 3 x^, 6 = 6 x ; for y = x^ — 12 X + 7, m = 3 x2 — 12, & = 6 X ; for any straight line y = kx+c, m = k, 6 = 0. The value of b is obtained by differentiating the given func- tion twice ; the result is called a second derivative, and is represented by the symbol : d^y _ d_ fdy\ _ dm _ , dJT dx \dxj dx Likewise, the tangential acceleration in a motion is d^s d /'ds\ dv df dt\dt) dt '"^'" If the relation between s and t is represented graphically, the speed is represented by the slojye, the tangential acceleration by the flexion, of the graph. Thus if s = gt'/2 be represented graphically, as in Fig. 6, p. 13, the slope of the graph is m = sloj)e = — = gt = speed = v, and the flexion of the graph is b = flexion = - — = — ^^ = — = 9 = tangential acceleration =jr- at at az * The word ciirvature is used in a somewhat different sense. See § 97, p. 1(59. t The flexion for this parabola is constant ; note that this means the rate of change of m per unit increase in x, not per unit increase in length along the curve. See §61, p. 106. IV, §45] SECOND DERIVATIVES 73 EXERCISES XV. SECOND DERIVATIVES — ACCELERATION [In addition to this list, the second derivatives of some of the functions in the preceding exercises may be calculated.] 1. Calculate the first and second derivatives in the following exercises. Interpret these exercises geometrically, and also as problems in motion, with s and t in place of y and x : (a) y = a;2 + 3a;-4. (A) j/ = Vx + >/^M^. (6) y=-x2 + 3x-4. (Z) y =(2 - 3x)2 (3 + x). (c) y = 2 x2 - X - 15. (,„) y=(x + 2)8 (x2 - 1). (d) y = - 2 x2 - X - 15. („) y = vT+x -^ Vl - x. (e) 2/ = x2-|x-21. (o) 2/ = ax + 6. (/) y = x8 — 3 x2 + 1. (p) y = c (a constant), (fir) y = 2x8-3x2-36x-20. (q) y = ax^ + bx + c. (A) y = X* - 8 x2 + 2. (r) y = c (x - a)». (0 y = x4-2x3 + 5x2 + 2. (s) y =(x - a)»(x - 6)» (j) y=(l + x)-^(l-x). (0 y = ^x-*. 2. Show that the flexion of a straight line is everywhere zero. 3. Show that if the distance passed over by a body is proportional to the time the tangential acceleration is zero. What is the speed in this case? 4. Show that the flexion of the curve y = ax2 + 6x + c is everywhere the same, and equal to twice the coefficient of x2. 5. Show that if the space-time equation is s = at^ + bt -\- c, the ac- celeration is always the same and equal to twice the coefficient of t-. Is such a motion at all liable to occur in nature ? 6. Find the flexion of the curve y = 1/x. Show that it resembles y itself in some ways. Does the slope also resemble y ? "Which one re- sembles y the more closely ? 7. Can you interpret Ex. 6 as a motion problem ? "What is true at the beginning of the motion (« = 0) ? Can a curve with a vertical asymptote represent a motion ? Can a piece of such a curve ? 8. Find the flexion of the curve y =(x - 2)8 (x + 3)2 (x - 4). Show that the flexion has a factor (x— 2), while the slope has a factor (X _ 2)2 (X + 3). 74 APPLICATIONS OF DIFFERENTIATION [IV, § 45 9. Show that the flexion of the curve y = (x - ay (yfi -ir 5) has a fac- tor (x - a). 10. If the function y = x^ + ay:^ + bx^ + ex- + dx + e = has a factor (a: — ay, show that dy/dx has a factor (a; — «)2, and d-y/dx"^ has a factor (x-a). 11. If the equation x^ + ax* + fex^ + cx^ + rfx + e = has a triple root X = «, show that the equation 20 x^ + 12 ax^ + 6 6.>: + 2 c = has a factor X- «. 12. Show how to find the double and triple roots of any algebraic equation by the Highest Common Divisor process. 13. If the equations of the curve in parameter form are x = J^, y = t^, find the slope m and the flexion b in terms of t. [Hint. First find m ; then use the values of 7n and x iu terms of t to find d7n/dx.] 14. Find m and b for each of the following parameter forms : t^ [See Ex. 13.] ^ 1 + «3 ' 15. If the equations of Ex. 13 express the position of a moving par- ticle at the time t, find the horizontal speed Vx — dx/dt and the vertical speed Vy — dy/dt. A second differentiation gives the time-rates of change of these component speeds : jx = dvx/dt = d'^x/dfi and jy = dvy/dt = d-y/dfi. Eind each of these quantities in Ex. 13. In each of the exercises in Ex. 14. 16. The total speed v = Vt^- + Vy- can be found as in Ex. 7, p. 49, from the values of v^ and Vy. Find v in each of the examples of Exs. 13 and 14. 17. The component accelerations jx ancb-^'^ of Ex. 15 may be com- bined to get the total acceleration .; ~ y/jx^ + j/ by the so-called paral- lelogram law of physics. Find j in each of the examples of Exs. 13 and 14. 18. The tangential acceleration jr can be found directly from Ex. 16, by means of its definition jy= dv/dt. Find j^ in Exs. 13 and 14, Show that j J, and j are different in every exercise except 14 (a). [The reason for this difference is not difficult : J^, is the acceleration in the path itself; j is the total acceleration, part of its effect being precisely to (a) x-a + bt,y = c + dt. (b) x = t\y^ (c) z = t,y = r2 -,1-1 and 2. id) X = 1 + «, «^=rii-4-'^'-- ^■^>^ = iT7. IV, § 47] CONCAVITY 75 make the path curved ; hence a part of J is expended not to increase the speed, but to change the direction of the speed, i.e. to bend the path. Notice that Ex. 14 (a) represents a straight line path; on it jj,=j; this holds only on straight line paths. In uniform motion on a circle, for example, jj.= 0.] 46. Concavity. Points of Inflexion. If the flexion b = dm/dx is positive, the slope is increasing, and the curve turns upwards, or is concave iipivards; if the flexion is negative, the slope is decreasing, and the curve is concave doicmcards. Thus y = z'^ is concave upwards everywhere, since b = 2 is positive. For y = x^ we find b = G .r, which is positive when x is positive, and nega- tive when X is negative ; hence y — x^ is concave upwards at the right, and concave downwards at the left of the origin. A point at which the curve changes from being concave up- wards to being concave downwards, or conversely, is called a point of inflexion. The value of the flexion h changes from positive to negative, or conversely, in passing such a point ; hence the value of h at (I point of injlexion is zero, if it has any value there.* Thus the origin is a point of inflexion on the curve y = x^, for the curve is concave downwards on the left, concave upwards on the right, of the origin. 47. Second Test for Extremes. In seeking the extreme values of a function y =f(x), we find first the critical j)oints (§ 38, p. 63), i.e. the points at which the tangent is horizontal. If, at a critical point, & = d-y/dx'' > 0, the curve is also con- cave upwards,^ and the function has a minimum there; if &<0, the curve is concave doicnwards, and f(x) has a maximuvi; that is, •/• dif ri 7 1 d-ri(>0] , r/ \ • (minimuin ] if m= -^=0 and 6 = — ^ [at x=a, f(a) is a i . \. dx dx-[<()j [maximum) * Points where the tangent is vertical, for example, may be points of in- flexion. t The curve is then also concave upwards on both sides of the point ; if the curve is concave upwards on one side and downwards on the other, b must be zero if it exists at the point. 76 APPLICATIONS OF DIFFERENTIATION [IV, § 48 Whenever the flexion is not zero at a critical point, this method usually furnishes an easy final test for extremes. If the flexion is zero, no conclusion can be drawn directly by this method.* (See, however, § 135.) 48. Illustrative Examples. Example 1. Consider the function y = x^ — 12 x + 7. See Ex. 3, p. 10, and Ex. 1, p. 65. The slope and the flexion are, respectively, dx dx- dx The critical points are (see Ex. 1, p. 65) x = ± 2. Since 6 a; is positive when X is positive, b is positive for x > ; likewise 6 < when x < 0. Hence the curve is concave upwards when x > 0, and concave downwards when X < 0. At x = + 2, & > 0, hence by § 47, y has a minimum at x = + 2; at x=— 2, &<0, hence y has a maximum (compare p. 10 and p. 66). To find a point of inflexion first set 6 = ; , dm d-y -j a • « 6 = — = — ^ = 6 X = 0, I.e. X = 0. dx dx^ Since dm/dx is negative for x < and positive for x > 0, the given curve is concave downwards on the left and concave upwards on the right of this point ; hence x = 0, j/ = 7 is a point of inflexion. (See Fig. 17, and § 49, p. 77.) Example 2. Consider the function y=Sx*-12 x^+50 (Ex. 2, p. 65). The slope and the flexion are, respectively, ,n = ^ = 12x3 ^ 36x2 ; b = ^ = '^ = 36x2- 72 x. dx dx dx^ The critical points are x = 0, x = 3. At x^ 3, 6 = 108 > 0, hence y is a minhnum there. At x = 0, 6 = 0, and no conclusion is reached by this method (compare, however, p. 65). To find points of inflexion, first set 6 = 0; 6 = — = ^ = 36 x2 - 72 X = 0, i.e. x = or x = 2. dx dx'^ * Even in this case one may decide by determining whether the curve is concave upwards or downwards ou both sides of the point ; but the method of § 40 is usually superior. IV, § 49] DERIVED CURVES 77 Near x = 0, at the left, dm/dx = 36x(x — 2) is positive, at the right, nega- tive ; the given curve is concave upwards on the left, downwards on the right, and (x = 2, j/ = 2) is a point of inflexion. (See Fig. 15, and § 49.) Example 3. For a body thrown vertically upwards, the distance s from the earth is : s = -l^gfi + vot, where Vo is the speed with which it is thrown. The speed and the tangential acceleration are, respectively, = -gt dt^' g- If we draw a graph of the values of s and t, the speed v (slope of the graph) is zero when »=—j — a- varies inversely as the cube of the abscissa x. 6. Show that the flexion of the conic Ax'^ + By- = 1 (ellipse or hyperbola) varies inversely as the cube of the ordinate y. 7. What is the effect upon the flexion of changing the sign of a in the equation y = ax^ + bx + c? 8. A beam of uniform depth is said to be of "uniform strenuth " (in resisting a given load) if the actual shape of its upper surface under the load is of the form y = ax- + bx + c, where x and y represent horizontal 80 APPLICATIONS OF DIFFERENTIATION [IV, § 49 and vertical distances measured from the middle point of the beam's surface in its original (unbent) position. Show that the flexion of such a beam is constant. 9. Show that the addition of a constant to the value of y does not affect the slope nor the flexion. 10. Show that the addition of a term of the form kx + c to the value of y does not affect the flexion. What effect does it have upon the slope ? 11. Show, by means of Exs. 9 and 10, that any beam in which the flexion is constant has the form specified in Ex. 8. 12. Show, by a process precisely similar to that of Ex. 11, that a motion in which the tangential acceleration is constant is defined by an equation of the form s = at- -\- ht + c. 13. What is the effect upon the graph of an equation if a constant is added to ?/ ? How are the positions of the maxima and minima affected ? [Take into account vertical as well as horizontal displacement.] 14. What is the effect upon the points of inflexion if a term kx -\- c'ls added to the value oi y '> Will this change in the original curve change the values of x which correspond to extreme values of y ? 15. Show that the curve (1 + x-)2/ = (1 — a;) has three points of inflexion which lie on a straight line. 16. Show that the graph of a polynomial of the ?i'i> degree cannot have more than n—2 points of inflexion. 17. Show that if a polynomial has a factor (a; — a)*, its flexion has a factor (x - a)*-2. 18. Find, by the methods of Exs. 9-12, what the form of y must be if the slope is: (a)^ = 0; (6)^/ = _3; {c)^ = Qx; (d) '^ = ax + b. ^ ^ dx ^ ^ dx ' ^ ^ dx ' ^ ^ dx 19. What is the form of y if the flexion i^ 6 ? if the flexion is 2 x + 3 ? y if the flexion is zero ? 20. If a beam of length I is supported only at both ends, and loaded by a weight at its middle point, its deflection y at a distance x from one end is y = k (3Z'^x — 4 x^), provided the cross section of the beam is con- stant. Find the flexion and show that there are no points of inflexion between the supports. 21. If the beam of Ex. 20 is rigidly fixed at both ends, and loaded at its middle point, the deflection of each half of the beam is y = k (SZx^— 4x^), where x is measured from either end. Show that there is a point of IV, § 51] TIME-RATES 81 inflexion at a distance Z/4 from the end, and that the greatest deflection is at the middle point. 22. Find the points of inflexion and the point of maximum deflection of a uniform beam of length I whose deflection is : (a) y = k(_3lx^-x^). [Beam rigidly embedded at one end, loaded at other end. Origin at fixed end.] (6) y = k(Sx-l^-2xi). [Beam freely supported at both ends, loaded uniformly. Origin at lowest point.] (c) 2/ = ^- (6 Z%-^ -ilx^-\-x*). [Beam embedded at one end only ; loaded uniformly. Origin at fixed end.] (d) y=k {fx - 3 ix3 + 2 a:*). [Beam embedded at one end, supported at the other end ; loaded uni- formly. Origin at free end.] 50. Angular Speed. If a wheel tvirns, a given spoke of it makes an angle 6 with its original position which changes with the time, i.e. ^ is a function of the time : d = f{t). Tlie time-rate of change of the angle 6 is called the angular speed ; it is denoted b>j w : o) = anqular speed = ^= lini — . 51. Angular Acceleration. The angular speed may change; the time-rate of change of the angular speed is called the angular acceleration ; it is denoted by a : ^ . Aw _ do) _ d^O a = angular acceleration = am T7 "~ ^ "~ '^ ' Example 1. A flywheel of an engine starts from rest, and moves for 30 seconds according to the law 18U(J 30 ■where e is measured in degrees, after which it rotates uniformly. 82 APPLICATIONS OF DIFFERENTIATION [IV, § 51 Then and de dt doj dt — <3 + l(2 450 10 150 This example furnishes an instance in which the derived curves, i.e., the graphs which show the values of w and of a are more important than the original curve; for the total angle described is rela- tively unimportant. In the figure a scale is chosen which shows partic- ularly well the variation of w ; ^ is allowed to run off of the figure completely, since its values are uninteresting. The acceleration a is so arranged that it does not suddenly drop to zero when the flywheel is allowed to run uniformly ; and the values of « are never large. Some- thing resembling this figure is what actually occurs in starting a large flywheel. In actual practice with various machines, curves of this type are often drawn experimentally ; the equations serve only as approximations to the reality ; but they are often indispensable in calculating other related quantities, such as the acceleration in this example. Curves which resemble the graph of w in this example occur frequently. (See §§ 87, 134.) , 52. Momentum. Force. As a further illustration of time- rates, we mention a statement often given as the definition of force: force is the time-rate of change of momentum. (Compare Newton's Second Law of Motion.) The momentum 3f of a body moving in a straight path is defined as the product of the mass m of the body times its speed v. M =m ■ v. \[ \ A^le 1 (degrees) | t -0 t 03 .0 -|^ ^ ''s IT J^ ^ t t V ^ 7 W 2n T Z T 7 ' 9i|lo5«* 4- —^f' t : y ^ = ^-i|! -- iV'' IT ^ 10 ^ ^ x = -4-i!j- \i h~l 1/^ . 2^^- '^- -^0 10 2 30^^ 40 Fig. 20. IV, § 52] TIME-RATES 83 The force acting on the body is therefore ^ clM V A 3/ d(m-v) dv . (Ps dt At^o ^t dt dt "^ de This law is often stated in the form : the force is the product of the mass times the acceleration; for the present the results are stated only for a body moving in a straight line along which the force itself acts. This consideration of time-rates makes clear that the two delinitions of force quoted above are equivalent. EXERCISES XVn.— TIME-RATES 1. Express as a time-rate the speed d of a moving body, and write the result as a derivative. 2. Express as time-rates the following concepts : (a) The tangential acceleration of a moving body. (b) The horizontal speed of a moving body. (c) The vertical speed of a moving body. ((i) The speed of evaporation of a liquid exposed to air. (e) The speed of formation of rust on iron. (/) The rate of growth of the height of a tree. (g) The rate of fluctuation of the value of gold, (/t) The rate of rise or fall of the height of a river. 3. In Ex. 2, which of the rates mentioned are surely constant ; which may possibly be constant in some instances ; which may be constant part of the time ? For which of them does a concept analogous to acceleration have a meaning ? 4. If such a rate is constant, how can the total amount (or value) of the changing quantity be computed ? Find the total amount of water in a tank which originally contained 2000 gal., after water has run into it for 10 min. at the rate of 10 gal. a second. 5. If a train, after it is 10 mi. from Chicago, travels directly away at 60 mi. an hour, how far is the train from Chicago 5 min. later ? 6. If y is any varying quantity, and if dy/dt = 7, express y in terms of « if y = 10 when t = 0. Again, if ?/ = 5 when t = 0. 7. If dy/dt = 2t + S, express y in terms ottiiy = when t = 0. [See Exs. 9, 10, List XVI.] 84 APPLICATIONS OF DIFFERENTIATION [IV, § 52 8. A flywheel rotates so that = t^ ~ 1000, where is the angle of rotation (in degrees) and t is the time (in seconds). Calculate the angular speed and acceleration, and draw a figure to represent each of them. 9. Suppose that a wheel rotates so that d = t^ ^ 1000 where 6 is measured in radians [1 radian = 180°/t]. Is its speed greater than or less than that of the wheel in Ex. 8 ? What is the ratio of the speeds in the two cases ? 10. Compare linear speeds in miles per hour with speeds in feet per minute. Reduce 60 miles per hour to feet per second. 11. Compare angular speeds in radians per second to speeds in degrees per second. Reduce 90° per second to radians per second. 12. Compare angular speeds in revolutions per minute (U. P. M.) with speeds in degrees per second. Express the angular speed in Example 1, §51, in R. P.M. 13. Reduce a linear acceleration 60 in./sec./sec. to ft./sec./sec. ; to in./min./min. ; to ft./min./min. Express the acceleration due to gravity (g = 32.2 ft./sec./sec.) in each of these units. 14. Reduce the angular acceleration in Example 1, § 51, to rev. /sec./ sec. ; to rev./min./min. 15. If a wheel moves so that 6 = — t*/16 — i/32, where 6 is measured in radians and t in minutes, find the angular speed and acceleration in terms of radians and minutes ; in terms of revolutions and minutes ; in terms of radians and seconds (of time). 16. If a Ferris wheel turns so that ^ = 20 f2 while changing from rest to full speed, where 6 is in degrees and t in minutes, when will the speed reach 20 revolutions per hour ? 17. If the angular speed is w = kt as in Ex. 16, show that the accelera- tion a is constant. Conversely, show that if a = k, and if t is the time since starting, u = kt. * 18. How far does a point on the rim of a wheel travel during one com- plete revolution ? Express the linear speed of a point on the rim of a wheel 10 ft. in diameter when the angular speed is 4 R. P. M. 19. Express in miles per hour the speed of a point of a wheel 2 ft. in diameter which is rotating with an angular speed of 10 revolutions per second. 20. If the Ferris wheel of Ex. 16 is 100 ft. in diameter, what is the linear speed of the rim at 20 R. per hour ? IV, § 53] RELATED RATES 85 21. Find the linear speed and the tangential acceleration of a poiiil on the rim of the wheel of Ex. 1, § 51, if the wiieel is lU ft. in diameter. What are they when « = 30 sec. ? 22. Find the linear speed and acceleration in Ex. 8, if the radius of tlie wheel is 4 ft. How large would the wheel of Ex. 9 have to be to make the linear speed of its rim the same ? 23. An engine with driving wheels 5 ft. in diameter is traveling 40 mi./hr. Express the angular speed of the rim in revolution per minute. 24. If a train starts from a station with speed v = t/2 + t'^/lOO (in feet and seconds), find the angular speed and hence the angular accelera- tion of drivers 6 ft. in diameter. What is the value of each of these quantities when t = 10? 25. Find the momentum ( = mass x speed) of a falling body, if the distance passed over is s = gt''/2. Find tlie force acting. [Note. If force is measured in pounds, mass = weight in pounds -T- g. Hence, force = mass • g.'] 26. If a body moves so that s = Zt- — 12, find the force acting if the body weighs 10 lb. 27. The hammer of a pile driver weighs 1000 lb. If it drops 15 ft. onto a pile according to the law of Ex. 25, what is the momentum of its impact? The average force of the blow is the average rate at whicli the momentum is destroyed. How much is this if the hammer is stopped in 1/1000 sec. ? 28. What is the average force of a hammer blow by a 2-lb. hammer moving at 30 ft./sec, stopped in 1/1000 sec. ? 29. The kinetic energy of a moving body \s E = mv^/2. Show that ilE/dt = mv ■ dv/dt = momentum x acceleration. 30. An electric current c (measured in amperes) is the (luantity q of electricity (in coulombs) which passes a given point per second. Express this fact in the derivative notation. 53. Related Rates. If a relation between two quantitie.s is known, the time-rate of change of one of them can be expressed in terms of the time-rate of change of the other. Thus, in a spreading circular wave caused by throwing a stone into a still pond, the circumference of the wave is (1) c = 2 7rr, 86 APPLICATIONS OF DIFFERENTIATION [IV, § 53 where r is the radius of the circle. Hence (2) '-^ = 2,r^; or, the time-rate at which the circumference is increasing is 2 7r times the time-rate at which the radius is increasing. (Compare Ex. 8, p. 27.) Dividing both sides by dr/dt, we find dc dr o ^c 7 ^ = = 2-77 = — =dc^ dr\ dt dt dr that is, the ratio of the time-rates is the derivative of c with re- spect to r; or, the ratio of the time-rates is equal to the ratio of the differentials. The fact just mentioned is true in general ; if y and x are any two related variables which change with the time, it is true (Rule [VII „], P- 40) that: ^^'I^ = 'M = dy^dx, dt dt dx that is, the ratio of the time-rates of y and x is equal to the ratio of their differentials, i.e. to the derivative dy/dx. Example 1. "Water is flowing into a cylindrical tank. Compare the rates of increase of the total volume and the increase in height of the water in the tank, if the radius of the base of the tank is 10 ft. Hence find the rate of inflow which causes a rise of 2 in. per second ; and find the increase in height due to an inflow of 10 cu. ft. per second. Consider the same problem for a conical tank. (u4) The volume Fis given in terms of the height h by the formula : V = irr"-h = im Tvh, hence ^=100:r^; dt dt or, the rate of increase in volume (in cubic feet per second) is 100 ir times the rate of increase in height (in feet per second). If dh/dt = 1/6 (measured in feet per second), dv/dt = 100 7r/6 = (roughly) 52.3 (cubic feet per second). If dv/dt = 10, dh/dt = 10-- 100 tt = (roughly) .031 (in feet per second) = 22.3 (in inches per minute). IV, § 53] RELATED RATES 87 (B) If the reservoir is coru'cal, we have V= J irr-/i = ^7r;inan2a, where r is the radius of the water surface, h the height of the water, and a the half-angle of the cone ; for ?• = h tan «. this case In = irh'^ tan^ a which varies with h. If « = 45° (tan a = 1), at a height of 10 ft., a rise 1/6 (feet per second) would mean an inflow of ttA^x (1/6)= 100 7r/6 =52.3 (cubic feet per second). At a height of 15 feet, a rise of 1/6 (feet per second) would mean an inflow of 225 tt/6 = (roughly) 117.8 (cubic feet per second). An inflow of 100 (cubic feet per second) means a rise in height of lOO/irh-, which varies with the height ; at a height of 5 ft., the rate of rise is i/w = 1.28 (feet/second). Fig. 21. the air resistance, Example 2. A body thrown upward at an angle of 45"^, with an initial speed of 100 ft. per second, neglecting etc. , travels in the parabolic path 10000 where x and y mean the horizontal and vertical distances from the start- ing point, respectively; g is the gravitational constant = 32.2 (about); and the horizontal speed has the constant value 100/ ■\/2. Find the ver- tical speed at any time t, and find a point where it is zero. The horizontal speed and the vertical speed, i.^. the time-rate of change of X and y, respectively, are connected by the relation (see §§ 8, 29.) dy . dx _dy q^ dt hence dt gx dx dx dt ^ 5000 (It This vertical speed is zero where 50 V2 V2 5000 gx + 1; ^100 50 V2 V2 5000 155.3 (about), which corresponds to y = 2500/gf = 77.7 (about). At this point the verti- cal speed is zero ; just before this it is positive, just afterwards it is nega- tive. When x = the value of dy/dt is 100/ \/2; when a: = 2500/fir, dy/dt = 50/\/2 ; when x = 7500/gr, dy/dt =- 50/\/2. 88 APPLICATIONS OF DIFFERENTIATION [IV, § 53 EXERCISES XVIII. — RELATED RATES 1. Water is flowing into a tank of cylindrical shape at tlie rate of 50 gal. per minute. If the tank is 8 ft. in diameter, find the rate of in- crease in the height of the water in the tank. 2. Water is flowing into a cone-shaped tank, 20 ft. across at the bottom and 15 ft. high, at the rate of 100 cu. ft. per minute. Calculate the rate of increase of the water level. How fast is the water entering the same tank when the height is G ft., if the level is rising 6 in. per minute ? 3. A funnel 8 in. across the top and 6 in. deep is being emptied at the rate of 2 cu. in. per minute. How fast does the surface of the liquid fall ? 4. A hemispherical bowl 1 ft. in diameter and full of water is being emptied through a hole in the bottom at the rate of 10 cu. in. per second. How fast is the surface of the water sinking when 100 cu. in. have run out ? When the bowl is just half full ? 5. If water flows from a hole in the bottom of a cylindrical can of radius r into another can of radius r', compare the vertical rates of rise and fall of the two water surfaces. 6. If a funnel is 8 in. wide and 6 in. deep and liquid flows from it at the rate of 5 cu. in. per minute, determine the time-rate of fall of the surface of the liquid. ^ 7. Compare the vertical rates of the two liquid surfaces when water drains from a conical funnel into a cylindrical bottle. Compare the time- rate of flow from the funnel with the time-rate of the decrease of the wet perimeter. 8. If a wheel of radius B is turned by rolling contact with another wheel of radius B', compare their angular speeds and accelerations. 9. If a gear wheel moves a toothed pck so that a point of the rack moves according to the equation s = 1 — t/2 + fi/3, what is the angular* velocity and angular acceleration of the wheel at any time t, expressed in revolutions and seconds? Express the angular speed and the angular acceleration in terms of radians and seconds. 10. Compare the speed of a train with tlie speed of a point on the rim of a wheel ; compare their accelerations. 11. If a point moves on a circle so that the arc described in time t is s = t'^ — 1/t- + 1, find the angular speed and acceleration of the radius • drawn to the moving point. IV, § 53] RELATED RATES 89 12. A point moves along the parabola y =i2x- — x in such a manner that the speed of the abscissa a; is 1 ft. /sec. Find the general expression for the speed of y ; and find its value when x = 2 ; when x = 4. 13. In Ex. 12, find the horizontal and vertical accelerations, the total speed, the tangential acceleration, and the total acceleration. [See Exs. 10-18, p. 74.] 14. A point moves on the cubical parabola y — t> in such a way that the horizontal speed is 3 ft./sec. Express the vertical speed when x = 0. Find its value. 15. Find the quantities mentioned in Ex. 13, for the problem stated in Ex. 14. ' 16. If a person walks along a sidewalk at the rate of 3 mi. an hour toward the gate of a yard, how fast is he approaching a house in the yard which is 50 ft. from the gate in a line perpendicular to the walk, when he is 100 ft. from the gate ? When 10 ft. from the gate ? 17. Two ships start from the same point at the same time, one sailing due east at 10 knots an hour, the other due northwest at 12 knots an hour. How fast are they separating at any time ? How fast, if the first ship starts an hour before the other? 18. H a ladder 8 ft. long rests against the side of a room, and its foot slips along the floor at a uniform rate of 1 ft./sec, how fast is the top . descending when it is 6 ft. above the floor? ' 19. The sides of a right triangle about the right angle are originally 3 ft. and 5 ft. long, and grow at the rates of 3 in. and 2 in. a second, re- spectively. Express the lengths of these sides in terms of the time «, and calculate the rates of change per second of the area, and of the tangents of each of the acute angles of the triangle. "Wliat are these rates when t = 1 sec. ? when t = 10 sec. ? At what moment is the triangle isosceles ? 20. If the radius of a sphere increases as the square root of the time ; ^ determine the time-rate of change of the surface and that of the volume ; the acceleration of the surface and that of the volume. 21. Express the area between the x-axis and the line y = x — 1 from X = 1 to X = xo in terms of xq. As Xy changes show that the rate of change of this area is measured by Xf, — 1 or y^,. 22. If the space-time equation of a motion is s=(ffl + ^0"'^ show that the speed varies inversely as the tangential acceleration. 23. What is the time-rate of change of the force acting on a body of mass m which moves on a straight line with the speed v = at- + It -\- cf 90 APPLICATIONS OF DIFFERENTIATION [IV, § 53 24. If a projectile is fired at an angle of elevation a and with muzzle velocity Vq, its path (neglecting the resistance of the air) is the parabola y = X tan a ^ , 2y(,2cos2a' X being the horizontal distance and y the vertical distance from the point of discharge. Draw the graph, taking g = B2, a = 20°, Vq — 2000 ft. /sec. Calculate dy in terms of dx. In what direction is the projectile moving when X = 5000 ft., 10,000 ft., 20,000 ft. ? How high will it rise ? 25. If in an experiment on compressing a gas it is known that pressure X volume = constant, and the time-rate of change of the pressure is 1 + t^, calculate the time-rate of change of the volume ; compare the acceleration of the pressure and that of the volume. 26. Ifp-v- k, compare dp/dt and dv/dt in general ; compare d^p/dfi and d-v/dt^. 27. If p • -y" = ^•, compare dp/dt and dv/dt. [For air, in rapid com- pression, n = 1.41, nearly.] 28. If q is the quantity of one product formed in a certain chemical reaction in time «, it is known that q = ckH/{l + ckt). The time-rate of change of q is called the speed v of the reaction. Show that V = ^^' = cCk - qy. (1 + ckty Show also that the acceleration a of the reaction is 2c2F ^_2cHk-qy. {1 + ckty^ CHAPTER V REVERSAL OF RATES — INTEGRATION — SUMMATION PART I. INTEGRALS BY REVERSAL OF RATES 54. Reversal of Eates. Up to this point, we have been engaged in finding rates of change of given functions. Often, the rate of change is known and the values of the quantity which changes are unknown ; this leads to the problem of this chapter: to find the amount of a quantity ivhose rate of change is knoivn. Simple instances of this occur in every one's daily experience. Thus, if the rate r (in cubic feet per second) at which water is flowing into a tank is known, the total amount A (in cubic feet) of water in the tank at any time can be computed readily, — at least if the amount originally in the tank is known : A^r-t + C, where t is the time (in seconds) the water has run, and C is the amount originally in the tank, i.e. C is the value of A at the time when t = 0. If a train runs at 30 miles per hour, its total distance d, from a given point on the track, is d = 30.t+ C, where t is the time (in hours) the train has run, and C is the original distance of the train from that point, i.e. C is the value of d when t = 0. (Notice that by regarding d as negative in one direction, this result is perfectly general ; C may also be negative.) If a man is saving $ 100 a month, his total means is 100 • ;i + C, where n is the number of months counted, and C is his means at the beginning ; i.e. C is his means when n —0. If the cost for operating a printing press is 0.01 ct. per sheet, tlie total expense of printing is T = 0.01 ■ n + O 91 92 INTEGRATION [V, § 54 where n is the number of copies printed, and where C is the first cost of the machine ; i.e. C is the value of T when n = 0. 55. Principle Involved. Such simple examples require no new methods ; they illustrate excellently the following fact : Tlie total amount * of a variable quantity y at any stage is deter- mined ichen its rate of iyicrease and its original value C are known. We shall see that this remains true even when the rate itself is variable. 56. Illustrative Examples. The rate R{x) at which any variable y increases with respect to an independent variable X is the derivative dy/dx ; hence the general problem of § 54-55 may be stated as follows : given the derivative dy/dx, to find y in terms of x. In many instances our familiarity with the rules for obtain- ing rates of increase (differentiation) enables us to set down at once a function which has a given rate of increase. Example 1. Thus, in each of the examples given in § 54, the rate is constant ; using the letters of this article : ^^ = Bix) = lc, dx where A; is a known fixed number ; it is obvious that a function which has this derivative is {A) y = kx+ C, where C is any constant chosen at pleasure. While the examples of § 54 can all be solved very easily without this new method, for those which follow it is at least very convenient. The value of C in any given example is found as in § 64 ; it represents the value of y when x = 0. Example 2. Given dy/dx — .r^, to find y in terms of x. Since we know that d(x^)/dx = 3 x^, and since multiplying a function by a number multiplies its derivative by the same number, we should evidently take : * This total amount is what is called in § 57 the integral of the rate ; the word integral means precisely the " total " made from the rate, by its English derivation ; compare the English words entire, entirety, integrity, integer, etc. V, § 56] REVERSAL OF RATES 93 i/=:-, or else y^^ + C; Tcheck : d (^ + c\ = x'^dx] , where C is some constant. As in § 54, some additional information must be given to determine C. In a practical problem, such as Ex. 3, below information of this kind is usually known. Example 3. A body falls from a height 100 ft. above the earth's sur- face ; given that the speed is u = — gt, find its distance from the earth in terms of the time t. Let s denote the distance (in feet) of the body from the earth ; we are given that (1) V = — = — gt, or ds - vdt = — gt dt, which is negative since s is decreasing. We know that d(t^) = 2tdt; hence it is evident that we should take : (2) s = ~^f^+ C; [check : ds = - gt dt]. As the body starts to fall, t = and s = 100 ; substituting these values in (2) we find 100 = + C, or C = 100. In this problem, therefore, we have s = _ 2^2 + 100. Example 4. Given dy/dx = x", to find y in terms of x. Since we know that d(a;"+i) = {n + \)x"dx., we should take (£) y = —i— x"+i + C ; [check : dy = x" dx]. 71 + 1 Since the rule for differentiation of a power was proved (§ 23, p. 38) for all positive and negative values of n, the formula (i?) holds for all these values of n except n = — \; when n = — 1 the formula {B) cannot be used because the denominator ji + 1 becomes zero. (See § 78, p. 136.) Special cases : i' „_i #_„ "^la;2+ C; check: (Zf-5c2'\=xcZx. X -{■ C ; check : d{x) = 1 • dx. di^xy2] = xyHx. = 1 dx = dx _1 2 dx = - dx = - 1 drj 3' dx 'M x~^, y = — x~' 4- C ; check : d{— l a—') = x--dx. a; 1/3 y z= ?a;2/3 ^ c ; check : di~x^'A = x-^^dx. ' ' 2 ' \2 I 94 INTEGRATION [V, § 56 Notice that these include Vx(= a;'/^), \/x^(= x~^), etc. ; other special cases are left to the student. Example 5. Given dy/dx =x^ + 2 x^, to find y in terms of x. Since d(x'^)/dx = 4 x^ and d{x^)/dx — 3 x^, and since the derivative of a sum of two functions is equal to the sum of their derivatives, it is evident that we should write * The check is # = Af^+2^+cUx3 + 2x2; dx dx\4: 3 / such a check on the answer should be made in every exercise. In general, as in this example, if the given rate of increase (derivative) is the sum of two parts, the answer is found by adding the answers which would arise from the parts taken separately, since the sum of the derivatives of two variables is always the derivative of their sum. EXERCISES XIX.— REVERSAL OF RATES 1. Determine functions whose derivatives are given below ; do not forget the additive constant ; check each answer. Ca)^ = ix. (6)^ = -5x. (c)^^ = 3x2. (d)^^ = 2. ^ ■* dx ^ ilx ^ ^ dx ^ ' dx ^ ^ dx ^^ dx ^^^ dx ^ dx 2. In the following exercises, remember that the derivative of a sum is the sum of the derivatives of the several terms ; proceed as in Ex. 1. (a) ^ = 4 + 5x2. (6)^=4x2-2x4-3. (c) ^ = «3 - 4« + 7. dx dx dt (d)^ = Sx5-8x\ (e)^=ax + b. (f) - = at'' + bt + c. dx dx dt I (^) ^ = .006 x2 - .004 x3 +.015 X*. (h) ~ = -t^ + bt^ - Gt'^ + 2. ^ dx dt (i) ^ = x-3. (j) - = t^ + 1/«2. (^•) ^ = 3 r2 + 4 r8. dx dt dt (l) ^ = x2/3. (m) '^y- = 2 xi/2 _ 3 x-i /2. („) ^ ^ kv-2-*\ dx dx dv * In all the Examples of this paragraph, we have had an equation which Involves dy/dx ; such an equation is often called a differential equation, be« cause it contains differentials. See also Chapter X. V, § 56] REVERSAL OF RATES 95 3. As a train leaves a station, its speed v is proportional to the time ; tind the relation between the distance s passed over and the time. [Hint, v = ds/dt = kt. Here and below, the unit of time is 1 sec.] 4. If in Ex. 3, A =1/4, find s when t = 10. What is the average speed during this time ? Is the actual speed ever equal to this average speed ? When ? Try to make a rough estimate in advance. 5. Compare the speeds and the distances passed over by an express train which leaves a station with an increasing speed v=t/2, with that of a freight train which stai'ts from a point 100 yd. ahead at the same in.stant with a speed v = t/lO. 6. Determine v and s in terms of t for a bullet shot vertically upward with a speed 2000 ft. /sec. [Hint. Acceleration =dv/d< = — ^=—32.2 ft. /sec. /sec. ; v = 2000 when ^ = ; s = when t = 0. Neglect the air friction.] 7. How high will the bullet in Ex. 6 rise ? How long will it remain in the air ? Make a rough estimate in advance. 8. A car starts with a speed v = t^/l2 ; find s ; how far will it go in 3 seconds ? 9. A flywheel starts with an angular speed w = .01 t- in radians per second. How long does it take to make the first revolution ? How long for the next ? 10. If a flywheel starts with a speed w = .001 1^, what is the time of the first revolution ? of the second ? of the tenth ? 11. If the angular speed in radians per second of a wheel while stop- ping is w = 100 — 10 1, how many revolutions will it make before it stops ? 12. Determine the form of the surface of water in a rapidly rotated bucket from the fact that any vertical section through its lowest point has a slope dy/dx = (o}'^/g)x, where x is measured horizontally and y vertically from the lowest point, w is the angular speed in radians per second, and g=^2.2. Plot the section when w = 8. 13. Determine a curve through (0, 0) whose slope is proportional to z; to x2 ; to 1 — x^. 14. Determine a curve through (0, 0) and (1,2) whose flexion is pro- portional to X ; to 1 -1- x2 ; one whose flexion is constant. 15. Determine the form of the upper surface of a beam if its flexion is constant, and if tlie beam rests on two fixed supports at a distance I from each other. See Ex. 20, p. 80. 96 INTEGRATION [V, § 57 57. Integral Notation. If the rate of increase dy/dx = R{x) of one variable y with respect to another variable x is given, a function y =. I (x) tvhich has i^recisely this given rate of increase is called an indefinite integral * of the rate li (x), and is repre- sented by the symbol t (1) !(•«)= ('jR{:Jc)dx; that is, <^2) *^ ^l^-^(-*')J=^(^)' then I(x)=j'll(x)dac, or, vyhat amounts to the same thing, (3) if d\_Iix)^ = B(x)dx, then I(x)= CB(x)dx. The results of Examples 1, 2, 3, § 56, written with the new symbol, are, respectively, [^] Ck dx = kx + C. jx'dx = x^S + a s = Jy dt+C = j~ gt dt + C= - gt-/2 + C. The first equation of Example 3 holds in general : [I] s =/^ dt + C, since — = v. dt The result obtained in (B), Example 4, § 56, gives jc^dx = '^ +C, n^-1, 71 + 1 * The common Eno;lish meaning of the word integrate is "to make whole again," " to restore to its entirety," "to ^ive the sum or total." See any dictionary, and compare §§ 54-55. To integrate a rate R{z) is to find its integral; the process is called integra- tion. Often the rate function Ii{x) which is integrated is called the integrand ; thus the first part of equation (2) may be read : " the derivative of the integral is the integrand." This is the property used in checking answers. The first equation in (2) and the first in (3) are differential equations. See footnote, p. 94. t Note that dx is part of the symbol. As a blank symbol, it is / (blank) dx ; the function R(x) to be integrated {i.e. the integrand) is inserted in place of the blank. The origin of this symbol is explained in § 67, p. 117. V, § 57] NOTATION 97 for all positive and negative integral and fractional values of n except 11 = — 1, for which see § 78, p. 13G. As examples of the many special cases, we write : « = 1, (xdx =^ + C. n = 0, (xMx, =(iax. = ffZx =x+C. n = \, ^x^'-dx = r Vx dx = f x^'- + C= f \/^+ C. M = - 2, (xr-dx = (\ dx = - x"^ + C = --+ O. n = - 1 r.v-i/''rfx = ^~dx = f x2/3 + C = f Vx2 + C. '^ ^ '^ Vx From Example 5 : r (..-3 + 2 x2)dx = (xMx + (-2 x^dx == ^ + ?-£!+ C. The general principle used in this example is that the inte' gral of a sum of two functions is the sum of their integrals : [C] fciJCx) 4- -SCx)] (Ijc = i B,{x) dx + f-SCir) dx, which is true because the derivative of the sura R(x) + S{x) is the sum of the derivatives : dlR + S^/dx = dli/dx + dS/dx. The rules {A), (B), (C) are sufficient to integrate a large number of functions, including certainly dM x>objnomials in x. EXERCISES XX. — NOTATION — INDEFINITE INTEGRALS 1. Express the value of y if dy/dz = 4x''^ + 3x by means of the new sign / ( )dx. Also find y. Check. 2. If dy/dx has any one of the following values, express y, first by use of the new sign J ( )dx and then directly in terms of x. Check the final answers. Do not omit the arbitrary constant. (a) x2. (c) X*. (e) x2 - 2. (g) xP- - 2 x + 3. (0 ax 4- h. (6) x8. (d)2x + 3. (/) 7. (K)x?-x\ (j)Vi. H 98 INTEGRATION [V, § 57 3. In many examples it is profitable first to expand the given expres- sion in a sum of powers ; proceed as in Ex. 2, and find y if dy/dx has any of the following values : (a)x(l+x). (e) (l + a;2)(l-x2). (i) (\ + x) (\ - x-^) . [b) (x3 + 4 a;2) -4- X. (/) (2 - 3x)(4 + x). (j) x'/s(x + x^). [c) 4(x + 2)2. (fj) xi/-'(l + aO- W (x3 + 2 x2 + x)V2. [d) 2x2(3-4x2). (h) (l-ix)Vx. (0 (a;2 + 2)2(2xV2 + 3). 4. Integrate the following expressions : rt) ijx^dx. (d) ij(l + v)dv. (g) ( xr^dx. b) (sthlt. (e) r(3x2-4x3)dx. (h) (sr^dt. c) (is-^ls. (/)(*( 10-9 x-3)dx. (0 Cfl/--dt. j) r (3 M + 5 ?«2 + 7 u^)au. (k) ( (7/x^2 + 8 x-io - 10/x'')(Zx. Z) r2^x2dx. C"0 r(5s*-3s2 + 2)ds. (?i) fCits/s + 10<5/2) (^^. . Integrate the following expressions, making use of the principle of 3: a) ((1-tydt. (g) ^Vx(a + bx)dx. b) (x{l + Vx)dx. (/i) (x''(a + bx)dx. c) rs(l-v^)2ds. (0 j'(a + 6x)2dx. d) (t%l-t'-)dt. U) j'(<2'5-2)«-5(?«. e) rx-i(l + X + x2) (te. (k) (x^*0 -{-x'^ydx. y) rx(ff + 6x)fZx. (?;> rV«(l + 2t-y-^dt. 6. Powers of linear expressions may be treated without expanding. Find a function whose derivative is (x + 1)2 by analogy with the function whose derivative is x2. 7. Can y be found when dy/dx = (x + 3)2 by the same analogy ? Can y be found when dy/dx = (2 x + 3)2 ? Be sure to check your answers. If they are wrong, put in the proper factor to correct the error. 8. Find y when dy/dx has one of the following values : (a) (3x-2)3. (6) (2x + 1)1/2. (c) (5x-4)-3. V, § 58] FUNDAMENTAL THEOREM 99 58. Fundamental Theorem. We have seen that such func- tions as x^, OCT + o, X- 4- C, where C is any constant, have the same derivative 2 x. If the rate of increase (derivative) of y with respect to x is given, there may be several answers for y in terms of x ; thus if dy/dx = 2x, the answers y = x^, y = 3r-\-o, y = X- + C are all correct solutions; to decide whicli one is wanted, additional information is needed, as in § 54 and§ 56 (Exs. 2,3, etc.). However, except for the additive constant C, all answers coincide; for practical purposes, there is but one answer. Stated precisely, this is the Fundamental Theorem of Integral Calculus : If the rate of increase (1) | = 7e(.) of a variable quantity y ivhich depends on x is given, then y is determined as a function of x, I(x), except for a constant term: (2) y =^R{x)dx +C=I(x)+ a Stated in different words this theorem is : T7ie difference betiveen any two functions I{x) and J{x) ivhose derivatives are equal, is a constant. Let this difference be D(x) = I(x)-J(x); then dD{x)^dI(x) dj(x) ^^^ dx dx dx Since the rate of increase (derivative) of D(x) is zero, D(x) neither increases nor decreases for any value of x; hence D(x) is a constant, as was to be proved.* The constant C which occurs in the answers is always to be determined by additional information, as in § 54 and § 56. ♦Graphically, the "curve" which represents I>{x) has its tangent hori- zontal at every point, — such a "curve" is necessarily a horizontal straight line : D(x)= constant. (See also § 131.) 100 INTEGRATION [V, § 59 59. Definite Integrals. In applications, we often care little about the actual total ; it is rather the difference between two values which is important. Thus, in a motion, we care little about the real total distance a body has traveled since the creation of the universe ; it is rather the distance it has traveled between two given instants. If a body falls from any height, the distance it falls is (Ex. 3, p. 93) s^(vdt+ C = ^gtdt+C = ^+ C, ■where s is counted downwards. The value of s when f = is s]«=o = C ; the value of s when f = 1 is s']i=i = g/2 4- C. The distance traversed in the first second is found by subtracting these values : =0 J<=1 J«=o \^ C'j-C = ^ = 16.1ft., where s~\ 'l! means the space passed over between the times t = and t = l. In this calculation, we care little about where .s is counted from ; or its total value. The result is the same for all bodies dropped from any height. Likewise, the space passed over between the times t = '2 and f = 5 is ^g^_25_g^ =^.21 ^338 (ft.). In general the distance traversed between the times t = a and t = b is i;::-L-a=.=("f The advantage realized in this example in eliminating C can be gained in all problems : TJie numerical value of the total change in a qtiantity between two values of x, x = a and x = b, can be found if the rate of change dy/dx = B{x) is given. For, if y = I(x)= (R(x) dx + C, V, § 59] DEFINITE INTEGRALS 101 the value of y for x = a is and the value of y for x = h is, The total change in y between the values x = a and x = 6 is This difference, found by subtracting the values of the indefinite integral at x = a from its value at x = b, is called the definite integral of R{x) between x=a and x = b ; and is denoted by the symbol : J It(x)dx = \j Kijc)djc\ -\jlt(x)ax\ = 1(b) -1(a). It should be noticed that, in subtracting, the unknown con- stant C has disappeared completely ; this is the reason for calling this form definite. Example 1. Given dy/dx = x^, fiud the total change in y from x = 1 to x = 3. Since y = (x^ dx = x*/4: + C, it follows that ,]•-=,] -,] =fl -fl =20. Ji=l Jx=3 Jx=l 4Jx=3 4Jx=l Interpreted as a problem in motion, where x means time and y means distance, this would mean : the total distance traveled by a body be- tween the end of the first second and the end of the third second, if its speed is the cube of tbe time, is twenty units. Interpreted graphically, a curve whose slope m is given by the equation TO = x^ rises 20 units between x = l and x = 3. The equation of the curve is ?/ = x^/i + C. 102 INTEGRATION [V, § 59 EXERCISES XXI— DEFINITE INTEGRALS 1. If water pours into a tank at the rate of 200 gal. per minute, how much enters in the first ten minutes ? how much from the beginning of the fifth minute to the beginning of the tenth minute ? 2. If a train is moving at a speed of 20 mi. per hour, how far does it go in two hours ? Does this necessarily mean the distance from its last stop ? 3. If a train leaves a station with a variable speed v = 1/4 (ft./sec), find s in terms of t. How far does the train go in the first ten seconds ? How far from the beginning of the fifth to the beginning of the tenth second ? 4. A falling body has a speed v = gt^ where t is measured from the instant it falls. How far does it go in the first five seconds ? How far between the times t = S and t = 7 ? 5. A wheel rotates with a variable speed (radians/sec.) w = t-/lOO. How many revolutions does it make in the first fifteen seconds ? How many between the times i =: 5 and t = 20? 6. From the following rates of change determine the total change in the functions between the limits indicated for the independent variable. Interpret each result geometrically and as a problem in motion, and write your work in the notation used in the text : (a) ^ = x, x = 1 to X = 2. dx (ft) ^ = i a:2, a; = - 2 to ic (0 (^d) !i^ = 1 - x2, X = to X = 10. dx Ax 4 dy_ X3 dx 12 dl__ dx = 1 - (/) ds _ dt ' _«i-2 t^ ' t = 1 to e = 3. (9) dv ^ dt ' _(l + 0- (3/2 , i = 16tot = 25. (h) dv_ dJt _ 2 ri - t- Sr -,< = 0.1to0.01 (0 de_ dt -Wtvi t = to « = .01. U) dd _ dt ~ _ a^ _ a^ t - = a to « =: 2 rt. -,t = lOtot = 100. dt f^ 7. Determine the values of the following definite integrals. [In cases where no misunderstanding could possibly arise, only the numerical values of the limits are given. In every such case, the numbers stated as limits are values of the variable whose differential appears in the integral.] V, § 60] AREA UNDER A CURVE 103 (a) i:> dx. (6) s:> xdx. (0 s:> C2 dx. (d) o x^dx. (»> X* Vxdx. (/) j*^V 3 dx. (k) ( V/3(s2-2 S) dS. /•r=(1 /• 0=100 (/i) i_2((-^ -!)(/( (m) j ^^ (.01 + .02^)djdx+C = jf{x)dx+C; and the area A between any two fixed values of a:, a; = a and x = b is the definite integral : (3) ^i:>^.j^LrX:' f(x) dx. Example 1. To find the area under the curve * y — x^ between the points where x = and x = 2. We have, by (2) A^(ijdx + C= (x^ dx + C = - + C, where A is counted from any fixed back boundary x = A; we please to assume, up to a movable boundary x = x. The area between x = and x = 2 is given by subtracting the value of A for x = from the value of A for x = 2 : aT" = A-] -a-] =j-'.=.x = f] -f] =|. Likevdse the area under the curve between x = 1 and a; = 3 is -j:;:=r:'^'-=f].7'i'i=.-- and the area under the curve between any two vertical lines x = a and X = 6 is b^ — a^ L_ 1 2/. '- i ^ t y^l A t ^ 2 -4 I 7t ^ 1 -.tr K 7 K^"^ if 1 ^ it It: Fig. 23. ♦The phrase "the area under the curve" is understood in the sense used in the first sentence of §60. When the curve is below the x-axis, this area is counted as negative. V, § 60J AREA UNDER A CURVE 105 EXERCISES XXn. — AREAS [Draw a figure and estimate the answer in advance, whenever possible.] 1. Find the area under each of the following curves between the ordinates x = and x = 1 ; between x = 2 and x = 5 : (a)y^3x\ (f0 2/=V7._ (r/) x-'^/ = 1. (See § 111.) (6) 2/ = x3. (e) 2/ = l/Vx. (See §111.) (h) y = x^ + -Sx-i. (c) 2/ = xV10. (/)y=(l-x2). (0 t/ = x(l-x)2. 2. Find the area between the line y = 2x and the parabola y = x^. 3. Find the area between y = x and y = s/x. 4. Show that y = x"^ and j/^ = x trisect the unit square whose diago- nal joins the points (0, 0) and (1, 1). 5. Find the area between y = x- and y = x^ ; and show that it is the same as that under the curve y = x- — x^. 6. Find the areas under each of the following curves : (a) 2/ = x3 + 6 x2 + 15x, (x = to 2 ; x = - 2 to + 2 ; x = - a to + a). (6) y = x'^'^, (x = to 8 ; X = - 1 to + 1 ; X = — a to + a), (c) y = X- + 1/-k'-^, (x = 1 to 3 ; x = 2 to 5 ; x = a to 6). 7. Find the area A under the line y = 2x + 3 between x = and X = X (any value) by geometry ; show directly that dA/dx = y. 8. Find geometrically the area A under the line x + y + 2 = between x = and x = x ; show directly that dA/dx = y. 9. Show that the area A* bounded by a curve x = ^(y) the j/-axis, and the two lines y — a and y = bi& Xy=h ^Hy)dy. 10. Calculate the area between the »/-axis, the curve x = y"^, and the lines 2/ = and y = I. Compare this answer with that of Ex. 3. 11. Find the area between the curve y — x^ and each of the axes sepa- rately, from the origin to a point (jfc, k^). Show that their sura is k*. 12. Find the area in the first quadrant between y = x^ + 3x, the j/-axis, and the lines ?/ = 0, ?/ = 4, by subtracting from a certain rectangle the area between y =x^ + 3 x, the x-axis, and the lines x = 0, x = 1. 13. Find the area in the first quadrant between the curve y=x^+2 x— 7, the 2/-axis, and the lines ?/ = 2, ?/ = 9, by the method of Ex, 12. M Ax t" / // ^ K > x=k x-x x=x+Ax Fig. 24. 106 INTEGRATION [V, § 61 61. Lengths of Curves. Let s represent the length of the arc FM of a curve C whose equation is y=f(x), between a Q fixed point F and a moving J^y point M, on the curve. As the point ikf moves on to N', the value of x increases by an amount Ao; = JK = ML, y by an amount Ay = LN, and s by an amount As = arc MN. The chord MN is given by the Pythagorean theorem : (1) [chord MNy = ML^ + LN" = Ax + Ky'. The instantaneous rate of increase of the arc s with respect to X is /ON ds T As V arc MN (2) — = lim — = lim • ; dx Ax=^ Ax Ai=y) Ao; this limit can be found by the fundamental fact (§ 12) that the limit of the ratio of an arc to its subtended chord is unity * ,os ds T arc MN ■,■ chord MN (3) — =lim = lim , dx Ax=y) Ax Ax^ Ax ,. arc MN ^ MN=o chord MN hence (4) W = [lim ("I'o-l^yn = ,i,„ g + ^ Ax^|_ \Ax) ]i' Ax {^■^) or It follows that the total are is (6) s=f^jl + (^ydcc + C=J^^+7It'dx+C, * This fact is the pith of the argument ; it contains the essence of the defi- nition of what we mean by the length of an arc of a curve. See § 12. V, § 62] LENGTH MOTION 107 and that the length of the arc between any two points at which X = a and x = b, respectively, is (7) ' = J V«+(^)'-^ Vl +ni-dx. The equation (5) 62. Motion on a Curve. Parameter Forms of § Gl is often written in the form (8) rfs2 = fix2 _|. ^y2, which is readily remembered through the suggestiveness of the triangles 3ILN and MLP of Fig. 25, in which ds = MP. Equation (8) will be called the Pythagorean differential for- mula. Since m = tan « (Fig. 12, p. 50), the quantity Vl + vi^ is equal to sec a. In particular, ds/dx = sec a, whence dx/ds = cos a. Likewise, dy/ds = sin a. If a point 3/ moves along a curve, its total speed v is ds/dt, its horizontal speed v^ is dx/dt ; its vertical speed v^ is dy/dt («)' (IT=(IJHi)'""' the square of the total speed is the sum of the squares of the hori- zontal and the vertical speeds; and u^= v cos «, v^ = v sin a. The equation (8) may be used in case the equations of the curve are given in parameter form (10) x=f{t), y = (t), whether the parameter t represents the time or some other convenient quantity. The length of an arc of a curve whose equations are given in the foira (10) is dx = Ax-^-^ Fig. 25. Dividing both sides of equation (8) by (dty, we find 2_,. 2_^^2_. (11) ^=/v(f)^^(fr-^^- 108 INTEGRATION [V, § 63 63. Illustrative Examples. Example 1. A point moves along a curve y = x"^. Express ds/dx and write the integral which represents the length of the curve. By the Pythagorean formula ds'^ = dx^ + dy^ ; but dy =2xdx; hence ds2 = (1 + 4 a;2) dx% or ^ = VH- 4 x^. dx The length s of any arc between points where x = a and a; = & is ']:::=jr V 1 + 4 x^ dx : but since we have never had a function whose derivative is v 1 + 4 x-, this integral cannot now be found. [See, however, Ex. 16, p. 129, and § 106.] The speeds v, Vj„ and Vy are given by the relations : dx v, = — , 'y„ = ^ = 2x — =2 xv^, V = VvJ + V = Vl + 4 x^ v^. dt dt dt If Vj. is given, the other two can be found ; thus, if Vj, is a constant k^ Vjc = k, Vy = 2 kx, V = k Vl + 4 x^. Example 2. Find ds, v, s for the curve y"^ = x^ and find the length of the arc from the origin to the point w^here x — -5. Since tj'^ = x^, we have 2y dy =3 x- dx, or 4 y^ dy'^ = 9 x* dx"^, or dy'^ = f X dx2 ; and ds- = dx^ + di/^ = (1 + | x) dx". It follows that the speed of a moving point is and that the length of the arc is s =JV1 + |X dx + C= 5V(1 + f xf^ + C, hence the length between the origin and the point where x = 5 is ]i=5 /»x=5 , 3/2-11=5 = \ Vl + |xdx = ^(l + |x) =W Example 3. Find f7s, v, s for the curve represented by the equations x = t^ + 5, y = ^(4 t + ly/-. We find ds^ = dx^ + dy'^ = (2 1 dVf + [(4 f + ly-l^-dtf = (2 « -f XydV^ ; whence ds = (2 « + 1) dt, or v = ds/<^^ = 2 « + 1, and s=V^J-dt^ C={{2t^V)dl-\- C^t- + t+ C. V, § 63] LENGTH MOTION 109 Between the points where t = and where t = 2 [i.e. the points (a; = 5; y = 1/6) and (x = 9, y = 27/6)], the length of the arc is ^-jr=2 ^ /.«=2 ^ _^ ^ ^^ = r«2 + f + c1'^^ =(4 + 2 + C) - C = 6. Our present ability to recognize derivatives enables us to integrate com- paratively few of the square root forms that occur in these length integrals. We shall be able to deal with these forms inore readily in Chapter "VI. EXERCISES XXIII— LENGTH — TOTAL SPEED 1. Determine by integration the lengths of the following curves, each between the limits x=:ltox = 2, x = 2tox = 4, x = rttox = 6. Check the first three geometrically : (a)y = 2x-l. (c) ?/ = TOX + c. (e) ?/ = ^ (2 x - 1)3/2. (6)y = 3+4x. (d) 2/ = |(x-l)-'/2. (/) ^ = i (4.,,_ ])3/2. 2. Find ds, and the length s of the path of each of the following mo- tions, between the given limits. Find the speed v at each end of the arc. (a) X = 1 + t, y -1 -t; t = tot = 2. (b) X = (1 + ty^, 2/ = (1 - 0^/- ; ^ =: to « = 1. (c) X = (1 - ty, y = 8 f^i"'!?, ; < = to « = 0. (d) x = \-\rt:-,y -t - t;^/'i ; « = to « =6. (e) X = 2A, y = t + 1/(3 «3) ; t ^ a io t = h. 3. If a point moves on the circle x^ + y- = 1, show that x{dx/dt) + yidy/dt) = 0, and that v^ = [dx/rfiJV?/^ ^ [dy/dtY/x-^- 4. If a point moves on the circle x^ + ?/2 = 1 with constant speed v = k, show that dx/dt = ± ky and dy/dt = ± kx, where the sign ± depends on the sense of the motion. 5. If a point moves on the hyperbola xy = 1, show that the horizontal and the vertical speeds v^ and Vy are connected by the relation xuy+yt\=0 ; and that v^ = Vr^(x'^+y^)/x^=Vy^(x^+y^)/y^. 6. If a point moves on the curve y^ = x, show that v-= (1 + 4 y-)Vi,'^. 7. Determine the path described when the x and y speeds are as be- low, if the point is at (0, 0) when t = 0. Find the length of the arc trav- ersed from t = to t = 9. What is the speed at each end of these arcs ? [dt [dt [dt ^^ '■ 110 INTEGRATION [V, § 64 PART IT. INTEGRALS AS LIMITS OF SUMS 64. Step-by step Process. The total amount of a variable quantity whose rate of change (derivative) is given [('.e. the integral of the rate] can be obtained in another way. For example, imagine a train whose speed is increasing. The distance it travels cannot be found by multiplying the speed by the time; but we can get the total distance approxi- mately by steps, computing (approximately) the distance trav- eled in each second as if the train were actually going at a constant speed during that second, and adding all these results to form a total distance traveled. If the speed increases steadily from zero to 30 mi. per hour, in 44 sec, that is, from zero to 44 ft. per second in 44 sec, the increase in speed each second (acceleration) is 1 ft. per second. Hence the speeds at the beginnings of each of the seconds are 0, 1, 2, 3, ••• , etc. Using the speeds as approximately correct during one second each, we should find the total distance (approximately) s = + l + 2 + 3 + -+42 + 43= ^A^ = 94(3, which is evidently a little too low. If we used as the speed during each second the speed at the end of that second, we should get (approximately) s=l+2+3+4+ which is evidently too high. But these values differ only by 44 ft. ; and we are sure that the desired distance is between 94G and 990 ft. If we reduce the length of the intervals, the result will be stiii more accurate ; thus if, in the preceding example, the distances be computed by half seconds, it is easily shown that the distance is between 957 ft. and 979 ft. ; if the steps are taken 1/10 second each, the distance is found to be between 965.8 ft. and 970.2 ft. Evidently, the exact distance is the limit approached by this step-by- step summation as the steps At approach zero : s =( vdt={ J(=0 Jt=o Jt f=44 fl-y '=44 ff?«=l- =968. V, § 65] APPROXIMATE SUMMATION 111 We note particularly that the two results for s are surely equal ; hence we obtain the important result : £ lim A« + r • A< + u I • A( + ... 1. J t=At J (=2A( I 65. Approximate Summation. This step-by-step process of sumiuatioii to find a given total is of such general application, and is so valuable even in cases where no limit is taken, that we shall stop to consider a few examples, in which the methods employed are either obvious or are indicated in the discussion of the example. Thus, areas are often computed approximately by dividing them into convenient strips. We have seen, § (50, that if A denotes the^ area under a curve be- tween X = a and x = b, then the rate of increase of A is the height h of the curve : clA dz Ji(x), whei-e Ii(x) is the rate of increase of A, and is also the height of the curve. For a parabola, h — x^, we may find the area A approximately between X — — 1 and X = 2 by dividing that interval into smaller pieces and com- puting (approximately) the areas which stand on those pieces as if the height h were con- stant throughout each piece. If, for ex- ample, we divide the area A into six strips of equal width, each 1/2 unit wide, and if we take the height throughout each one to be the height at the left-hand corner, the total area is (approximately) (+l)2i + (|)H = li»/8, whereas, if we take the height equal to the Pjq 07 height at the right-hand corner we get :il/8. The area is really 3, as we find by § 00. Tak- ing still smaller pieces the result is of course better ; thus with 30 pieces Xj- 5 it 3 -3 "^/ ii^i^ii;ii --k'-f--- 1 1 i 1 112 INTEGRATION [V, § 65 each jJjj unit wide,* the left-hand heights give 2.855, the right-hand heights 3.155. With still more numerous (smaller) pieces these approximate re- sults approach the true value of the area. (See § 67, p. 116.) EXERCISES XXIV. STEP-BY-STEP SUMMATION — APPROXIMATE RESULTS 1. Approximate to about 1 % the areas under the curves below, be- tween the limits indicated. Estimate the answers roughly in advance. Use judgment with regard to scales to gain in accuracy by having the figure as large as is convenient. Check results by integration where possible. (a) 2/ = 1 + x2 ; x = to 3. (/) 2/ = x-i ; X = 10 to 100. (6) 2/ = ™ ; X = 5 to 10. (9) 2/ = (1 + x)/^ ) a; = 2 to 4. 100 (c) 2/ = x2-2x;x = lto3. (h) y = V9 + x ; x = to 7. (d) 2/ = 4x2 - X* ; X = to 2. (0 y = V9 + x^ ; x = to 4. (e) 2/ = x-2 ; X = 1 to 10. (j) y = V9 + x* ; x = to 2. 2. Approximate to about 1% the distance passed over between the indicated time limits, where the speed is as below ; when possible check by integration. (a) v=l+ Vi; t=Oto 100. (ft) v=2t + t^; « = 1 to 4. (c) V = ^ ■ t — to 100 / j\ „, i . / 1 +/-> A 1 + Vi' ^'^^"-2i+e2'^-l*°4- (e) v = '^;t = ltolO. (in^ = f^f' «=itoio. {g) V = l±^ ; « = 4 to 9. Vt (A) V = ^^ ; i = 4 to 9. 1+ v< (i) v = ^~^; t = 0to50. ( j) v=y/l + f^; « = 10 to 20. 3. The volume of a metal casting is often found by dividing the entire pattern into parts, each of which can be computed readily. Show how to find the volume of a flat casting shaped like the letter H, if the thick- ness and the width of each portion is given. * In this computation the formula 12 + 22-1-32H h n2= n{2n+ 1)(h+1)/6 is convenient. For any reasonable degree of accuracy, this method, in this example, is longer than that of § (10, but fur other examples, especially when the curve is drawn and we know no equation for it, this method is often con- venient. Notice that the average of the two last results found above is reasonably accurate ; it is 3.005 ft. (See § 66, p. 114.) V, §65] APPROXIMATE SUMMATION 113 4. Show how to calculate approximately the volume of a dumbbell whose ends are spheres. Notice that a small volume at the intersection of the spheres with the cross-bar is neglected. 5. Show how to find the volume of a cone approximately, by adding together layers perpendicular to its axis. 6. Find the volume of a sphere by imagining it divided into small pyramids with their vertices at the center and their bases in the surface, as in elementary geometry. 7. Discuss the approximate evaluation of areas in a plane by counting the squares in a figure drawn on cross-section paper. Would still more finely ruled paper be more accurate ? Show that the area of any closed figure may be defined by extending this process indefinitely. 8. The volume of a ship is computed by means of the areas of cross sections at small distances from each other ; show how the result is cal- culated. Show how to make a more accurate computation by the same method. 9. In shipments of ores or coal, it is usual to sample each car ; show how to obtain the total amount of metal in a shipment of several car-loads of ore. Is the result accurate or approximate ? Show how a more accu- rate result can be found. 10. The number of bacteria in a river is computed by sampling at various distances from the shore. Show that the total thus computed is reasonably accurate, on the assumption that the bacteria per cubic foot is approximately constant for short distances. 11. The total sales of a given stock or bond in one year on the New York Stock Exchange can be computed from the record of the number sold each day and the price on that day. Show that the result lies be- tween that found by using the highest and the lowest daily prices. Would the average of the latter two be more accurate ? 12. The number in 100,000 persons alive at any given age who die be- fore they are one year older is important in life insurance ; show how to compare the actual death rate of a given group of people, — say of the students in a given university, — with the published figures showing the normal expectation of death during each year of age. 13. The amount of cement used in concrete varies in different portions of the same building from one part in two to one part in six. Show how to find the entire amount of cement used in the work from the specifications. 114 INTEGRATION [V, § 66 66. Exact Results. Summation Formula. As in the preced- ing articles, given the rate of increase ]i(x) of a variable quantity y we can always compute the total difference in the values of y between two values of x, x — a and a; = 6, [ i. e. the integral ilzlR(x) dx^ : Let us break up the interval x = a to x = b into n portions, each of size Ax ; the first interval is from a to a + Ax, the second from a-{- Ax to a + 2 Ax, and so on. The change in y during each interval can be computed approximately by taking the rate of change as constant and equal to its value at the be- ginning of the interval ; doing so we would obtain, in the first interval a change Ax • E(a); in the second Ax • E(a-\- Ax) ; in the third Ax • E (a -{- 2 Ax) ; etc., so that the total change is (approximately) the sum : (1) s = AX' E (a) + Ax- E(a + Ax) -{-Ax- E(a + 2 Ax) + ••• +AX' E[a + (71 - 2) Ax^ + Ax ■ E[a + (n~l)Ax']. If we take the constant rate as the rate at the end of the interval, we get the sum (2) S = Ax- E(a + Ax) + Ax-E (a + 2 Ax) + AxE (a -|- 3 Ax) -f \-Ax-E[^a + (n — 1) Ax] -|- Ax-E (a + n • Ax). The first of these sums contains the term Ax • E (a), the second the term Ax • 72 (a -}- w • Ax) ; their difference is D = S -s = Ax . [E(a-{-7i • Ax) - i? (a)] = Ax [72 (6) -7? (a)] since b = a-\-7i • Ax. If, for example, the rate E(x) is increas- ing, the correct answer evidently lies between S and s. S is too high, s is too low. As we make the intervals smaller and more numerous, Ax will approach zero, n Avill become infinite,* and D = S — s = Ax[E(b) — E(a)^ will approach zero, since E(b) — E{a) is a constant. Hence it is evident that the correct value of the total change iti y is the li7nit of the simi s (or of the sum S, since the difference between S and s approaches zero) ; that is: * For the meaniug of this phrase, see § 14, p. 19. V, § 66] EXACT SUMMATION 115 (^)I R (x) dx lim \i^x ■ R (rt) + Aa^ ■ -B (ffi + Ax) + Aa>=0 / + A.r ■ltia+ (n - 1) Aj-] This foniuila will be called the Summation Formula of the Integral Calculus. Interpreted as a motion problem^ B (x) means the speed, x denotes time, y distance ; the intervals A.c ai'e small intervals of time during which we conceive the speed as sensibly constant ; Ax • R (a) is the dis- tance (approximately) traversed in the first interval, during which the speed is supposed to remain approximately equal to the speed S (a) at the beginning of the interval ; and so on, as in the example of § 65. Graphically, if x and y denote any concrete quantities one pleases, drawing z horizontally as usual, we may represent the rate R{:i-) by a h=Ii{x) ¥ui. 28. curve whose height is 7i : h = B (x). The intervals Ax are small intervals along the .r-axis ; PPu PiP-z, P^Ps, •••, in each of which we think of the height h = B(x) as sensibly constant. The product Ax • B (a) is the area 7'/'iA'iJ/of the rectangle whose base is Ax and whose height is PM = h]^^a = Bia). The next term of the sum s is Ax ■ Bia + Ax), which is the area of the rectangle P\P.K>L\, and so on ; the whole sum s is the area of the polygon PQKsLiJuLsh'iLiKzLih'iM in Fig. 28, in which 7i is taken equal to 5. 116 INTEGRATION [V, §66 Likewise the sum ^S* is the area (i'ig. 29) of the polygon PQNJ^LtJi L3J3L2J2L1J1, which is exterior to the curve. .7, IZ^^^ h=R{x) h lim S-- Ax=0 lim s lim {Ax B{a) + +^xB\_a-\-{n—\)^x]} Fig. 29. The difference D — 8 — s := 6.x[^B {h) — B{a)'] is the area of a rec- tangle whose base is Ax and whose altitude [_B{h) — B{a)'\ is the differ- ence between PM and QN ; and it is evident that this area approaches zero with Ax. The area of either polygon, s or *S', evidently approaches the area A under the curve between x = rt and x = 6 as Ax approaches zero : (4) ^]^ which agrees with our previous formulas since = \ hdx= \ B(x)dx, and the two values of A agree, by (3). This agreement maybe regarded, however, as a new proof of (3), since the two formulas for^ are obtained independently ; but attention is called to,^ the fact that this argument is simply a special case of the general argument used above. It is evident from the figures that (3) holds also if R(x) is decreasing, or indeed even if R(x) changes from increasing to decreasing, or conversely. 67. Integrals as Limits of Sums. By far the greater number of integrations appear more naturally as limits of su7ns than as reversed rates. v,§ EXACT SUMMATION 11? Thus, as a matter of fact, even the area A under a curve, treated in § 60 as a reversed rate, probably appears more naturally as the limit of a sum, as in (4), § 66. Of course the two are equivalent, since (3), § 66, is true ; in any case the re- sults are calculated always either approximately, as in the exercises under § 65, or else precisely by the methods of §§ 58- 59. Hence the method of § 59 was given first, because it is used for each calculation even when the problem arises by a summation process. On account of the frequent occurrence of the summation process, we may say that an integral really means* a limit of a sum, but when absolutely precise results are wanted it is calcu- lated as a reversed dijferentiation.'f The symbol J is really a large S somewhat conventionalized, while the dx of the symbol is to remind us of the Ax which occurs in the step-by-step summation. 68. Water Pressure. As another typical instance, consider the water pressure on a dam or on any container. The pressure in water increases directly with the depth s, and is equal in all directions at any point. The pressure p on unit area is (1) p^k.s where s is the depth and k is the weight per cubic unit (about 62.4 lb. per cubic foot). I,, .;,, Suppose water flowing in a para- bolic channel (Fig. 30), the parabola being defined by the equation 1 1 1 100 ! 1 1 ' 1 i II Ml i-UJ I 1 ■ ' ' /■ Tvo --\ / . " * It is really a waste of time to discuss at great length here which fact about integrals is used as a definition, and which one is proved ; to satisfy the demand for formal definition the integral may be defined in either way, — as a limit of a sum, or as a reversed differentiation. The important fact is that the two ideas coincide, which is the fact stated in the Summation Formula. t Later we return to approximate methods of calculation. (See § 125.) Ah 118 INTEGRATION [V, § 68 (2) 10-^ = 225 h, where to is the width and h is the height above the bottom. J^et a be the total depth of water in the channel. Then the depth s at any point is s = a— h, and the pressure is (3) p = k(a-h). If the water is stopped by a cut-off gate, the total pressure on the gate is most easily computed by dividing the gate into horizontal strips of height Ah each ; throughout one of the strips the pressure is very nearly constant ; the total pressure on a strip is (approximately) the product ol its area and the pressure per unit area : (4) pressure on each strip = {^o • Ah] • {p} = p -w • Ah, so that the total pressure P on the gate is (approximately) (5) P = fp . tol . A/i + fn . Mj] ■ Ah+ •■■ +\p- w] [L Jh=Ah L Ja = 2AA L jA=n. where n is the number of strips. The exact value is therefore (6) P] """ = lim \ [inv'] ■ Ah + [ pio] + pio\ ■ Ah\ = \ . picdh, ^y (•^)) P- 115. In the problem before us, w — 15 /i^/- and p = k(a — h) ; hence = \ Vo k{a - h)hy^ dh = 15 A- 1^ - '4-1 = -i ^•«"'. that is, the total pressure P on the gate increases as the fifth power of the square root of the total depth a of water in the channel ; e.g. four times the depth of water would mean 32 times the pressure. Note that the formulas (3), (4), (5), (6) apply in any similar example. It is important to notice that the total pressure up to any height h = h is a function of h whose rate of change is p-w. Thus, if the gate be made in two parts, the lower portion, of height 7t, bears a pressure P,.pT='^=304^-^r^:.30A:(^-^^-^% J;,=o L 3 5 J;,=o \ 3 5 y The rate of change of P,, as h increases is dP,Jdh = l^i k{n—h)h'^^- —p-w. In general, if the height of the lower portion of the gate be incrt'ased by an amount Ah, the pressure P^ on the portion is increased by an amount ■Ah V, §68] EXACT SUMMATION 119 ^P^ = p ^nAh, approximately, so that A P^/ Ah = p ■ to (nearly) ami (ir,,/dh = i^ .w where p is the pressure at the upper edge of the lower por- tion. The integral in (G) may be thought of as the reversal of this rate, as ill §§ 64, ()6. This argument is, however, by no means so natural as the above argu- ment by summation. The important thing to notice is that even in this case the integrated function is really the rate of increase of P as a func- tion of h. But in some problems it is difficult to show directly that the integral is a reversed rate, except by using (3). The great value of the summation formula (.•>), § 66, is that it makes it unnecessary for us to express each problem as a reversed rate. EXERCISES XXV. — INTEGRALS AS LIMITS OF SUMS Determine the following quantities, (a) approximately by step-by-step summation ; (h) exactly by integration between limits : 1. The area under the curve ?/ = z- from x = 1 to x = 3 ; from x = a to X = 6. 2. The area under the curve y = x^ from x = to x = 2 ; from X =— 1 to X =-f 1. 3. The area under the curve x-y = 1 from x = 2 to x = 5. 4. The distance passed over by a body whose speed is u = 2 « -|- 10 from « = to f = 3. 5. The distance passed over by a falling body (v = gt) from < = 2 to t = 5. 6. The increase in speed of a falling body from the fact that the acceleration is ^ = 32.2, from « = to « = 3. 7. The increase in the speed of a train which moves so that its accel- eration is j = *' a _-H*'^ \ Fig } / / ^s^ 1 r . 33. 1 / F G. 32. Example 1. To find the volume of a sphere of radius a. The sphere may be thought of as located between two parallel tangent planes at a distance 2 a from each other. A section parallel to one of the planes is a circle of radius r = CN ; its ayea is but, in the triangle OCN, OC-a-s, 0N= a, whence r^ = CN^^ ON^ - OC^ =^ a^ -(a - sy = 2 as - s^ It follows that In any case, s may be counted from the lower bounding plane, if conven- ient. V, § 70] VOLUMES OF SOLIDS 123 - f{a) + 4/(ff + A.r) + 2/(a + 2 Ax) + if {a + 3 Ax) Ax + ■■' +/(6) J 3' [This rule is called Simpson's Rule ; see § 12.5.] 13. Calculate the following integrals approximately by the process suggested in Exs. 11-12. Notice that some of them cannot be evaluated otherwise at present : («) Cx^dx. (c) CVxdx. (e) fvi + x^dx. (6) £(l/x)dx. (d) j^'^VTT^fZx. (/) j^" sm X di 14. Show that the area of any surface of revolution, formed by revolv- ing a curve y =/(x) about the x-axis, is the limit of a sum of terms of the form 2 Try As, where s denotes the length of arc, as in § 61. Hence show, by § (31, that the area is given by the integral 2.j-,*=2,j-,v^+(iy^. 15. In a manner analogous to Ex. 14, show that the area of a surface of revolution formed by revolving a curve about the y-axis is2ir ixds. 16. Find approximately the length of the arc of the curve y = x^ from X = to X = ^ ; from x = ^ to x = 1. (See Ex. 1, p. 108.) 17. Find approximately the area of the convex surface of that portion of the paraboloid formed by revolving the curve y = Vx about the x-axis which is cut off by the planes x = and x = ^ ; by x = J and x = 1. K CHAPTER VI TRANSCENDENTAL FUNCTIONS PART I. LOGARITHMS — EXPONENTIAL FUNCTIONS 72. Necessity of Operations on Transcendental Functions. The necessity for the introduction of transcendental functions in the Calculus depends not only on their own general impor- tance, but also upon the fact that integrals of algebraic functions may he transcendental. Thus, in § 57, in the case n = — 1 the integral J.r" dx could not be found, although the integrand 1/x is comparatively simple. We shall see that this integral, Ja;~'f/a;, results in a logarithm. (See § 78, p. 137, Ex. 3.) We shall see also in § 81 that nu- merous cases arise in science in which the rate of variation of a function /(a;) is precisely 1/x. In Ex. 1, p. 108, the integral J VT+Ta? dx could not be evaluated ; throughout Chapter V, integrals involving radicals were avoided except in special cases, because such integrals usually result in transcendental functions. 73. Properties of Logarithms. The logarithm X of a number N to any base B is defined by the f a^t that the two equations (1) N = B^, logsN = L are equivalent. Thus if i = log^ N and I = log^ n, the identity B^ • B^ = 5^+^ is equivalent to the rule (2) log^ (^V . n) = log^ N + log^ n, where n and iVare any two numbers. Likewise B^-~B^=B^~' gives (3) log,(N^n) = \og,N-log,n; 130 VI, §74] LOGARITHMS AXD EXPONENTIALS 131 and (B'-y = B'" becomes (4) log^ ^Y" = n log^ iV, where n may have any value whatever. Another fundamental rule results from the application of (4) to the equation (5) x=B^, i.e. y=\ogj,x. For if h is any other base, (6) log, X = log, {By) = y log, B ; [by (4)] but since y = log^ x, we have (7) log,x = logj,X'\og,B. In particular if x = b, since log,& = 1, we have (8) 1 = log« b . log, B, or log, i5 = 1 h- log^ b. The equations (1), (2), (3), (4), (7), (8) are the fundamental rules for logarithms. (See Tables, II, A.) 74. Graphical Representation. A fairly accurate graph of the equation (1) y = \ogsX is obtained by writing the equation in the form (2) X = B", and plotting a few points given by taking integral (positive and negative) values of y. Thus y = 0,l, 2, •••, —1, —2, ••• give a- = l, B, B-, ■', 1/B, l/S^, •... The student should draw a figure from such values, for several different values of B, taking 5 = 2, then B = S,5, 10, etc. When B = l, the equation (2) degenerates into the horizontal straight line x = l, while (1) degenerates completely and becomes meaningless; for this reason, tJie number 1 is never vsed as a base of logarithms. To make these graphs accurately, more points are necessary. The easiest method is to calculate the desired values by com- mon logarithms, i.e. logarithms to the base 10. Taking the common logarithms of both sides of (2), we find 132 TRANSCENDENTAL FUNCTIONS [VI, § 74 logio X = logjo B' = y • logio B, (3) or y = logio x^ login B. It should be noticed that (3) is equivalent to (1) and therefore to (2) ; the curves for B = 1.5, B = 2,B = 3, B = 4.5, 5 = 9 are shown in the figure. They should be carefully drawn on a much larger scale by the student, by use of (3). See Tables. -1 1 ^ — P ^ P ^ <^ - -4- y = lo f^n X --' ^ - ^ J y ^ — — ' b _ ^ ^ / ^ ■^ ' ,, s _ —i -J ~ J ^ , f 4 5 / / ' __. . , ^ / y ^ -^ •— 2_ Lj . — ' — ■p r ^ •^ f0 1 ) 3 4 : /hi i I (\ Fig. 36. EXERCISES XXVm. - LOGARITHMS AND EXPONENTIALS 1. Find the value of 10^ when x = 2; ; 1.5 ; 2.3 ; - 1 ; - 1.7 ; 0.43. 2. Plot the curve y = 10* carefully, using several fractional values of x. 3. Plot the curve y = logio x by direct comparison with the figure of Ex. 2. Plot it again by use of a table of logarithms. 4. Plot the graph of each of the following functions : (a) logio a;2. (&) logio (1/x). (c> logio Vx. ((?) logio «2/8. Do any relations exist between these graphs ? 5. Plot the graph of each of the following functions and explain its relation to graphs already drawn above : (a) logio (1 + a;). (t>) logio Vl + x. (c) logio (xVl+x). 6. Plot the graphs of each of the following functions and show the relations between them. (a) logaa;. (b) logi x. (c) logsxz. (d) 2*. VI, § 75] DIFFERENTIATION OF LOGARITHMS 133 7. Show how to calculate most readily the values of the following expressions, and find the numerical value of each one : (a) logiiT. (c) (5.4)6-2. (e) 10>-5+ 10-1-5. (g) [ogslO. (6) 2*-53. (d) logs 8. (/) 5 1og4 6. (h) lOiogio'. 8. Draw each of the following curves : (a) y = 10' + 10~'. (c) y = xlogiox. (e) y = logio cos ac. (6) p^i-" = const. (d) y = 2' sin x. (/) 2/ = 10 ^'^^ 75. Slope of y = log^o x a,tjc = 1. The slope M of the curve (1) y = logioo; at the point (1, 0) can be approximated very closely. Let (1, 0) be called P, and let (1 + A.r, + Ay) be called Q ; then + Ay= logio (1 + Ax), and the slope m^g of PQ is (2) m,Q = ^ = ^-^^^^±M, ' ^ Ax Ax If Ax is given in succession the values .1, .01, .001, we find '^^Jax=i = 10 logio (1-1) = 0.4139; mpoj ^ = 100 logio(1.01j = 0.432} Wpg] _^^^= 1000 logio (1.001) = 0.43 [using five-place tables] = 0.434 [using six- or seven-place tables]. Still smaller values of Ax would give the same result by the usual interpolation rules, so that for values of Ax less than .001 a table of more than seven places would be needed ; and even then the result would be changed at most in the fourth place of decimals. 134 TRANSCENDENTAL FUNCTIONS [VI, § 75 The slope M oi the curve (1) at (1, 0) is the limit of these slopes as A.i; approaches zero ; hence (3) M=^^~\ = lim mpy = 0.434 ••• (approximately).* 76. Differentiation of logjo^?. It is now easy to find the derivative of logio re. Let P, (a;, ?/), be any point for which (1) . 2/ = logio X, o\ x = 10^ ; and let Q, {x + Aa;, y + A?/), be any other point on the curve ; then (2) 2/ + A?/ = logiu (.T + Aa-), or a.- + Ax = 10"+-^^'. Subtracting the second form of (1) from the second form of (2), Ax- = 10"+^^ - 10^ = 10»'(10^^ - 1) and ^ ' Ay Ay Ax a; lU^«' - 1' since a; = 10«'. In particular at x = 1, Ay _ A?/ which, by § 75, approaches the limit 3/= 0.434 •••. In general,! therefore, (4) flu — (Hogioa? ^ j.jjj ri Ay 1 = M = 0. 434 •» ^ * This assumes only that the ordinary interpolation scheme for common logarithms is approximately correct. The number M is so important that its value has been calculated to a large number of decimal places ; to ten places it is 0.4342944819. An independent method of calculating it is given in § 134. Logically, the present approximate determination of M could be omitted entirely until that time, and M could be carried through all the work as an unknown constant. Practically, it is very desirable to have an approximate value of M at once. t The difficulties ordinarily met in proving this formula are here avoided by placing the burden of any difficulty where it should be, — upon the read* VI, § 77] DIFFERENTIATION OF LOGARITHMS 135 77. Differentiation of log^ jc. Since by (3), § 74, the equation (1) y = log J, X can also be written in the form (2) y = logio a; -=- logw B, it follows that (3) ^ = ^1^ ^ ^ ^ logjo 5 = — -=- logio 5. dx clx dx X Since the number Af which occurs in all these formulas is an inconvenient decimal, it is useful to find a value of B, for which (4) logio i3 = 3/= 0.434 ••■; this value is readily found from a logarithm table, and is denoted by the letter e: (5) e = 10^ = 2.72 ••• (approximatelj). If i^ = e, the formula (3) becomes [VIII J ^io&^ = L. On account of the simplicity of this formvla the base e will be vsed hoiceforth in this book for all logarithms and exponentials unless the contrary is exjilicitly stated;* it is called the natural base, or tlie Napierian base. If B has any value whatever, (3) becomes [V..I] 'l^ = l.-K^ = Vj^«Ji = l.io,,e; dx X log JO B X logio B X ing of ail ordinary table of logarithms : for the essence of the difficulty lies in the lack of accuracy of the usual elementary definition of logarithms. No pretense of rigorous logic in the proof of (4) is justified unless a proof that the common logarithm of any number exists is given. * The value of e to ten places is 2.718281H285. Another method of com* puting its value is given iu § 134 ; see also § 142. d\og,ox_ ^M _ logio e _ _ 0.434 ... clx X X X 136 TRANSCENDENTAL FUNCTIONS [VI, § 77 for theoretical purposes, the last form is used; for practical computations, the next to the last. If 5 = 10, we find [vni,] These three Rules, of which [VIII] is the general form, are added to the list of seven Rules in Chapter III. While the common base 10 is exceedingly convenient for computations, the new base e is simpler in all theoretical discussions, chiefly because [VIII„] is simpler than [VIIIj]. Logarithms to the base e are called natural, or Napierian, or hyperbolic logarithms. See Tables, V, C. 78. Illustrative Examples. We may now combine Rule [VIII] with [I]-[VII], and with the reverse differentiation (integration) formulas of Chapter V. Example 1. Given y = logio(2 x^ + 3), to find dy/dx. Method 1. Derivative notation. Set m = 2 x^ + 3, then di_dy du _ dloginu _ d (2 x'^ + 3) _,M^^^_ 4 Mx ^ dx du dx du dx u 2 x^ + 3 Method 2. Differential notation. d2/ = d logxo(2 x-^ + 3) = ^-ilL_ d(2 x2 + 3) = ^i^ (Zx. Example 2. Find the area under the cm-ve y = l/x from x = 1 to X=:10: ]x=10 /•^=10 1 , -1 x=10 , 1 = ( ifZx = log.x| = log, 10 = —!— = -. ;r=l Jx=l X, Ja-=1 ^ lOgj^e M * The number lege 10 = 1 -4- iV = 2.302585 is important because common logarithms (base 10) are reduced to natural logarithms (base e) by multiplying by this number, since log«iV=logio -ZVx loge 10. Similarly, natural loga- rithms are reduced to common logarithms by multiplying hy M= logio e ; since logio iV = 3/ -=- loge JV. It is easy to remember which of these two multipliers should be used in transferring from one of these bases to the other by remem- bering that logarithms of numbers above 1 are surely greater when e is used as base than when 10 is used. VI, § 78] DIFFERENTIATION OF LOGARITHMS 137 Example 3. If the rate of increase dy/dx of a quantity y with respect to X is 1/x, find y in terms of x. Since dy/dx = 1/x, y = \ -'dx = logeX + c, where c is a constant, — the value of y when a; = 1. It should be noted that logarithms to the base e occur here in a perfectly natural manner ; the same remark applies in Example 2. Note that loge x = logio x -i- M. This case arises constantly in science. Thus, if a volume v of gas ex- pands by an amount Ay, and if the work done in the expansion is A W, the ratio A ir/Au is approximately the pressure of the gas; and dJI'/tZy = p exactly. If the temperature remains constant pv = a. constant ; hence dW/dv = k/v. The general expression for W is therefore J V expanding from one volume '1'^=' = (^'^ dv = k loge v\ "^ = k log. ^-^ = ^logjoH2. and the work done in expanding from one volume vi to another volume V2 is W EXERCISES XXIX. - LOGARITHMS 1. Calculate the derivative of each of the following functions ; when possible, simplify the given expression first : (a) logioa;2. (6) logio Vx. (c) logio (1+ Sx). {d) logio (1 + x"-). (e) log, (1 + x)2. (/) log. VrT2^. {g) log. (1/x). {h) logio (x-2). (0 a;log.x. c;) log. (i^^). W log. (2 + ^). (0 log. V^. \ra)'^2EA. (u) log, {log. x}. (o)(log.O'^. 2. Evaluate each of the following integrals : ■(^^r^-^'^- wX'^*- (oJ-;a-"-)(i + «-v» 138 TRANSCENDENTAL FUNCTIONS [VI, § 78 3. Calculate the area between the hyperbola ocy = I and the a;-axis, from a; = 1 to 10, 10 to 100, 100 to 1000 ; from x=ltox = k. 4. Show that the slope of the curve y = logio x is a constant times the slope of the curve y = log^ x. Determine this constant factor. 5. Find the flexion of the curve y = log, «, and show that there are no points of inflexion on the curve. 6. Find the maxima and minima of the curve y = log^ (x^ — 2 x + 3). 7. Find the maxima and minima and the points of inflexion (if any exist), on each of the following curves: (a) 2/ = 2 a;2 - log, x. (b) y = x + log, (1 + a;-^). (c) y = x^- log, x3. (d) 2/ = (2 X + log x)'^. 8. Find the areas under each of the following curves between x = 2 and X = 5 : (a) y = x + 1/x. (&) y = (x^ + l)/x3. (f) y = (xV2 _ x)/x^. 9. Find the volume of the solid of revolution formed by revolving that portion of the curve xy- = 1 between x = 1 and x = 3 about the x-axis. How much error would be made in calculating this volume by the prismoid formula ? 10. If a body moves so that its speed v = t + l/t, calculate the distance passed over between the times t = 2 and t = 4. 11. Find the work done in compressing 10 cu. ft. of a gas to 5 cu. ft., ifjsy = .004. 12. Find the areas under the hyperbola xy = k^ between x = 1 and X — c, c and c^, c'^ and c^, c' and c*. 79. Differentiation of Exponentials. Since the equations y = logg X and a; = B'' are equivalent, Rule [VIII] gives ^ = ^=l-^^ = -^_=5-log,B. ay ay ax log^ e If we interchange the letters x and y, for convenience of memory, we obtain the standard forms : y = B'^ (or X = log^ y) dx dx log„e VI, § SO] DIFFERENTIATION OF EXPONENTIALS 139 of which the two special cases B = e and jB = 10 are : This formula [IX] can be combined with all the preceding rules, as in § 78. 80. Illustrative Examples. Example 1. Given y = e"-, to find dy/dx. Method 1. Set x^ — u ; then ax du dx du dx Method 2. dy = de""" = e=^' d(x^) = 2 x e*' dx. Example 2. Find the length I of the arc of the catenary y = (e*+e~')/2, between the points where a; = and where x = 1. By § 61, p. 107, we have = (•'=' Jl + (JIjuJ:^ dx = i f ^' {e' + e-') dx ^Ife-l^^ (2.718 -0.368)72 = 1.175 (nearly). 2\ e / This curve is very important because it is the form taken by a perfect inelastic cord hung between two points. Tlie given function is often called the hyperbolic cosine of x, and is denoted by cosh cc, so that coshx =(e* + e-^)/2. Example 3. If a quantity y has a rate of change dy/dx with respect to X proportional to y itself, to find y in terms of x. Given dx 140 TRANSCENDENTAL FUNCTIONS [VI, § 80 we may write dy y' hence kx= \ - dy = loge + 0, by § 78, Ex. 3. Transposing c, we have loge y = kx — c, or 2/ = e*^-" = g-^e*^ — Ce**, where C(= e~'^) is again an arbitrary constant. The only quantity y xohose rate of change is proportional to itself is Ce*^ where O and k are arbitrary, and k is the factor of proportionality . This principle is of the greatest importance in science ; a detailed dis- cussion of concrete cases is taken up in § 81. EXERCISES XXX. — EXPONENTIALS 1. Show that the slope of the curve y = e^ is equal to its ordinate. 2. Show that the area under the curve y = e^ between the y-axis and any value oixis y — \. 3. Find the derivative of each of the following functions : (a) e^-. (d)(e- + l)'^. ^„^ ^ - e-' (j) e^-H*. (6) e'\ 2 (c) ire*. 2 4. The expression (e*— e-^)/2, used in Ex. 3 (/) is called the hyper- bolic sine of x ; and (e^ -\- e"^)/2 is called the hyperbolic cosine of x ; they are represented by the symbols sinh x- and cosh x respectively. See Tables, II, H, Show that d sinh ,T. = cosh x dx, d cosh x — sinh x dx. 5. Show that 1 -i- sinh^x = cosh^a; ; hence find the length of the arc of the curve y — cosh x from x = to x = 2. [The curve 2/ = coshx, ory = (e^ -f- e-^)/2, is called a catenary (§80).] 6. Find the area under the catenary from x = to x = 3 ; from x = — 1 to x = -|-l; from x = to x = a. [See Tables, V, C] T. Find the area under the curve y = ainhx from x = to x = 3; from X = to x = a. "" e- + e-- W I'+x^- {k) 10^'. (i) x'^^K (0 X • 102^+8 VI, § 81] COMPOUND INTEREST LAW 141 8. Find the maxima and minima and the points of inflexion (if any exist) on each of the following curves: (a) y = sinh x. (6) y = cosh x. (c) y = tanh x = sinh x/cosh x. (d) 2/ = e-^^ (e) y = e-^*. (/) y = sech x = 1 -^ cosh x. 9. Show that the pair of parameter equations x = cosh t, y = sinh t represent the rectangular hyperbola x^ — y- = l. Hence show that the differential of arc for this hyperbola is ds= (cosh 2 ty/- dt, and find the speed at the point where « = 0, it t denotes the time. 10. Show that the area under the hyperbola x^ — y"^ = I from x = 1 to X = a is represented by the integral £ sinh'- tdt= \ [(cosh 2 t - l)/2] dt where cosh c = a. Hence show that this area is (sinh 2 c)/4 — c/2. 11. Show that the area of a triangle whose vertices are the origin, the point (x, 0), the point (x, y) on the hyperbola x^ — y^ — 1, is xy/2 = (sinh 2 0/4- [Ex. 9.] Hence show by Ex. 10 that the portion of this triangle outside of the hyperbola is t/2. [Note. The parameter equations are often written in the form X — cosh 2 A, y = sinh 2 A, where A is the last area mentioned.] 12. Calculate the following integrals : (a) ]■/%' + i)2dx. (Ii) ( (e' + S) e-' dx. (0 j {e'-^+^ + 1) dx. 81. Compound Interest Law. The fact proved in the Ex. 3 of § 80 is of great importance in science : If a variable quantity y has a rate of increase ivith respect to an independent variable x proportional to y itself, then (2) y=Ce^, ichere C is an arbitrary constant. (a) £e^dx. ((7) rsiuh2xdx. (b) |;.-^dx. (e) CcoshSxdx. (c) ^\^dx. (/) Csinh'^xdx. 142 TRANSCENDENTAL FUNCTIONS [VI, § 81 For this reason the equation (2) between two variables x and y was called by Lord Kelvin the "Compound Interest Law," on account of its crude analogy to compound interest on money. For the larger the amount y (of principal and in- terest) grows the faster the interest accumulates. " Compound interest " is, however, only a convenient name, since interest is really compounded at stated intervals (e.g. each year) and not continuously. A more suggestive name might be the snowball law, since a snowball grows more rapidly the larger it becomes, and its rate of growth is roughly propor- tional to its size. In science instances of a rate of growth which grows as the total grows are frequent.* Example 1. Work in Expanding Gas. The example used to illns- trate Ex. 3, § 78, can be put in this form. Since, in the work W done in the expansion at constant temperature of a gas of volume -y, we found dW/dv = k/v, it follows that dv/dW=v/k; hence v = Ae^/'', which agrees with the result of § 78. Example 2. Cooling in a Moving Fluid. If a heated object is cooled in running water or moving air, and if 6 is the varying difference in temperature between the heated object and the fluid, the rate of change of d (per second) is assumed to be proportional to 6 ; dd/dt = — kd, where t is the time and where the negative sign indicates that d is decreasing. It follows that 0=0- e"*'. [Newton's Law of Cooling.] Such an equation may also be thrown in the form of § 78 ; in this example, dt/dS =— l/{kd), whence t = — ('i./k) ■ log^ + c, and the time taken to cool from one temperature 0i to another temperature 02 is Je=e, Je, k0 k °" je^ k ^'^i where is the temperature of the body above the temperature of the surrounding fluid. * The common expressions "grows like a snowball," "gathers momentum as it goes," "wealth breeds wealth," "it grows by its very growth," "the rich grow richer, the poor poorer" illustrate the frequent occurrence of such VI, §81] COMPOUND INTEREST LAW 143 The law for the dying out of an electric current in a conductor when the power is cut off is very similar to the law for cooling in this example. See Ex. 17, p. 146. Example 3. Bacterial Growth. If bacteria grow freely in the pres- ence of unlimited food, the increase per second in the number in a cubic inch of culture is proportional to the number present. Hence '— = kX, N- = Ce", t = - log, .V + c, dt A ° ' where iVis the number of thousand per cubic inch, t is the time, and k is the rate of increase shown by a colony of one thousand per cubic inch. The time consumed in increase from one number iVi to another number tT= ri^=iiog..v7^=iiog.^. \ J.Yj k iV k * J.\ k Xi If iV2 = 10j\"i, the time consumed is (1/A-) loge 10 = l/(i.V). This fact is used to determine k, since the time consumed in increasing iV ten- fold can be measured (approximately). If this time is T, then T = \/(JcM), whence k = 1/{TM), where Tis known and M = 0.43 (nearly). Numerous instances similar to this occur in vegetable growth and in organic chemistry. For this reason the equation (2) on p. 141 is often called the "law of organic growth." (See Exs. 18, 19, p. 146.) Example 4. Atmospheric Pressure. The air pressure near the surface of the earth is due to the weight of the air above. The pressure at the bottom of 1 cu. ft. of air exceeds that at the top by the weight of that cubic foot of air. If we assume the temperature constant, the volume of a given amount is inversely proportional to the pressure, hence the amount of air in 1 cu. ft. is directly proportional to the pressure, and therefore the weight of 1 cu. ft. is proportional to the pressure. It follows that the rate of decrease of the pressure as we leave the earth's surface is propor- tional to the pressure itself : ^P=-kp, p= Ce-*\ h=-'^ \og,p + c, dh k where h is the height above the earth ; and, as in Exs. 2 and 3, the dif- ference in the height which would change the pressure from pi to p^ is Since /i]p^ and p2 and pi can be found by experiment, k is determined by the last equation. 144 TRANSCENDENTAL FUNCTIONS [VI, § 82 82. Percentage Rate of Increase. The principle stated in § 81 may be restated as follows : In the case of bacterial growth, for example, while the total rate of increase is clearly- proportional to the total number in thousands to the cubic inch of bacteria, the percentage rate of increase is clearly constant. In any case the percentage rate of increase, r^,, is obtained by dividing 100 times the total rate of increase by the total amount of the quantity, 100 • (dy/clx) -j-y; and since the equa- tion dy/dx= ley gives (dy/dx)-^y = 1c, it is clear that the per- centage rate of increase in any of these ptrohlems is a constant. The quotient (dy/dx) -f- y, that is, 1/100 of the percentage rate of increase, will be called the relative rate of increase, and will be denoted by 7\. In some of the exercises which follow, the statements are phrased in terms of percentage rate of increase, r^, or the rela- tive rate of increase, r^ = ?"p -r- 100. EXERCISES XXXI. — COMPOUND INTEREST LAW 1. If 2/ = 5 e^^, find chj/dx, and show that {dy/dx) ^ y = 2. 2. Find dy/dx and (dy/dx) -=- y for each of the following functions : (a) 7e3z. (fZ) e'\ (g) (ax + 6)e*^. (6) 4e-2-5«. (e) e*'+\ (A) {x^+px + q)e^. (c) xe'. if) (a;2 + 2)e». (i) (Sx + 2)e-^\ 3. If a body cools in moving air, according to Newton's law, dd/dt = _ Jc0^ where t is the time (in seconds) and is the difference in tempera- ture between the body and the air, find k if 6 falls from 40° C. to 30° C. in 200 seconds. 4. How soon will the difference in temperature e in Ex. 3, fall to 10^ C? 5. If a body is cooled in air, according to Newton's law, find k if the changes from 20° C. to 10° C. in five minutes. How soon will d reach 5°C.?^ 6. If a body cools so that the percentage rate of cooling is 2 % (in de- grees C. and minutes), how long will it talce to cool from a difference 20° to a difference 10° (with respect to the surrounding air)? VI. § 82] CO:^IPOUND INTEREST LAW 145 7. In measuring atmospheric pressure, it is usual to express the pres- sure in millimeters (or in inches) of mercury in a barometer. Find C in the formula of Ex. 4, § 81, ii p = 762 mm. when h =0 (sea level). Find Cifp = SO in, when h=:0. 8. Using the value of C found in Ex. 7, find k in the formula for at- mospheric pressure if p = 24 in. when h — 5830 ft. ; if p = 600 mm. when h = 1909 m. Hence find the barometric reading at a height of 3000 ft. ; 1000 m. Find the height if the barometer reads 28 in. ; 650 mm. [Note. Pressure in pounds per square inch — 0.4908 x barometer reading in inches.] 9. If a rotating wheel is stopped by water friction, the rate of decrease of angular speed, dw/dt, is proportional to the speed. Find w in terms of the time, and find the factor of proportionality if the speed of the wheel diminishes 50 % in one minute. 10. If a wheel stopped by water friction has its speed reduced at a con- stant rate of 2% (in revolutions per second and seconds), how long will it take to lose 50 % of the speed ? 11. The length I of a rod when hfeated expands at a constant rate per cent ( = 100 k). Show that dl/dO = kl, where 6 is the temperature ; if the percentage rate of increase is .001 % (in feet and degrees C), how much longer will it be when heated 200° C. ? At what temperature will the rod be 1 7o longer than it was originally ? [Note. This value of k is about correct for cast iron.] 12. The coefficient of expansion of a metal rod is the increase in length per degree rise in temperature of a rod of unit length. Show that the coeflScient of expansion of any rod is the relative rate of increase in length with respect to the temperature. (See Ex. 12, p. 27.) 13. A chimney is designed so that the pressure per square inch on each horizontal cross section is a constant k. If the outer surface of a section at a height h is a circle of radius B, and if all the cross sections are simi- lar, including the flue holes, show that the total pressure on a cross sec- tion is proportional to kE-, and that k(B + AR)'^ = kB'^ - pR-Ah, where p is the weight per cubic inch of the material. Hence show that dB/dh = — pB/{2 k) and that B = i?oe"P*'''-*\ where B^ is the radius of tiie bottom section {h = 0). 14. Assuming that the form of a chimney is given by the equation i? = i?oe"P*'^'*' [Kx. 13], show that the total weight (neglecting the flue holes) is kTBQ-{l — e-p"''') , where H is the total height. Hence .show that L 146 TRANSCENDENTAL FUNCTIONS VI, §82 the pressure per square inch on the bottom section is k(l — e-p^"'), and that it approaches the theoretical limit A; as if increases. 15. Show that the results of Ex. 14 are the same when the flue holes are taken into account, with the assumptions made in Ex. 13. [Note, The pressure per square inch depends solely on the height, for the same material. The height is limited by the crushing strength of the material.] 16. When a belt passes around a pulley, if T is the tension (in pounds) at a distance s (in feet) from the point where the belt leaves the pulley, )• the radius of the pulley, and /j. the coefficient of friction, then dT/ds = fiT/r. Express T in terms of s. It T= 30 lb. when s = 0, what is T when s = 5 ft., if r = 7 ft., and /i = 0.3 ? 17. "When an electric circuit is cut off, the rate of decrease of the cur- rent is proportional to the current C. Show that C = Coe-**, where Co is the value of C when t = 0. [Note. The assumption made is that the electric pressure, or electro- motive force, suddenly becomes zero, the circuit remaining unbroken. This is approximately realized in one-portion of a circuit which is short- circuited. The effect is due to self-induction : k = B/L, where B is the resistance and L the self-induction of the circuit.] 18. Radium automatically decomposes at a constant (relative) rate. Show that the quantity remaining after a time t is q = qoe~^, where go is the original quantity. Find k from the fact that half the original quantity disappears in 1800 yrs. How much disappears in 100 yrs. ? in one year ? 19. Many other chemical reactions — for example, the formation of in- vert sugar from sugar — proceed approximately in a manner similar to that described in Ex. 18. Show that the quantity which remains is q = qoe-'^ and that the amount transformed is A=: qo — q = qo{^ — c— **). Show that the quantities which remain after a series of equal intervals of time are in geometric proportion. 20. The amount of light which passes through a given thickness of glass, or other absorbing material, is found from the fact that a fixed per cent of the total is absorbed by any absorbing material. Express the amount which will pass through a given thickness of glass. 83. Logarithmic Differentiation. Relative Increase. In § 82 we defined the relative rate of increase ?v of a quantity y with respect to x as the total rate of increase (dy/dx) divided by y. VI, §84] RELATIVE RATES 147 If y is given as a function of x, (1) y=A^), the relative rate of increase ?v = idy/dx) -=- y can be obtained by taking the logarithms of both sides of (1),* (2) \o^,y = \og,f{x), and then differentiating both sides with respect to x-. (3) r =l-.*Iy= ^^ ^^^^ y = dlogeAx) ^ ij dx^ dx dx This process is often called logarithmic differentiation : the logarithmic derivative of a function is its relative rate of increase, Tr, or 1/100 of its percentage rate of increase. Example 1. Given y = Ce>^, to find Vr = {dy/dx) -4- y. Taking log- aritlims on both sides : loge y = loge C + kx; differentiating both sides with respect to x ; dx dx The resnlt of Ex. 3, p. 139, may be restated as follows: the only function of x whose relative rate of change (logarithmic derivative) is constant is Ce'". Example 2. Given j/ = x^ + 3 a; + 2, to find r^. Method 1. ^ = 2 X + 3, hence r^ = -^ -^ 2a; + 3 dx dx ^ xi + ^x + 2 Method2. .^ = ^^^2/=^Mi/=.^nogixi+3x + 2)^_lx+3_, ' dx dx dx x2 + 3 X + 2 84. Logarithmic Methods. The process of logarithmic dif- ferentiation is often used apart from its meaning as a relative rate, simply as a device for obtaining the usual derivative. This is particularly useful in the case of variables raised to variable powers, and it is at least convenient in such other examples as those which follow. * Since log ^is defined only for positive values of N, all that follows holds only for positive values of the quantities whose logarithms are used. 148 TRANSCENDENTAL FUNCTIONS [VI, § 84 Example 1. Given y = Vx, to find dy/dx. Method 1. Ordinary Differentiation. dy^ _ dVx _ dx^^' _ 1 1/2 _ 1 ^ dx~ dx ~ dx ~ 2 ~ 2 x^'^ ' Method 2. Logaritlimic Method. Since for positive values of Vx, « log y = log x'^^ = ^ log X, we have Example 2. Given ?/ = (2 a;2 + 3) 10*^-^ . Method 1. Ordinary Differentiation. ^ = (2 a;2 + 3) f (lO*^"') + lO^-i A (2 x2 + 3) dx dx dx '^ = (2x2 + 3). 4. ll0'"-Vl0*""'-4x = 4 . 10i^-ir(2x2 + 3)/Jtf + x], where itf = logioe = 0.434. ilfeifeoc? 2. Logarithmic Method. Since log y = log (2 x2 + 3) + (4x-l)logl0, we have r,. = ^^y= ^^ + 4 • log 10, (Zx 2x-' + 3 ° ' or ^ = y r_i^_ + 4 log lOl = 4 . 10*^-^ [x + (2 x2 + 3)log 10], dx L2 x2 + 3 J which agrees with the preceding result, since loge 10 = 1/logio e = \/M. Example 3. Given y = (S x~+ 1)2*+*, to find dy/dx. Since no rule has been given for a variable to a variable power, ordinary differentiation cannot be used advantageously. Taking logarithms, however, we find log?/=(2x + 4)log(3x2 + lX whence r;- = ^ ^ y = 2Iog (3x2 + 1) + ^^ (2x + 4), dx 3 x- + 1 or ^ =(3^2 + i)2x+4J21og(3x2 + l) + ;^4^ (2^ + 4)! • dx ( 3 x^ + 1 ] The use of the logarithmic method is the only expeditious way to find the derivative in this example. VI, § 84] RELATIVE RATES 140 EXERCISES XXXn. — LOGARITHMIC DIFFERENTIATION 1. Find the logarithmic derivatives (relative rates of increase) of each of the following functions, by each of the two methods of §§ 82-83: (a) e-2«. (e) 0.1 gio'-s. (i) (r^ + i) e-'-=. (6) 4e« (/) 102-+3. (j) (2 - 3 «2) c2<2-i. (c) e»+2. (gf) e-«^+*^'. (A,-) (1 _ <■-' + <4) lo^'+s^. (d) e-»*. (A) 2f%-5'. (Z) ger. 2. Find the derivative of each of the following functions by the logarithmic method : (a) (l + a:)i+x. (c) x^. (e) (1 + x)(l + 2 a:)(l + 3 x). 3. If y = WW, show that dy -^ y = du ^ ti + dv -i- v. In general show that the relative rate of increase of a product is the sum of the relative rates of increase of the factors. 4. If a rectangular sheet of metal is heated, show that the relative rate of increase in its area is twice the coefficient of expansion of the material [see Ex. 12, List XXXI]. 5. Extend the rule of Ex. 3 to the case of any number of factors. Apply this to the expansion of a heated block of metal. 6. Show directly, and also by use of Ex. 5, that the relative rate of increase of x" with respect to x, where n is an integer, is n/x. 7. Compare the functions e^^ and e2i+3 . compare their relative rates of increase ; compare their derivatives ; compare their second derivatives. 8. Compare the following pairs of functions, their logarithmic deriv- atives, their ordinary derivatives, and their second derivatives: (a) e^ and 10^. {d) e-"^ and 6+"^. (6) e" and 6"+*. (e) e-*^ and sech x. (c) e« and 10"^. (/) e-^' and 1 ^ (a + hx?). 9. Can k be found so that ke" and lO*"^ coincide ? Prove this by com- paring their logarithmic derivatives, and find h in terms of a. 10. If the logarithmic derivative {dy/dx) ^ y is equal to 3 + 4 x, show that log 2/ = 3 a; + 2 X'^ + const., or y = ke^^+^'^. 11. If {dy/dx) ^y=f{x) show that y = ke^^^'^'"'. 12. Find y if the logarithmic derivative has any one of the following values : (a) 1 - X. (c) n/x. (e) e'. (b) ax + 6x2. (d) a + n/x. (/) e* + n/x. 150 TRANSCENDENTAL FUNCTIONS [VI, § 85 Fig. 37. (1) PART 11. TRIGONOIMETRIC FUNCTIONS 85. Introduction of Trigonometric Functions. The way in which trigonometric functions enter in the Calculus is illus- trated by the following simple case of uniform rotation : A point Jfmoves with a constant speed of 1 ft. per second on a unit circle. Let be the center of the circle, and let x and y be the horizontal and vertical dis- tances, respectively, of the moving point M from 0. The equation of the circle ic^ + 2/2 = 1 may also be written in parameter form (2) x = cos^, ?/ = sin^, where 6 = Z. XOM, as is evident from the figure. If is measured in circular measure, 6 = s, where s is the arc AM, since the radius is 1. Moreover, since M is moving with a constant speed of 1 ft. per second, s = t, where t is the time measured in seconds since Jf was at A, and s is measured in feet. The equation (2) may be written in the form : (3) X = cos t, y = sin t, (where $ = s = t). The horizontal speed of M, v^, and its vertical speed, Vy, are, respectively : (4) dx dcost dy dt' d sin t dt dt " dt dt to find these we need precisely to know the derivatives of cos t and of sin t with respect to t. 86. Bifferentiation of Sines and Cosines. These derivatives may be found directly from the example of § 85. To do so, we need to find two eqtiations for the two xmlinown quantities dx/dt and dy/dt; one of these is given by differen- tiating (1), § 85, with respect to t : VI, § 86] TRIGONOMETRIC FUNCTIONS 151 the other is found from the fact that the sum of the squares of v^ and v^ is equal to the square of the total sj^eed (§ 62) : (fJHIJ = 1, since the speed ds/dt = 1 in § 85. Either unknown can now be found by solving (1) and (2) simultaneously : -ox dx • . dy , , . (3) — = -y = -s\nt, -^ = + .T = + cos^ dt dt since x^ + y' = 1. In extracting square roots in this solution the negative sign is attached to the value of dx/dt because x is decreasing when y is positive. The signs in (3) are easily seen to be correct for both positive and negative values of x and y. Comparing (3) with equation (4) of § 85, we find: [XI] ^^ = -smt, [X] ^^« = + co8«, or, in differential notation : [XI]' dcos< = — sinf<«, [X]' rf sin « = + cos « rt«. These two formulas are the basis of all work on trigonometric functions. Circidar measure of angles teas used in obtaining them, and this system of measurement will be used oi all that follou's* A direct proof of these two important formulas is easily made. For, let y = sinx; then y + Ay = sin(.r + Ax), 3^ + -^ jsin— . Hence ^ = cos(x + ^) • 5^1^^, Ax V 2 / Ax/2 * Circular measure of angles is used in the Calculus for the same reason that Napierian Logarithms are used for logarithmic and exponential functions: in each case the standard formulas for differentiation are simplest in the system adopted. 152 TRANSCENDENTAL FUNCTIONS [VI, whence dy dx = lim A?/ Ai=y) Ax lim(sin «)/« = !. d sin X : cos X, or — : = cos X, dx The proof of [XI] is exactly analogous. See also Ex. 7, p. 154. 87. Illustrative Examples. The formulas [X] and [XI] may be combined with other standard formulas. Some of the results are themselves worthy of mention as new standard formulas; these are numbered below in Roman numerals and printed in black-faced type. Example 1. Given y — sm2 6, find dy. dy = d (sin 26') = cos 2ed(2 0) = 2 cos 2ed9. Example 2. Given y — tan 6, find dy/dd. dd [XII] J sin ^ „ „ o f? sin d cos 6 cos e dd . dd diand dO 1 C0S2^ C0S2^ sec2^. , d COS 6 dd _ 1 ~C0S2d' Example 3. Given y = ctn e, to find dy/dd. ,cos^ [XIII] Similarly, [XIV] [XV] Example 4. dy _ de" g, dx dx d ctn ~~dd~' ds&cO ' dd descO sin^ dO dd , 1 sin^ 1 sin^ sm cos- = sec 9 tan 0. — cos< = — CSC ^ ctn 6. dd dO sin- tf Given y = e" sin x, to find dy/dx. X + ^ ^'" -^ . e* = e* sin x + e* cos x = e*(sin x 4- cos x). dx VI, §87] TRIGONOMETRIC FUNCTIONS 153 Example 6. Given y = cos^ (2 ("- + 1), to find dy/dt. Let u = 2 t- + 1, aud v = cos m, then dt dl dt dt dt dt z= — 3 1)2 • sin M ■ 4 « = — 12 ( • sin (2 «^ + 1) • cos- (2 t- + \). Example 6. To find the area under the curve y = sin x from the point where x = to the point where x = 7r/2. ]x=ir/2 rx=JT/2 -\x=Tr/2 = \ sin x dx = — cos x \ =— cos 7r/2 + cos = 1, x=0 Jx=0 Jx=0 since d ( — cos a;) = sin x dx. Comparatively few of the trigonometric in- tegrals can be found by simple inspection ; a detailed treatment of them is given in Chapter VII. EXERCISES XXXin. — TRIGONOMETRIC FUNCTIONS 1. Find the derivative of each of the following functions : (a) sinSx. (e) sinx^. (i) xsinx. (b) cos ((9/2). (/) tan(2< + 3). (j) e'tan«. (c) tan(— e). (g) cos (—it). {k) log cos x. (d) cos^x. (h) sec(x/3). (0 sin e^. («i) sin X + 3 cos 2 X. (p) e-'cos- (1 + 3«), 00 e-' sin (2 « + tt/IO) . (ry) e-^+^sm(3t-ir/i). (o) (1 + x2)sin(2x + 3). {>•) e-'/iO[cos« + 4sin3<]. 2. Find the area under the curve y = sin x from x = to x = tt ; test the correctness of your result by rough comparison with the circumscribed rectangle. 3. Find the area bounded by the two axes and the curve y = cos x, in the first quadrant. 4. Find the maxima and minima, and the points of inflexion (if any ^xist) on each of the following curves : (a) 2/ = sinx. (d) y = xsinx. (ff) y = e-*s\nx. (b) y-cosx. (e) y = I + sin 2 x. (h) y = e-^s'mx. (c) 2/ = tanx. (/) y = sin x + cos x. (i) y = cos(2x + w/6). 5. Find the derivative of each of the following pairs of functions, and draw conclusions concerning the functions : (a) sinx and cos(7r/2 — x). (d) sin 2 x and 2 sin x cos x. (b) sin- X and I — cos^ x. (e) cos 2 x and 2 cos^ x. (c) cos X and cos ( — x) . (/) tan'^ x and sec^ x. 154 TRANSCENDENTAL FUNCTIONS [VI, § 87 6. Integrate the following expressions ; in case the limits are stated, evaluate the integrals, and represent them graphically as areas : (a) y sin X da;. (c) f" sec'^xdx. (e) \ cos {S t +ir/ 6) dt. (6) I '^ cosxdx. (d) \sin2xdx. (/) i ta.n t sec tdt. (g) jj (I + sin x) dx. Uj) (cos'^xdx. (h) \{cosx + 3sm2x)dx. [hint. 2 cos^ a; = 1 + cos 2 x. (i) i (cos2a; — l)c?x. (^•) ( '' sin^xdx. 7. Find the derivative of sin x directly by shovying that sin (x + Ax) — sin X = sin x (cos Ax — 1) + cos x • sin Ax and remarking that lim [(cos Ax— 1)-;- Ax]= and lim [(sin Ax)-=- Ax] = 1. [See § 13, p. 19; Ex. 8, List V; and § 96.] 8. Find the derivative of cos x directly as in Ex. 7. 9. Find the derivatives of the two functions (a) vers X = 1 — cos X. (6) exsec x = sec x — 1. 10. Differentiate each of the ansioers in the list of formulas. Tables, IV, Ea, Ej. What should the result of your differentiation be ? [The teacher will indicate which formulas should be thus tested.] 11. Find the speed of a moving particle whose motion is given in terms of the time t by one of the pairs of parameter equations which follow ; and find the path in each case : ' X = 2 cos 3 ^ f X = sin « + cos «. (") [?/ = 2sin3f. (^) ly = sin«. ' X = 2 cos 4it. f X = sec t. 2/ = 3 sin 4 «. ^^^^ [ ?/ = tan i 12. A flywheel 5 ft. in diameter makes 1 revolution per second. Find the horizontal and the vertical speed of a point on its rim 1 ft. above the center. 13. A point on the rim of a flywheel of radius 10 ft. which is 6 ft. above the center has a horizontal speed 20 ft. per second. Find the angular speed, and the total linear speed of a point on the rim. VI, § 88] SIMPLE H.\RMOXIC MOTION 155 14. The cycloid (Tables, III, Gi) is defined by the equations sc = a (« - sin <), y - a(l — cost). Find the horizontal and the vertical speeds if t represents the time in the motion of a particle for which these equations hold. Find the total speed ; the tangential acceleration. Find the values of each of these quantities when t = 7r/4. 15. Find the area of one arch of the cycloid. [See Ex. 6, (j) ] 16. Show that the differential of the arc, ds, of the cycloid is ds = aV2 — 2costdt = 2asin((/2). Hence find the length of one arch of the cycloid. 88. Simple Harmonic Motion. If, as in § 85, a point 3/ moves with constant speed in a circular path, the projection Pof that point on any straight line is said to be in simple harmonic motion. Let the circle have a radius a ; let the constant speed be v ; and let the straight line be taken as the cc-axis. We may suppose the center of the circle lies on the straight line, since the projection of the moving point on either of two parallel straight lines has the same motion. Let the center of the circle be the origin. Then we have (1) X = OP=a cos 0, or x = a cos (s/a), where s = arc AM, since = s/a. Moreover, since the speed v is constant, v = s/T, if T is the time since M was at ^; or v = s/{t— to) if t is measured from any instant whatever, and ^0 is the value of t when M is at A. We have therefore g(.-.)]=« (2) x = a cos - = where k = v/a, and e = — 7it^ = — Wo/a. From (2), the speed clx/cU of P along BA is cos[A^< + e] (3) dx d\acoMlct + .)-] ^ _ ^^ gi^ ^^.^ ^ ^^^ dt 156 TRANSCENDENTAL FUNCTIONS [VI, § 88 and the acceleration of P is (4) ^V=^ = -«^'cos(A;« + e)=-A;2.a;, or, (5) jV-^ = ^'-^i« = -^'; that is, the acceleration of x divided by x, is a negative constant, — !2, p. 82.) In the case of vibration, therefore, the acceleration, being proportional to the force, is proportional to the distance, x, from the central position ; it follows that, in ordinary vibrations, the relative acceleration is a negative constant, — negative, because the acceleration is opposite to the positive direction of motion. For this reason, each particle of a vibrating body is supposed to have a simple harmonic motion, unless disturbing causes, such as air friction, enter to change the result. Neglecting such frictional effects temporarily, the distance x from the central position is, as in § 88, X — a cos (kt + e) = A sin kt + B cos kt, where t denotes the time measured, from a starting time to seconds before the particle is at x — a, and where e = — tok. Moreover, from § 88 and also from what precedes,* The quantity a is called the amplitude, 2 tr/k is called the period, and to = — e/k is called the phase, of the vibration. 91. Waves. Another important application of S. H. M. is in the treatment of wave motions. Thus the form of a simple vibration of a stretched cord or wire is assumed to be y = asiuyTT, , * Electric vibrations follow this same law if the resistance is negligible. If V represents the electromotive force in volts, d^v/dt'^= —k'h, where k is a constant. The sudden discharge of an electric condenser by a good conductor would give such an electric vibration. But tlie etTect of the electric resist- ance (which corresponds to the friction in mechanical vibrations) is very marked, and the vibrations die out with extreme rapidity. 158 TRANSCENDENTAL FUNCTIONS [VI, § 91 where I is the total length of the cord between the fixed ends and n is the number of arches in the wave. y ^ V ^ v^ X ^ ^ ^ , ^ S^ x *> J _ - _ _ _j y = sin^-7rfor ?=5?i; i.e. y = sin -— . I 5 Fig. 39. A compound vibration of such a stretched cord is thought of as made up by combining several such simple vibrations simultaneously : 2/ = ai sin ^?^ IT + ffla sin ^^ TT + + ap sin "P^ TT. An alternating electric current varies with the time in a similar man- ner ; for a simple alternating current, C = a sin ^t, where C is the current in amperes and t is the time measured in seconds from a time when C = ; or the sum of several such terms for a com- pound current. In general, a sum of several simple harmonic terms : a\ sin {kxt -\- ei) -|- as sin (^•2^ + 62)+ •■• 4- «;, sin {kj,t + e^) is called a compound harmonic function. See Tables^ III, F. EXERCISES XXXrV. — SIMPLE HARMONIC MOTION — VIBRATIONS 1. Find the speed and the acceleration of a particle whose displace- ment X has one of the following values ; compare the acceleration with the original expression for the displacement : (fl) « = sin2«. (e) a; = sin2 « -|- 0.15sin6«. (6) a; = sin (//2 - 7r/4). (/) x =sini;- ^sin3«-F ^sin 5«. (c) a; = sin « — I sin 2 1. (g) x = a sin {kt + e) . (d) r = cos « -f i cos 3 1. (h) x- A cos kt + B sin kt. 2. Determine the angular acceleration of a hair spring if it vibrates according to the law e = .2 sin 10 irt ; what is the amplitude of one vibra- tion, the period and the extreme value of the acceleration ? VI, § 91] SIMPLE HARMONIC MOTION 159 3. Show that each of the following functions satisfies an e(iuation of the form d-u/dt- + k-u = or d^u/dt- — k'-ii = ; in each case determine the value of k : (ffl) ti = lOsmSt. (/) ?( = 5cos(j is equal at any Fig. 40 « instant t to the value given by a formula like (2) in which a diminishes, the relative rate of decrease in a being + 1. The curve is shown in Fig. 40 (c) ; it may be obtained by drawing the ordinates in (1) multi- plied by the corresponding ordinates in (2). ^ — — — — y y-- -- 1 s ll, t N / r \ y r N / \ t \ y \ y V y \ -J \ 7/- fip't sin Kl -\~~---------------- :]-::S;:::;::::::7 ::;==:=====:===:== Fig. 40^ 162 TRANSCENDENTAL FUNCTIONS [VI, § 92 Likewise, (4) y = ae-*' sin {kt + e) is a damped vibration, wliich may be written (5) y = A sin {kt + e) , wliere A = ae-K Here A is a variable decreasing amplitude, whose relative rate of decrease is — dA/dx -i-A—b; that is, the relative rate of decrease of A is constant.* The successive derivatives of y, by (4), are: ^ = rte-6' [- b sin(A-« + e)-\-k cos {kt + e)], ^ = ae-" [(6- - ^•■') sin (A.-« + e) - 2 hk cos (^•« 4 e)], whence it follows that Equations which contain derivatives are called differential equa- tions ; thus (6) is the fundamental differential equation for damped vibrations. EXERCISES XXXV. — DAMPED VIBRATIONS 1. Each of the following equations represents a damped harmonic vibration ; find the speed and the acceleration in each case ; and write an equation connecting the acceleration, the speed, and the value of y. Draw the graph of each equation. {a) y -e-*sva.2t. (d) y = 2 e-io« cos 5 «. (6) y = e-2< cos 4 t. (e) ?/ = 2 e-5« sin (2 t + v/S). (c) y = 5e-^sin"t. (/) ?/ = 4 e-i«cos (3«-5 7r/12). * In common language, this is often expressed by saying that " the vibra- tion dies away regularly," or "fades out uniformly." The fact that the relative rate of decrease of A is constant is the fundamental assumption. t The equation (6) is often obtained directly and solved to obtain (4) as in Chapter X ; the assumptions made in this work are equivalent to the assump- tion just mentioned, — that the relative rate of decrease of A is constant; this assumption is really the fundamental one, and its reasonableness is the real justification of the assumptions made when (6) is obtained first. The term in dy/dt, or v, proportional to the velocity, occurs only when " damping" Cor friction) is considered. A similar equation governs electric vibrations. VI, § 93] INVERSE TRIGONOMETRIC FUNCTIONS 163 2. The factor e-'' produces a more rapid damping effect. Draw y — g-'^sini, and compare it with y — e-' sin t. Find the speed and the acceleration in each case. 3. The factor (1 + «2)-i (Example 2, p. 1G5) produces an effect similar to that of the factor e-«'. Draw y —{\-\- 1-)-'^ sin t ; find the speed and the acceleration. 4. Show that y = e~''sin t satisfies the equation d-y/dC^ + 4 t (dy/dt) + (3 + if^)y = 0. 5. Draw the curve y = sech t-s'mt; compare it with y = e-''sin t ; find the speed and the acceleration. 93. Inverse Trigonometric Functions. Since the equations * (1) y = sin X, x = sm~^y (=arc sin y) are equivalent, it follows that rXVIl = 2 ctn 2 ^. ^ de Hence ctn i// = 2 when e = v/8, whence ^|/ = 26° 34'. EXERCISES XXXVII.— POLAR COORDINATES 1. Plot each of the following curves in polar coordinates ; find the Talue of ctn f in general, and the value of \p in degrees when ^ = 0, ir/6, 7r/4, 7r/2, tt. (/) p = e. (k) p = sin 2 e. (g) p = e^. (?) p = 2 cos 3 e. (h) p = 1/e. (m) p = 3 sin (3 e + 2 7r/3). (f ) p = e-«. (n) p = 3 cos (? + 4 sin d. (i) p=e-'e, (o) p = 2/(1- cos ^). 2. Show that ctn \p is constant for the curve p = ^•e«*. 3. Show that ctn \{/ for the spiral p = A;^ is greater than ctn \j/ ior p = e^ when e < 1. Hence show that the former winds up more rapidly than the latter, as p ^ 0. 4. Show that if the curves p = e^^ are supposed drawn, for various values of a, any function p =/(^) whose relative rate of change (loga- (a) P- = 4 sin 0. (&) P- = 6 cos ^ — 5. (<') P- = 3 + 4 cos (9 (d) P- = tan 0. (e) P - = 2 + tan^ 0. VI, § 97] CURVATURE 169 rithmic derivative) is variable crosses them ; shov? that the new curve moves across the othei-s away from the origin if its relative rate of change is increasing as increases. 5. Find ctn \p for each of the following curves : (a) p =p/(\ - e cos 0) (conic). (c) p = a(l + cos 0) (cardioid). (6) p-as.ecd±b (conchoid). (d) p^ = 2 a- cos 2 (lemniscate). 6. Find the value of tan « [Fig. 41] in terms of the angles and f. Find tan a for each of the curves of Ex. 1, at the points specified. 97. Curvature. An important application of these formulas consists in finding a more accurate measui-e of the bending of a curve. The Jlexion (§ 45, p. 71), dm _ dx daf' is a crude measure of the bending ; but it evidently depends upon the choice of axes, and changes when the axes are ro- tated, for example. If we consider the rate of change of the inclination a = tan~^ w with respect to the length of arc s, that is, A a da (1) 7 Uiiio HI/ (2) lim A8=o As ds' it is evident that we have a measure of bending which does not depend on the choice of axes, since A« and As are the same, even though the axes are moved about arbitrarily, or, indeed, before any axes are drawn. The quantity da/ds is called the curvature of the curve at the point P, and is de- noted by the letter K: the curvature is the instantaneous rate of change of a per unit length of arc. Since « = tan~^m, and since ds- — dx- -^ di/ (§ 62, p. 107), we have. 170 TRANSCENDENTAL FUNCTIONS [VI, § 97 da = d tan-^?n= 1 +m -dm, ds=Vl + m' dx, where m = dy/dx; hence the curvature K is (3) K da. -ds- --^^-dm 1 + m"" dm dx b V'l + m^ dx (1 + m'f/' (1 + m2)3/2 where b = d^y/dx^ (= flexion), and m = dy/dx (=s\o-pe). It appears therefore that the flexion b when multiplied by the corrective factor 1/(1 + m^f^^ gives a better measure of the bending, since K is independent of the choice of axes. The reciprocal of K grows larger as the curve becomes flatter; it is called the radius of curvature, and is denoted by the letter R: (4,) it=± = ^ = (l + OTt2)S/2 ^ ^ K da b It should be noticed that this concept agrees with the elementary con- cept of radius in the case of a circle, since As = /• Aa in any circle of radius r. Substituting the values of b and m, formulas (3) and (4) may be written in the forms d-y dx2 K: Since Vl + m- = K^l/B^bcossa. dry dx' It is preferable, however, to cal- culate m and b first, and then sub- stitute these values in (3) and (4). sec a the formulas may also be written in the form VI, § 97] CURVATURE 171 It is usual to consider only the numerical values of A' that is I K I, without regard to sign. Since K and b have the same sign, the value of K given by (3) will be negative when h is negative, i.e. when the curve is concave downwards (§ 46, p. 75). The same remarks apply to R, since R = 1/K. EXERCISES XXXVin. —CURVATURE 1. Calculate the coi'vature K and the radius of curvature i? = \/K for each of the following curves : (a) y = x\ Ans. i? = (1 + 4 a-2)3/2/2. (6) y = x3. Ans. i? = | (1 + 9 .r*)3/-76 x|. (c) 2/ = ax2 + 6x + c. Ans. i? = |[1 + (2 ax + 6)2]3/2/2 a\. (d) t/^ = 4 ax. ^HS. i? = (y2 + 4 a2)3/2/4 a2. (e) xy = a2. Ans. R = (3? + y^yi^/2 a^ (/) y = 3 62a;2 _ 2 a;*. (^r) 2/ = 6 62a;2 _ ft^ + x*. (/i) 2/ = sin X. (t) 2/ = cos X — (cos 2 x)/2. U) y = e'. (^-) 2/ = (e^ + 6-^/2 = cosh X. 0-2 ,,2 /3.2 „2\3/2 (m) Vx+V2/ = Va. ^»is. B = (x + 7jy^y(2Va). (n) x3 + 2/3 - «^- ^"s- -?^ = sVaxy. (0) 2/ = ^Ce^''" + e-^''«)/2. ^?is. i2 = 2/VI«l- 2. The center 0/ curvature § of a curve, corresponding to a point P, is obtained by drawing the normal at P and laying off B on this normal toward the concave side of the curve. Show that the coordinates of Q are a = x — B s'lTKf); p = y + B C0B (csc'^ic in 1st or 3d quadrant). [XXII] d vers~^ x = :> (vers~^x in 1st or 2d quadrant). Other rules : See also algebraic forms (p. 40 or p. 52), hyper- bolic function forms (Ex. 4, p. 140, Ex. 8, p. 141, and Exs. 11- 13, p. I(i6) ; Gudermannian (Ex. 10, p. 166). Litegral formulas : See Chapter VII, and Tables, IV, A-H. CHAPTER VII TECHNIQUE — TABLES — SUCCESSIVE INTEGRATION PART I. TECHNIQUE OF INTEGRATION 99. Question of Technique. Collection of Formulas. The discovery of indefinite integrals as reversed differentials was treated briefly, for certain algebraic functions, in Chapter V. We proceed to show how to integrate a variety of functions, but the majority are referred to tables of integrals, since no list can be exhaustive. See Tables, IV, A-H. To every differential formula (pp. 52, 173) there corre- sponds a formula of integration : if d(x) = /(ic) dx then Cfix) due = (oc) + C The numbers assigned to the following formulas corre- spond to the number of the differential formula from which they come. The most important ones are set in black-faced type, except that black-faced type is not used when the formula is easy to remember intuitively. Certain omitted numbers correspond to relatively unimportant formulas. FUNDAMENTAL INTEGRALS mif ^ = 0, then y = constant. [See § 58, p. 99.1 » doc [The arbitrary constant C in each of the other rules results from this rule.] [II]i jk f(x) dx = k^f(x) dx + a [III]i J" \f{x) + {x)\dx = j*/(a;) dx + 1«^ {x) dx + C, 174 VII, §99] FUNDAMENTAL FORMULAS 175 [IV] i Ja;» dx = ^~ + C, ivhen n^-1. (See VIII.) [VI]i uv = Cd (uv) = ju dv + Cv dii + C. [" Parts"] [The corresponding formula [V]i for quotients is seldom' used. See § 103.] [VlIJi i'f{ii)du] . =.r/[^(a-)]rf^(ir) + C x/ J.l« =-(/>(«) «/ [Substitution] = Cf[{x) ] ^?^ dx + C. J * dx [Vlllji p^ = log ar + C. [IX], Jfe«^ dx =e^+C, [X]i j cos X dx == sin « + C [XI], Jsin xdx = -(i<^x+C. [XII]i Jsec2 a; dx = tan a; + C [XIII]i j*csc2a;f?x- = -etna;+a [XlVJi Tsec X tan a- dx = sec x + C. [XV]i J CSC X ctn a; dx = — esc x + C. [XVI]i f-^^^= = sin-i x + C = - cos-^ x + O. [XVII], [XVmji f^^^ = tan-i « + C = - ctn-i x + C. [XIX]i »/ 1 + X" [XX]i r ^^:^^ = sec-i x-\-C=- csc'^ x + C. [XXIJ, •^ X Vx^ — 1 [XXIIJi f—^^-- = vers-i x + O = - covers-* x + C". *^ V2x*— ar^ The remaining differential formulas referred to on p. 173 give rise to other integral formulas ; these will be found in the short Table of Integrals, Tables, IV, A-H. 176 TECHNIQUE OF INTEGRATION [VII, § 100 100. Polynomials. Other Simple Forms. The rules [II], [III], [IV] are evidently sufficient without further explana- tion to integrate any polynomials and indeed many simple radical expressions. This work has b&en practiced in Chapter V extensively. Attention is called especially to the fact that the rules [II] and [III] show that integratior* of a sum is in general simpler than integration of a product or a quotient. If it is possible, a product or a quotient should be replaced by a sum unless the integration can be performed easily otherwise. Thus the in- tegrand (1 + af)/x should be written \/x -\- x\ (1 + ^Y should be written 1 + 2 cc- + aj* ; and so on. This principle appears frequently in what follows. 101. Substitution. Use of [VII]. As we have already done in simple cases in Chapters V and VI, substitution of a new letter may be used extensively, based on Rule [VII]. Example 1. To find ^ ^^ Va- - x^ Set u = x/a, then du = dx/a, ov dx = a du, and r^^=sm- dx _C adu _. .. -"—'„+ (7= sin-i--fC. Va2 - x2 •^ Va-^ - a^u^ Check. d sin-i - —. dx/a _ dx V-S "" V' Va^ - x2 Example 2. To find j sin 2 x dx. Method 1. Direct Sxchstitution. j sin 2xdx = | J sin (2 x) d (2 x) = - | [cos 2 x -|- C] = — | cos 2 x -f- C. Check, d (— I cos 2 a;) = - ^ (? cos (2 x) = -f- 1 sin (2 x) d (2 x) =sin 2 x dx. Method 2. Trigonometric Transformation and Stihstitution. i sin 2x dx = j 2 sin x cos xdx ——2 I cos x d(cos x) = — (cos x)'^ + /r= — cos^x 4- K. Vll, § 102] SUBSTITUTION 177 Notice that cos- a; + ^ = 1/2 cos 2 a; + C since cos 2 x = 2 cos^ x — \. Do not be discouraged if an answer obtained seems different from an answer given in some table or book ; two apparently quite different answers both may be correct, as in this example, for they may differ only by some constant.* Whenever a prominent part of an integral is accompanied by itn derivative as a coefficient of dx, there is a strong indication of a desirable substitution ; thus if sin x occurs prominently and is accompanied by cosxdx substitute w = since; if log x is prominent and is accompanied by (l/x)dx, set u = \ogx; if any function f(x) occurs prominently and is accompanied by df(x), set u=f{x). This is further illustrated in exercises below. 102. Substitutions in Definite Integrals. In evaluating defi- nite integrals, the new letter introduced by a substitution may either be replaced by the original one after integration, or the values of the new letter which correspond to the given limits of integration may be substituted directly without returning to the original letter. Example 1. Compute J ^~o sin x cos x da;. 3Iethod 1. fi=n/2 . rx=TT/2 -I y2-lx=»r/2 SlVl^ Xl'='^/^ \ sinxcosxc?x= \ udu =— ^ ^HLjf J-»=0 Jx=0 Ju=sinx 2j,=o 2 Jx=0 1 "2' Method 2, rx=7T/2 . , ru=l -1 M2-lt4=l 1 \ sm X cos xdx= \ n du = ^ = -, J»=0 J«=0 J«=8inx 2j„=o 2 since ?<(= sin x) = when x = 0, and u = l when x = 7r/2. Care must be exercised to avoid errors when double-valued functions occur. The best precaution is to sketch a figure showing the relation be- tween the old letter and the new one. In case there seems to be any doubt, it is safer to return to the original letter. * Occasionally it is really difficult to show that two answers do actually differ by a constant in any other way than to show that the work in each case is correct and then appeal to the fundamental theorem (§ 58, p. 90). 178 TECHNIQUE OF INTEGRATION [VII, § 102 EXERCISES XXXIX. — ELEMENTARY INTEGRATION SUBSTITUTION 1. Integrate the following expressions : (a) ((l + x)(l+x^)dx. (e) j (e^ + e-'^ydx. (c) j*(a + te)2dx. (g) ((l + 2x)Wxdx. (d) J^^^^^^dx. (h) r(x2-2)(a;i/2 + a;2/3)dx. 2. In the following integrals, carry out the indicated substitution ; in answers, the arbitrary constant is here omitted for convenience in printing. (a) r V2X + S dx; setu = 2x + 3. Ans. u^'^/S =(2x + 3)3/2/3- (b) C-^ = ilog(2a: + 3)=logV2xT3. (c) C ^ =itan-i(2x-f 3). ^ ■' Jl+(2x + 3)2 ^ ^ ^ (d) (x VT^dx ; set M = 1 + x2. Ans. u^/ys = (1 + x2)3/2/3. (6) j'^^=|l0g(l+x2)=l0gVrT^^. (/) fsinxVcosxdx; set tt=cosx. Ans. —2u^/^/S=-2(cosy^x)/S. (^) rcosxViSxdx = 2(sin3/2a;)/3. (h) ^ e^+^'x dx = I e^+-\ (i) Tcos X (1 + 2 sin X + 3 sin2 x)dx = sin x + sin^ x + sin^ x. (j) rsin3xdr.= CsinxCl - cos2x)dx =- cosx +(cos3x)/3. (A:) fcos (2 X + 3) sin (2 x + 3) dx = ^ sin2 (2 x + 3) . (0 rsin(l-3x)cos3/2(l-3x)dx = T2jCos5/2(l-3x). (m) f_^_;setM=-- Ans. - tan-i m = i tan-i - . ^ J a- + x'^ a a a a VII, § 102] SUBSTITUTION 179 3. In the follo^Ying integrals, find a substitution by inspection and complete the integration : («) fT-^=Tlog(3a; + 4)=log(3x + 4)i/3. J 8 X + 4 3 (6) CV\-2xdx = -(1-2 a;)3/2/3. (c) rsin(2x-3)dx = -^cos(2x- 3). (e) I cos'' X dx = sin X — (sin^ x)/3 . (/) Tecs xsinSxdx = (sin* a-)/4. (g) p^^ dz = j (log xy. (ft) j'2xcos(l +x2)dx = sin(l +x2). (i) i tan X sec2 xdx = (tan^ x)/2. (j) C(e2' + e-^)dx =(e'-^dx+(e-'-^dx = (e^' - e-'^) /2. {k) fcos^ .r dx = j* [ (1 + cos2 x)/2] rfx = x/2 + (sin 2 x)/4. (l) fsin2xdx= (*[(! - cos 2x)/2] dx = x/2 -(sin2x)/4. (m) jcoss X dx = sin x — 2 (sin^ x)/3 + (sin^ x)/5. (?j) fctnxdx = i (cosx/sinx) dx = log sin x. (o) jtan X dx = — log cos X = log sec X. dx 1 ■tan-i 2 + x- Sy/2 VV2/ *^V'l2-4x2 2Jv'3-x''^ ^ \V3/ 180 TECHNIQUE OF INTEGRATION [VII, § 102 4. Compute the values of the following definite integrals : «) r'^ = ^tan-^-^1^^ = J-tan-iJ- = ^^. b) r' ''" =sin-i(-^^1-=^^sin-i-L = !r. c) f '^'^i^o = i ^og, (3 + a:2)1'=' = i (log,4 - log,3) = .1438. d) r^' ^lfl_ :=-v/23T^1^'^-(Vl-V2)^.4142. -'xM) V2 - X2 Jx=0 e) \'~ ' sin^xdx = ["— cosx + (cos3a;)/3T"'''' =5/24. /) C^\^e'' dx = e^ysl'"^ = e/3 - 1/3 = .5728. •^1=0 Ji=0 >^x=i ^ J^=o6 + 2x2 Cx=Tl% /• 1=77/4 f) i sin^xcosxdx. (m) ( cos^xdx. rx=:-ir/6 /•x=TT/i j) \ sin2xdx. (n) \ Jx=0 •^z=0 cos^xdx. x=0 5. Find the area under the witch y = l/(a + bx^) for a = 9, & =1, from X = to X = 1 ; for o = 8, 6 = 2, from x = 1 to x = 10. 6. Find the volume of the solid of revolution formed by revolving one arch of the curve y — sinx about the x-axis. 7. Find the area under the general catenary y = a cosh (x/a) = a(e^''«4-e-*/'»)/2 from X = to X = ffl. 8. Find the area of one arch of the cycloid x = a{9 — sin^), y = a{l — cos^). 9. Find the volume of the solid of revolution formed by revolving one arch of the cycloid about the x-axis. VII, §103] INTEGRATION BY PARTS 181 10. Compare the area of one arch of the curve y — sinx with that of one arch of the curve y = sin 2 a; ; with that of one arch of j/ = sin2 x. 11. Show how any odd power of sin.x or of cos a; can be integrated by the device used in Ex. 2, (j). 12. Show how any power of sin x multiplied by an odd power of cos x can be integrated. 103. Integration by Parts. Use of Rule [VI]. — One of the most useful formulas in the reduction of an integral to a known form is [VI], which we here rewrite in the form vdu [VI'] \udv = uv— j called the formula for integration by parts. Its use is illus- trated sufficiently by the following examples : Example 1. J x sin a; dx. Put u — x,dv = sin x dx ; then du = dx and r = J sin X dx = — cos x ; hence, J X sin X dx = — x cos x + J cos x dx = — x cos x + sin x ; (check). Example 2. J log xdx. Put log x = u, dx = dv; then dzi = (l/x)dx, V = x, and flog xdx=x logx— rx---dx = X logx— fdx =xlogx— x +C; (check). ^ X Example 3. J Va-—x- dx. Put u = Va^—x'^, dv = dx; then v=x, and du = -^=3; dx, and \ Va-^ - x^ dx = x Va^ - x^+ \ , ; Va^ - x2 -^ -^ Va2 - x2 but, by Algebra, C ^^ =- C V^^TZT^^ dx + f-^^^ ; 'hence a^dx Va2 - ic'- dx = xVa^ — x^ + j — = This important integral gives, for example, the area of the circle x2 + 2/2 — (i2, since one fourth of that area is 182 TECHNIQUE OF INTEGRATION [VII, § 103 EXERCISES XL.— INTEGRATION BY PARTS 1. Carry out each of the following integrations : a) ix Gosxdx = X sin x + cos a; + C. 6) j xe^ dx = e^(a: — 1) + C [Hint, u = x, dv = e' dx.] c) Ix log xdx=— xV4 + (x2 log x)/2 + C. d) rx2 log xdx = - xV9 + (x3 log x)/3 + C. e) fxV d-K = e^(x2 _ 2 X + 2) + C. [Hint. Use [VI] twice.] /) I sin-i X dx = X sin-i X + Vl — x^ + C. [Hint. M = sin-ix.] g) \ tan-^ xdx = x tan-i x — log(l + x-)'^'^ + C h) Cxs tan-i x dx = (x^ tan-i x)/3 - xV6 + log(l + x-y^ + C i) i x(e^ — e-*)/2 dx = | x sinh xdx = x cosh x — sinh x + C. j) rx2 e2x (^a; = e2^(x2/2 - x/2 + 1/4) + C. k) fe^ sin x dx = e^(sin x — cos x)/2. [Set m = e=^ ; use [VI] twice.] I) Ce^"" cos 2 X dx = e^{2 sin 2 x + 3 cos 2 x)/13. m) I e-^ sin 4 X dx = — e-"^(4 cos 4 x + sin 4 x)/17. «) I e"^ cos MX dx = e«^(n sin nx + a cos nx)/(a'^ + n'^) . 2. Show that | P(x) tan-i x dx, where P(x) is any polynomial, re- duces to an algebraic integral by means of [VIJ. Show how to integrate the remaining integral. VII, §103] INTEGRATION BY PARTS 183 3. Show that i P (x) log x dx, where P(x) is any polynomial, re- duces to an algebraic integral by means of [VI]. Show how to integrate the remaining integral. 4. Express | x" e<" dx in terms of | x"-i e<" dx. Hence show that I P{x)e'"dx can be integrated, where P{x) is any polynomial. 6. Carry out each of the following integrations : («) fci 4-2x-x2) logxcZx. (c) r(.>;2-2.T + 3)e-2'dx. (6) r(3a:2 + 4x-l)tan-ia:rfx. {d) ("(xs - 5)e*^dx. 6. From the rule for the derivative of a quotient, derive the formula ( {l/v)du = u/v + I {u/v-)dv. Show that this rule is equivalent to [VI] if u and v in [VI] are replaced by 1/v and u, respectively. 7. Integrate i e^ sin x dx by applying [VI] once with u — e^, then with M = sin x, and adding. 8. Integrate i e"" sin nx dx by the scheme of Ex, 7. 9. Find the values of each of the following definite integrals : (a) C^logxdx. (d) r^\l +3x2) tan-ixrfx. {h) J^^'xe-dx. (e) r*^^ V2^ cos 3 X f?x. (c) \ sin-i X dx, Jx=Q (/) 1^^%' - e-) dx. 10. Find the area of one arch of the curve y = e~' sin x. 11. Find the area beneath each of the following curves : (a) y = xe-', (6) y = x-e-', (c) 2/ = x^e~^, from x = to x = 1. 12. Compare the area beneath the curve y = log x from x = 1 to x = e with the approximate result obtained by using the adaptation of the prismoid formula, Ex. 6, p. 128. 13. Show that the sum of the area beneath the curve y = smx from X = to X = A; and that beneath tlie curve y = sin-i x from x = to x = sin k is the area of a rectangle whose diagonal joins (0, 0) and (k, sin k). 184 TECHNIQUE OF INTEGRATION [VII, § 104 104. Rational Functions. To integrate any polynomial, rules [II], [III]) [IV] are sufi&cient. When the integrand is rational, but fractional, the formulas [VIII] and [XVIII] are required. Example 1. /(^ 2x 1 \dx = x^ — log x-\-2, arc tana; + C. X 1 + x^J Moreover, it may be necessary to prepare the expression for integration. Example^. ^^2.^ + ar- + 2.-1^3^_l 2 ^ D Q^ + X X l+x" The method of preparation is as follows : Any rational frac- tion N/D in which the degree of N is greater than or equal to the degree of D may be replaced by an integral quotient Q to- gether with a proper fraction R/D, where R is of lower degree than D. This results from ordinary algebraic division, where Q is the quotient and R is the remainder. Thus, N^ 2x*-\-x^ + 2x-l ^^^ x^-2x-\-l D x^-\-x x" + X The rational fraction R/D can then be broken up, by algebra, into a sum of proper partial fractions, whose denominators are the real factors of the first or second degrees of D, and integral powers of these factors. If D has only simple factors, there are just as many proper partial fractions as there are factors, each partial fraction taking as its denominator one factor. Thus in Example 2, the factors of D are x and x^ + 1; hence we write B _ x^-2x + 1 ^A J Bx+ O D x^ + x X \+x^ where A^ B, C are at present unknown constants.* * Notice that the uumerators inserted are just one less in degree than the correspouding denominators : this is because it is known that the resulting partial fractious will be proper fractions. The numerator should be written in the most geueral form for which the fraction is a proper fraction. VII, § 104] RATIONAL FUNCTIONS 185 Clearing of fractions, we find comparing the coefficients of like terms, A + B=l, C=:-2, A = l, whence ^ = 1,B = 0, C=-2 aud - - = ^"-2^ + 1 =1 + _JIl1_ , (Check by addition) D a^ + x X 1 + a;2 ^ ^ whence the example can be completed as above. If D consists of one factor raised to an integral power, and if the degree of R is not less than the degree of that factor, then B/D can be simplified still further by ordinary division, using the factor to the first power as a divisor. Example 3. Given - = ^x + 2 ^^ ^^^ C R ^^ D (x+l)2 JD Dividing i? by (x + 1), we find x+l~ x + 1 x + l' hence ^ = -^^ ■ — !— = -^ ; (check). D x + l x + l x + l (x + l)2' ^ ^ Therefore r|dx=r-^-r-^^ = 31og(x + l)+^ + C; {check). J D J X + 1 J (x + 1)-^ ^ ^ X + 1 Example 4. Given ^ ^ 3:^ + 2x2 + 2 ^^ ^^^ C^ ^^^ D (x2 + 1)2 ' J D i Dividing B by (x^ + 1), we find R = x + X2+1 " ■ x2+r D (X2+1) (x2 + l)2' and (^ dx - f -^-^ + {^^ - r ""^^ JD Jx2 + l^J.r2+l J(X2+1)2 = I log (x2 + 1) + 2 tan-1 x + J ; {check). 186 TECHNIQUE OF INTEGRATION [VII, § 104 If, finally, D has several factors, some repeated and some simple, we proceed as before to make as many distinct proper partial fractions as there are distinct factors, using as denomi- nators the simple factors as before, and the repeated factors raised to the same poiver as that in zvhich they occur in D.* Example 5. Given ^ = i^i±l^i±l^i + l^±^ , to find (^dx. D (3:k + 2)(x2 + 1)2 Jd Set — = a , 'bv^ + cofi + dx + e D 3x + 2 (x2 + l)2 clear of fractions, and equate the coefficients of like powers : a + 36 = 4, 26 + 3c = 8, 2« + 2c + 3cZ=6, 2d + 3e = 6, a + 2e = 5; whence a = 1, 6 = 1, c = 2, d = 0, e = 2 j and B^^^}_ x^ + 2x^ + 2^ D 3 X + 2 (x2 + 1)2 The integration is then completed as in Ex. 4. EXERCISES XLI. — RATIONAL FRACTIONS Carry out the following integrations : ^Jx2-l 2JVx-l x + iy 2 "x+l c) r_^^ 1 log^ZlJ?. J X- — a- 2 a x + a d) r ^^ =r ^ = ltan-i^+i. ^ J x2 + 2 X + 5 J (a; + 1)2 + 4 2 2 ') f ., , f , = tan-i(x + 2). J X- + 4 X + 5 * This simple rule, together with the algebraic reduction by long division just mentioned, is perfectly general and is always successful whenever the denominator D can be factored. VII, §1041 RATIONAL FUNCTIONS 187 ^ Ja:-'2 + 2x-3 J (a: + 1)2 _ 4 4 "x + s' C/i) r (?x _ 1 ^p„ 2x-2 _ 1 ^Q X- 1 J4x-^ + 4x-8~12 °2x + 4~12 ^x + 2' 2. In the following integrals, first prepare the integrand for integra- tion as in § 104 ; then complete the integrations. (a) r_^^^ll_cZx = log(x - 2) + 2 log(x-3) = log [(x-2)(x-3)2]. ^ X- — 5 X + b (P) f -"^ "" V'V ^ = 3 logx - |log (3 X + 6). -^ X + X'' 6 (c) i • "^ fZx = X + tan-1 x. ^ X- + 1 (d) f , ^~^ dx = log Vx2 + 2 X + 2 - 2 tan-i(l + x). J X- + 2 X + 2 (e) r ^:^ =J__tan-i-^-ltan-i*. ^ ^ J X* + 7 x-^ + 12 V3 V3 2 2 ' ^ Jx3 + 1 ^ ^ Jx2 + 10x + 25 (i) r 8x8-36x2-2 (2x + l)(2x-5) x2 - 20 dx. 1) (a;2 _ 4) x2-45 2 x3 - 18 X dx. J x3 + x^-- 2 X ^ X2 + X + 1 2x + r 2x2-x + 3 ^,. J X + X2 + X3 . r dx ^^ Jx3 + x2 + X+l' r T^-dx . ^^ J:(.-* + x2-2 ^ JX2-X J (X + 1)" '^L- (fx. 3x =Ca:+l;2 dx. 188 TECHNIQUE OF INTEGRATION [VII, § 104 3. Derive the following formulas : (a) j ^- r = — — ^ log- (rtx + b) {mx + n) an — hm ax + h (^'fe X dx _ 1 J (x + aY _ a)(x + 6) ~ a-b "^ [x + by' xdx _ b 1 V'x^ + a . Va t„Q-i « (c) f ^i?^!^ _=_A_log X + 6 a + 62 ^- 4. Derive each of the formulas Nos. 18-24, Tables, IV, A. 5. Evaluate each of the following definite integrals : p28-3x-x2^^^ p-i5_^^+3x2_3^3 ^ ^ Jx=C/5 (5 X - 2)3 [Note. Further practice in definite integration may be had by insert- ing various limits in the previous exercises.] 6. Carry out each of the following integrations after reducing them to algebraic form by a proper substitution : r sinx ^^ (r^^^dx. (c) f-^ ^ ^ J 1 + cos2 X ^ ^ J 4 - sin2 x ^ W 1 - e' (d) fsecx^x^r ""^^ dx = llogl±^''^. ^ ^ J J I- sin2 X 2 ^ 1 - sin X (e) jcsc X dx. (g) i csch x dx. (j) i f ,.. fsinxcosx ,^ .,.-. re-' — e-^ (/) Jsech X dx. (0 J i^,,,3^^ ^^- (^) J ,-7^:7="^ ^^^^^-^ — dx. tan X — tan2 x dx. 105. Rationalization of Linear Radicals. If the integrand is rational except for a radical of the form -x^ax -f b, the sub- stitution of a new letter for the radical, r — Vax + b, renders the new intesrrand rational. VII, §106] QUADRATIC RADICALS 189 Example 1. Find ( ■'' + ^^ + ^ dx. Setting r = Vx + 2, we have x = r- -2 and dx = 2rdr; hence J 1 + a; J ,.- — 1 J r + 1 ?-dr = 2J'fr + 1 ---^Vzr = j-2 + 2r-21og(r+ 1)-|- G = X + 2+ 2 Vx + 2 -21og ( Va-. + 2 + 1)+ C. The same plan — substitution of a neiv letter for the essential radical — is successful in a large number of cases, including all those in which the radical is of one of the forms : a-^/", (ax + by/'^, /^^^f^J+l^Y'; where 7i is an integer. Integral powers of the essential radical may also occur in the integrand. 106. Quadratic Irrationals: Va+bx± x-. If the integral involves a quadratic irrational, either of several methods may be successful, and at least one of the following always succeeds : (A) If the quadratic Q = a-\-bx±x- can be factored into real factors, we have ±x_ VQ = ^{a + x){^3±x) = (a + x)^J^^^ ^ a + x and the method of § 105 can be used. The resulting expres- sions are sometimes not so simple, however, as those found by one of the following processes. (B) If the term in x^ is positive, either of the substitutions VQ = t-\-x, ^Q = t-x, will be found advantageous. One of these substitutions may lead to simpler forms than the other in a given example. 190 TECHNIQUE OF INTEGRATION [VII, § 106 (C) Completing the square under the radical sign throws the radical in the form VQ = V±k±{x± cY; the substitution x ±c=y certainly simplifies the integral, and may throw it in a form which can be recognized instantly. Example 1. Let VQ = Va^ ± a^ ; show the effect of substi- tuting ^Q = t — x. If Va?±a- = t — x, we find ^=^'-='-^'*'^— ^=^^ and the transformed integrand is surely rational. Carrying out these transformations in the simple examples which follow, we find (i) J V^c^±^ J\ 2f 2t J J t ^ ^ = log (£c + VQ) + C, where Q = x?±a^. ai) fv^..^J'lffJ:±f.u=fa.£^f^y. ^^'±|log« = ^±fl»«(xWO)H-C. These integrals are important and are repeated in the Table of Integrals, Tables, IV, C, 33, 45 a. Many other integrals can be reduced to these two or to that of Ex. 3, p. 181, or to Rules [XVI] or [XX] by process (C) above. VII, §106] QUADRATIC RADICALS 191 EXERCISES XLU.— INTEGRALS INVOLVING RADICALS 1. Verify the following integrations : (a) ^xVTT^x dx =^^=^ (1 + x)3/2. V'2 - x - 1 ^'^'^7= = log- ) \/2 — X \/2 - X + 1 2 tan-Vx-2. (x-1)- ■ W f -^ (x-1 (0 f ^i-= -^ (x-DVx-i (,) C ^^ ^-l,tan-iA/«^±I. •^ (ax + 2 6)\/ax+ 6 aVft ^ (/) r dx ^ Vx + 1 I 1 jQg Vx + 1 + 1 (g) f (« + to)3/2(?x = A (a + 6x)5/2. ^ 5 & 2. Carry out the following integrations : («) f-^- (&) f^l^- (c) f -' x-Vx - 1 •' Va -X '' •^ Vx + 1 -^ Vl - X -^ (x + 2) v^^T (^)i'S^3-- Wj*^.^^- « 1(3^3- (,•) 1^^ .X. (.) J^^^x. CO J^ ^x. ^^•J^'x + Vx ^' -^^^^^ + 2 ''J^x + 1 V ax + 6 192 TECHNIQUE OF INTEGRATION [VII, § 106 3. Carry out the following iutegrations by fii-st making an appropri- ate substitution : (a) f-^ J 1 + e=" (0 (^ dx. 2Vcosa; ^xdx 3 tan X id) 1^^ (e) r+'^ cosxdx. '^ 1 — Vsin x ,^x r sin X dx ^' ' J (2-3cosx)3/2' , 4. Substitution of a new letter for the essential radical is immediately successful in the following integrals : (a) i xVl + x'-^dx a-s d: ^ \/2+2x+x2 •^ Vl + x2 ^ ^ J (a + 5x-^)3/2 (0 |x(,+x=)-.x. (/) St^^^^, 5. Cany out the following integrations : («) r-4^ = log(x+V^;2Zl). Va + hx^ dx hx^ (6) r ^^'^ =iiog ^-^"+^'- (c) r — ^£^,^=i^iz«. (d) r ^^£^ •^ (1+2X2)V1 +X2 -^ Vl - X2 Vx2 + 1 dx = sin-1 X — Vl — x2. (/) j'.^^L^^^^V2tan-i- ^ V4x^- I *^x\/a2_x2 •V2 <'•' J^ dx. W _r ^^^^^- (,;) f '^-==. (I) (^^- -^(x-l)Vl-x2 *^ (l + x)Vl + x-^ •^Vx^ + 4 VII, § 106] QUADRATIC RADICALS 193 6. The following integrations may be performed by the methods of § 106; note especially method (C), which consists in completing the square under the radical. (a) r '^^ = Iog(l + 2 .r + 2 Vx- + x + 1 )• •^ Vx- + X + 1 (^,) r ^^ ^sin-i^^^. *^ Vl +x — X- V5 (c) f ^^ = sin-i ^— ^ = vers-i x [ + const.]. •^ V2 (Tx — x2 a •^ ./-Vl + X + X- ^ (.) r ^^ ^Isin-i^::^. •^xV3x2 + 4x-4 2 2x -^ V2x--^ + x+l -^ Vl+x-2x'-2 -^ \/l-2 (?X + 1 -^ Vl+x-2x'-2 -^ Vl- (0 r ^^'^ — • (0 f Vl + X + x^ dx' •^ VO X - x"^ - 5 •' (;) r ^-^ (m) r V;3 X-' + 10 x + 9 dx. •^xVx-^ + 2x + 3 *^ ^'^ i'(x + 4)V^T3¥^4* ('^) I'^v/BT^^^dx. 7. Integrate by " Parts," [VI], the following integrals : (a) i X sin-i x dx. (d) i (3 x — 2)sin-i x dx. (l.) J^-^^Zx. (.) J'i^^^os-ixrfx. (c) r.r cos-ixdx. (/) r(sin-ix + 2xcos-ix)dx. 194 TECHNIQUE OF INTEGRATION [VII, § 106 8. Show that \ P(x) s'm-^xdx reduces by means of [VI] to an in- tegral whose integrand contains no other radical than Vl — a;^, if P{x) is any polynomial. 9. Show that by means of the substitution a; = sin ^ the integrals (dx/Vl-x^ and (dd are equivalent. 10. Reduce the following integrals to trigonometric integrals ; then complete the integration : i) r(l + a^)(? 3g =(*(!+ sin 6) dO, if a; = sin Ans. 6 — cos 6 = sin-i x — v 1 — x^. (J)) C da: ^ f-^, if x = sing. Ans. tan 6 = (c) (xVx:^ - 1 dx = Itan^ 6 sec^ 6 dd, if x = sec 0. Ans. (tan3 0)/3 = (x^ - l)yyS. •^ Vl + a;2 -^ sec tan ^ d^, if a; = tau 0. Aiis. sec tf = Vl + x^. 11. Reduce the following integrals to algebraic integrals ; then com- plete the integration : fa) ( — ^ — =f ^^' ,iix = sin 0. [See 5 (/i).] J 1 -sing -^ (l-x)Vl-x-'' (6) Csec edd= ( ^•'' , if X = sec 0. [See 5(a).] •^ "^ Vx- — 1 (• secgdg ^ r dx , if a: = tang. [See 5(d).] ^^JH-2tan2g J (i + 2x-^)VrT^^ « JttI^- <" SVT^. <-^' J"fT sec g d0 VII, § 108] TABLES OF INTEGRALS 195 107. Elliptic and Other Integrals. If the essential radical in the integrand is the square root of a cubic or of a polynomial of higher degree, or a cube root or higher root, the integrals are usually beyond the scope of this book. If the only irrationality is VQ, where Q is a polynomial of the third or fourth degree, the integral is called an elliptic in- tegral. While no treatment of these integrals is given here, they are treated briefly in tables of integrals, and their values have been computed in the form of tables.* See Tables, V, D, E. 108. Binomial Differentials. Among the forms which are shown in tables of integrals to be reducible to simpler ones are the so-called binomial differentials : I (ax" -f 6) "a;"' dx. It is shown by integration by parts that such forms can be replaced by any one of the following combinations, where u stands for (okc" + 6) : (1) r Wx"" dx = (^i) mPx'"+i -f (Si) r jfP-'iC" dx, (2) j* u-^x^dx = (A) ?^p+ia;'"+^ + (Bs) j* u^+^x'" dx, (3) f mpx"* dx = (As) t S"ii^ dx. ips + l dx. dx. (a; -2)2 Jx*-3xa + 3x2-x .,. f Sx^- nx + 21 ^^ .. r 7a:2 + 7x-176 p3_2x2 + 7x + 4^_ r 0:^-3x4-3 ^^ ^^J (x2-l)2 ^•'Mx3-4x2-7x + 10 2. (a) r ^^ + \ dx. (e) r_6^i^^. r27xMx. r^-d^. ^ ^ J 2 + 3x2 ^"^ J2 + 5x* ^^^ Js-IT^- ^'^ Jx(3 + 5xe)- 3. (a) rxVT+^t^x. (c) T-^dx. (.) (■^=^.<'- «o I— :M=. J V3 X + o -' xVx - a ^ 4. (a) rx\/^^r4dx. (c) )- „, ^^, • ^ ^ J -^ V(a + 6x)p (6) rx\/a + ?>rdx. (d) f-^^^- *^ ♦^3\/x + 2 ;va + bxdx. J oX (h) Tl + Vx - V^ 3f7x 5 — 7 X + 2 x2 x2dx •^^ J5 + 2X + X2 ^ J X2 + 4 X 4- 2 x2 dx + 2x /) j*x2V5 5 + 2 X dx. ) rxv'3x + 7dx. V3-X dx. 1+Vx (0 f , -J'\^ •^ v'l + X + VI + , •^ Vl +X + Vl+; 198 TECHNIQUE OF INTEGRATION [VII, § 109 J -i/n J- h'r J (g) ra;3(a + x2)i/3 5.* (a) r — ^^ ^ ^ J (X2 + 5)3 6.* (a) y J Vx^ — a f \ C x^dx ^""^ J (-7 +4x3 dx x3 v'x^ — 4 x^cZx 7.* (7+4x3)2/3 (a) jsin* X dx. "^ ^ J2 + sin^ iff) fcos^ a da. Va + bx n) CA±^dx. (p) (xy/a + bx:^dx. ! (r) rx5(l +x3)i/3dx. ^ J (x2 + 3)3 ^ ^ J(x2 + 2x+5) r 5x-3 ^^ r_i2x+l)_<^ ^ J (2x2-1)-^ J (x2+6x + 10)3 dx '^1 tan3 X dx. _de^ 2 - 3 cos « (c) j sin2 X cos* X dx. 5sin^ {h) rctn2 3xdx. (i) fsinS/s^cos^^ ^ ^ Jo (x-2)2 Jo2x2-3 Jo V4-x2 ^^^ 3i (2x-l)3''''- •^2^V2^+^ -^1 VxM^ ^''^ Jo 2x2 + 3 ^2 Vi^^ITs Ji Vx2 - 1 ^•''' Jo (2 X + l)(x2 + 2) J-2 v/x2 - 8x dft ^ VII, §109] GENERAL EXERCISES 199 9.* Find the values of the followiug definite integrals by using the tabulated numerical values : Tables, V, A-H : Xx=1.5 rx=\A /•i=4.1 1 (.9) T'"" ^— ^t_^ dx = p"^' cosh X dx. (A) p~"sinh.rdx. (i) ('^' ^•'' == cosh-i x1"^". (j) r^^ <^-^ =sinh-ix1^\ XT=3.6 /7r /J.-] 1=3.6 /^^- = cosh-1 ? /•i=14.4 /7r 7.-|ar=M.4 (0 i ^-^ = sinh-1 ^ Jx=fl Vx2 + 9 3j^=o (m) r^^° <^g i^(i^ 30°), Tables, V, D. Je=fl° vi - (V-1) sin2 tf (?i) r^*'° Vl-(l/4)sin2^f7^ = £r (^, 45°), Tafe/es, V, E. r^^' ^^ _. (^,) f-^°°V l-.25sin^^ dg. ^^15° Vl - .04 sin-^ d •^«=^° ^ X=o Vr^^ Vl - .25 x2 *^«=*' VI - .25 sin2 6 =^ , if X = SU1 (r) f" ^ .1 1 - .SQx^ dx = r^^° Vl - .36siu2^d^, if x = sin 6. ^ ' Jx=l/2\ i_3.2 J9=30» ^x=l/2 Vl - X-^ Vl - .49 X2 •^x=V'272 ' 1 - X^ [Note. Many of the exercises in Lists XXXIX-XLII may be used for additional practice in use of the tables.] 200 TECHNIQUE OF INTEGRATION [VII, § 109 10.« Show that the even powers of sin x can be integrated by reduction to the integral | sin^ x dx. 11. Show that odd powers of cos x can be integrated readily without a table. 12.* Show that any power of tan x can be integrated by reduction to I tan X (Zx or to i tan^ x dx. 13. Show that any even power of sec x can be integrated by splitting off one factor sec^ x and then using the relation sec^ x = 1 + tan^ x. 14. Show that \x'^e''dx can be integrated by the repeated use of Rule [VI]. Hence show that j P(x)e^ dx can be integrated, if P (x) is any poly- nomial. 15. If \ f{x)dx = (p(x) show that \f(x) tan-ixdx can be reduced to ^[0 (x)/(l + x'^)] dx by Rule [VI]. State a similar result for the integral | / (x) sin-^x dx. 16. Show that the integrals which result from breaking up a rational fraction whose denominator has only simple linear factors can be expressed in terms of simple powers and logarithms. 17. Show that a ■ simple quadratic factor in the denominator of a rational fraction gives rise to a term in the final answer which contains an arc tangent or a logarithm. 18. Show how to integrate terms of each of the following types, and show that no others arise in integrating rational fractions : ■' ^ ^ J ax + b ^ ^ J x'^ + a' ^•'^ J ar^ + hx + c (6) r ^^ . (.)* C ^^ + S dx. (hr r -^^ + ^ dx. (c)* r ^' • (n*C ^^ + B dx. CO* r ^^ + -^ dx. VII, §110] IMPROPER INTEGRALS 201 PART ir. T^IPROFER AXD MULTIPLE INTEGRALS 110. Limits Infinite. Horizontal Asymptote. If a curve approaches the a^axis as an asymptote, it is conceivable that the total area between the x-axis, the curve, and a left-hand vertical boundary may exist; by this total area we mean the limit of the area from the left-hand boundary out to any vertical line X = m, as m becomes infinite. Example 1. The area under the curve y=e^'' from the y-axis to the ordinate x — m is J 1=0 t/i=0 ■ dx = 1 — 6" As m becomes infinite e""* approaches zero; hence i i . 2/=^- Y qi ^5 '^. //kS-4y ^ U J X=--in A\ = j e~' dx= lim j -1 x=0 t/x=0 „,=x,/^=0 and we say that the total area under the curve y = e a; = to a; = + oc is 1. Fig. 44 e-'(?a;= lim (1 — e"'") = 1, from Example 2. The area under the hyperbola y = 1/x from a; = 1 to a; = m is a\ ^^ = f"^"'^ = log x\ ^" = log m. Jx=l c/x=l a; Jar=l As m becomes infinite, log m becomes infinite, and I -| x=m 1 lim 1-1 = lim log in does not exist ; hence we say that the total area between the a;-axis and the hyperbola from a;= 1 to a; = co does not exist* * This is the standard short expression to denote what is quite obvious,— that the area up to x — m becomes infinite as in becomes infinite. This result makes any consideration of tlie area up to ^ = x perfectly useless ; hence the expression " fails to exist," which is slightly more general. 202 GENERALIZED INTEGRALS [VII, § 111 111. Integrand Infinite. Vertical Asymptotes. If the func- tion to be integrated becomes infinite, tl\e situation is precisely similar to that of § 110 ; grapliically, the curve whose area is represented by tlie integral has in this case a vertical asymptote. Itf(x) becomes infinite at one of the limits of integration, jc = 6, we define the integral, as in § 110, by a limit process : Xx=l> /•6-c f(x) dx = \im. I f(x) dx. A similar definition applies if f{x) becomes infinite at the lower limit, as in the following example. Exam2:)le 1. The area between the curve y = l/-\/x and the two axes, from a; = to cc = 1, is ^1 ^^ = r^'—^dx = limf r^'— =da;1 J ^=0 *^x=o Va; ^^ [yx=c -Vx J = limr2\/^1" =limr2-2Vc =2. y V = X- i \ ^\ - 'y/\. ^/^z s y^. __ c X Example 2. The area be- tween the hyperbola y = 1/x, the vertical line x = 1, and the two axes, does not exist. For, I -dx== log x\ = — log c, Jx=c X Jx=c but lini (— log c) as c = does not exist, for — log c becomes infinite as c = 0. VII, § 112] IMPROPER INTEGRALS 203 Examples. The area between the curve y = l/-^x—i, its asymptote x = 1, and the line x = 2 is f --^^ = lim r-^^ = ?lim(l_c^/3)=3. Jl ^X-1 c^uJl+c^/x-1 ^e=y, ^2 112. Precautions. It is dangerous to apply limits of in- tegration between which the integrand becomes inhnite or is otherwise discontinuous. Example 1. Show that I 1/x^dx does not exist. The ordinate y = 1/xr becomes infinite as x approaches zero, i.e. the y-axis is a vertical asymptote. Hence to find the given integral we must proceed as in § 111, breaking the original integral into two parts : Jx=c ar x_\x=c c ii=f-i..=-il~-=i-i. *^'x=-i or a;Jx=-i c ^ I t\ >y. k; \ >'<>. :^ V/A |f^ -4 ■ /: \ y \ y = 7 'i / Vj - — -^ - 1 J 1 The limit of neither exists since 1/c becomes infinite as c = 0; hence the given integral does not exist. Carelessness in such cases re- sults in absurdly false answers ; thus if no attention were paid to the nature of the curve, some person might write : Fig. 40 Jz=-1 Jx=-\ X- ^ ' X J^_i (sic!)- 1-1 = which is ridiculous (see Fig. 46). The only general rule is to follow the principles of §§ 110- 111 in all cases of infinite limits or discontinuous integrands. Such integrals are called improper integrals. 204 GENERALIZED INTEGRALS [VII, § 112 EXERCISES XLIV.- IMPROPER INTEGRALS "Verify the following results : . ri dx . j determinate if n < 1, Jo a?* \ non-existent if n ^ 1. ) C" ^^ =2\/a. (o) P ^^ is non-existent, -^o V^^^ A52X-3 M f^ dx ^ Ji.5(2x-3)» y/a- *2 f^x ,■„ f determinate if n < 1, non-existent if n ^ 1. •i dx State a similar rule for f {hx + A:)" f" rfx ^^. (;;.) r_ ^^^^ =§Zg!. Jo V^i^TT^ 2 ' ' Jo V^^^:^ 8 . C ^dx _^ n) C+^ dx is non-existent. ^ Jo vr^^2 J 1 x'^ + 5 X -t- 4 2. Show that the integrals j ' tan x dx, j'cotxdx, J secxdx, Jo X 3. Verify each of the following results : j ^^ rdx 1 /„N p t?x _ g ") ji x^ = 2- ^^ Jo (l + x)3/2-2- 6) r — is non-existent. (-f^) C^ — '^ — is non-existent. ^ Jl ^ ^ ^ Jo (l-fx)2/3 r" dx • f determinate if n > 1, Jo (1 -t- x)" I non-existent if 71 -^ 1. ^' rrf;;r»=r ® I, . f" dx IT /.oo ^^ Ja ^M^ = 4^* 0) jo ^-^^^^ = ~ i? dx is non-existent X non-existent. VII, § 112] IMPROPER INTEGRALS 205 4. Determine the a,rea between each of the following curves, the x- axis, and the ordinates at the values of x indicated : (a) 2/3(x _ 1)2 = 1 ; X = to 9. Ans. 9. (6) xy-{\ + .<■)- = 4 ; x = to 4. Ans. 4 tan-i 2. (c) ?/'-x*(l + x) = 1 ; X = to 3. ^ns. qo. ((?) x22/2(x2 - 1) = 9 ; X = 1 to 2. Ans. 2 jr. (e) 2/3(x-l)'- = 8x3; x = 0to3. Ans. 9S/2 + 9/2. (/) x2r/2(x2 + 9) = 1 ; X = 4 to ». Ans. % log 2. ((x) is asymptotic to the y-axis, and if /(x) > 0(a;) ^ 0, show that J /(x)(?x cannot exist unless ^^(f>{x)dx exists. Hence show that J^x-2rfx does not exist by comparing it with jjx-i dx. 6. If each of two curves y =/(x) and y = <^(x) is asymptotic to the X-axis, and if /(x) > 0(x) ^ 0, show that J f(x)dx cannot exist unless J^ x" for large values of X by showing that J*e' ■ e-' dx does not exist. (a) j'^'-dx = l. (c) j|"x2e-dx = 2. (^) r (ft) P .re-' dx = 1. (d) C^^^'^-=|^^^]n x=i 4 (1-A), hence this volume, by (1), is ""-A' 2/Ja=o Notice that h, during the first integration, was essentially constant. Notice also that the volume of the wedge is one third the volume of the circumscribed rectangular parallelopiped ; and that since ^5 is a linear function of A, the prismoid rule (§ 71, p. 125) gives the volume precisely. Combining the formulas used in this example, the volume Fmay be written V~\ '^' = C^'\2 (^^ (1 - h) V^ clAlh = ? . Ja=o »/A=0 [^ ^x=0 J 3 Such successive integrals in two letters are very common in all applications. 208 GENERALIZED INTEGRALS [VII, § 114 EXERCISES XLV. — SUCCESSIVE INTEGRATION 1. Determine a function y = f (x) whose second derivative d-y/dx- is 6 X. Alls. y = xs+ Cix + Co. 2. Determine tlie speed v and tlie distance s passed over by a particle whose tangential acceleration d-s/dt- is (3 1. Find the values of the arbi- trary constants if v=0 and s=0 when t=0; if v=:100 and s=0 when t=0. 3. Find the general expressions for functions whose derivatives have the following values : (a) d'-y/dx^ = 6 x^. (d) d'^r/dd^ = 1/ VI^. (p-) d^y/dx'-^ = e^ (6) d2s/d«2 = 1 + 2 «. (e) dV/d^s = ^2 _ 2 ^. (/i) d'^s/f^^" = sec^ t. (c) d2s/d«2=Vl-«. (/) d%/rfw3 = 1 + M'-2. (j) d^dir^ = l/w'-. 4. Determine the speed v and distance s passed over in time t, when the tangential acceleration jy and initial conditions are as below : (a) Jt = sin ^ ; w = and s = when « = 0, (6) jj.— t + cos ^ ; ■?; = and s = when « = 0. (c) jj. = VT+1 ; V = 3 and s = when « = 0. (d) ^V = ^/Vl + t^ ; V = 1 and s = when t = 0. 5. Evaluate each of the following integrals, taking the inner integral sign with the inner differential : (h) \ \ Gx^{l-y)dydx. (g) \ \ {x+y)dydx. r- p=3 (^2^1) (4-2/2)d2/ dx. (/t) r=' r'=''^' (X + ?/)'^ dydx. Jx=0 Jy=2 Jx=0 Jy=\ (d) r~ r^ Vrt + u du dv. (j) ^ P'' p"'"^(x + ?/+2)d2 di/d.r. 6. Find the volume of the part of the elliptic paraboloid 4 x^ + O y'^-Zd z between the planes z = and z = \; between the planes z = a and z =.b. VII, §114] SUCCESSIVE INTEGRATION 209 7. Find the volume of the part of the cone 4 x- + 9 y- = 36 s- between the planes 2 = and z = 2 ; between z = a and z = b. 8. Find the volume of the part of the cylinder x"^ + y^ = 25 between the planes z = and z = x ; between the planes z = x/2 and 2 = 2 x. 9. A parabola, in a plane perpendicular to the x-axis and with its axis parallel to the 2-axis, moves with its vertex along the x-axis. Its latus rectum is always equal to the x-coordinate of the vertex. Find the volume inclosed by the surface so generated, from 2 = to 2 = 1 and from x = to x = 1. 10. Find the volume of the part of the cylinder x'^ + 2/2 = 9 lying within the sphere x^ + y- + z- = 16. 11. For a beam of constant strength the deflection y is given by the fact that the flexion is constant : b = d-y/dx- = const, if the beam is of uniform thickness. Find y in terms of x and determine the arbitrary con- stants if y = when x = ± 1/2. [This will occur If the beam is of length 1, and is supported freely at both ends.] 12. Determine the arbitrary constants in the case of the beam of Ex. 11, if y = and dy/dx = when x = 0. [This will occur if the beam is rigidly embedded at one end.] 13. For a beam of uniform cross section loaded at one end and rigidly embedded at the other, b = d^y/dx^ = k{l — x) where I is the length of the beam, x is the distance from one end, and A; is a known constant which is determined by the load and the cross section of the beam. Find y in terms of x, and determine the arbitrary constants. 14. Find y in terms of x in each of the following cases : (a) d^y/dx:^ = Jc{r^ - 2 Ix + x'^); y = 0, dy/dx = wiien x = 0. [Beam rigidly embedded at one end, loaded uniformly.] (6) d^y/dx^ = a + bx; y = 0, dy/dx = when x = 0. [Beam of uniform strength of thickness proportional to (a + 6x)-i, em- bedded at one end.] (c) d^y/dx- = ^•(^V8 - x'^/2) ; y = when x = ± 1/2. [Beam supported at both ends, loaded uniformly.] (d) dhj/dx- = ^•/x2 ; 2/ = 0, dy/dx = at x = ?. [Beam of uniform strength of thickness proportional to x^, embedded at 15. Find the angular speed w and the total angle d through which a wheel turns in time ^, if the angular acceleration is a = dr0/dfi = 2t, and if ^ = w = when t = 0. 210 GENERALIZED INTEGRALS [VII, § 115 (1) Fig. 48 115. Double Integrals. It is often convenient to restate such problems as that solved in § 114 in somewhat different form. In obtaining "the area Ag we originally (§ 66, p. 115) cut the area into strips of width Aa;; their length is 2 y each, since they reach from one side of the parabola to the other. We then showed that As=2 \ y doc = 2 lim [Sum of terms like y Ace.] Jx=0 Aa;=0 We may as well proceed to set up both integrations at once, as follows : let us consider the small column whose face is 2 1/ Ax and whose thickness is Ah ; its volume is (2) 2yAxA7i; the volume of the whole layer whose base is As is (3) As- Ah = 2 Ah . r~\j dx =2Ah- lim T"=^ (y Ax), where 2 stands for " the sum of terms like " ; hence (4) As -Ah = 2 lim Ah ■ V'=' (yAx) = 2 lim V ^' (y Ax Ah). The entire volume is, however (§ 70, p. 121) : (5) Fl *^' = f^'Asdh = lim V'^AsAh = 2 lim y*=ilimy^\ Ax Ah). VII, §116] DOUBLE INTEGRALS 211 The expression which occurs in (o) is equal to the double limit which follows ; it is called a double integral, and is denoted by \\ydxdh: («) is2;:::2::^^-^'-Xrxr* dx dh. In the particular example in hand, y = (1 — ^)Vic, and the limits are (h = 0, h = l) and (x = 0, x = l); but the argument was not affected by our knowledge of these values. It follows that the successive integrals mentioned in § 114 are always equal to the double limit in (6), where y is any function F(x, h) of X and h we please : (7) f^r r"^''i^(x,/i)ri^1rZ7i = limT';:^ V^:%A»Afc I Fix, h) dx dh. h = a Jx = c This is the fundamental summation formula for double in- tegrals. In writing it, any letters, not necessarily x and /i, may be used. Moreover c and d may depend on /i, as we shall see in numerous examples. It is used exactly as we have used the original summation formula : quantities we desire to measure often appear most naturally in the forms of approximate double sums like (6). The accurate evaluation is done by suc- cessive integration by means of (7). 116. Illustrative Examples. In this paragraph, several applications of double integration are worked out. These should not be memorized, but rather the formulas should be built up by the student each time they are used. 212 GENERALIZED INTEGRALS [VII, § 116 [A] Volumes by Double Integration.* A problem which is essen- tially the same as that of § 114 is to find the volume under any surface whose equation is given in the form z = F(x, ?/), where F(x, y) is any function of x and y. Consider for example the volume V bounded by the surface, a;5;-plane, the planes x=a &nd x=h and the right cylin- der whose base is a given curve y =f(x) of the a:2/-plane. If the volume is divided into layers by planes parallel to the ys-plane, equally placed at intervals Ax ; and if these layers are them- selves divided into small columns of width Ay, the volume of any one column is approximately z Ay Ax, and the total volume is lira '%-^ = Ay Ax =cz F(x, y)dydx. Thus the volume under the surface z = x^ + y" between the a-^-pkne, the planes x = and x = 1, and the cylinder whose base is y = Vx is Jx=0 Jy= (x2 + y^) dy dx x:'[^ •■y + y-\ dx 3 J»=o =£;■(-- 105* [JB] Area in Polar Coordinates. The area A bounded by a curve whose equation in polar coordinates is p=f{d), and two radii vectors 6 = a, d = ^ is approximated by dividing it into trian- gular strips by radii vectors spaced at equal angles A^. If we then draw cir- cles with centers at 0, equally spaced at intervals Ap, the whole area A is divided into small curvilinear "squares" like the one shaded in ^ig. 50. * Formulas from Solid Analytic Geometry are to be found in Chapter IX. VII, § 116] DOUBLE INTEGRALS 213 The straight line side of one of these is Ap, while the circular side has a length pAe, where p is the value of p along that side. Hence the area of the shaded " square " is, approximatelj^ pApAd and the area to be found is, precisely, lim xzx p=/(e) pApAe \ \ p dp dd. Je=^a Jp=0 The first integration Jp dp can always be performed, since I pdp=p-/2; but it is best not to burden the memory * with this, since it is evident each time such an area is to be found. Thus the area bounded by the curve p = sec 6 (draw it) and the lines d = 0, 6 = 7r/4 is Je=o x: p dp dd : Je=o L2jp=o 1 n e=T/4 1 - tan 6' = . 2 Jd=i> 2 Je=o dd [C] Moment of Inertia of a Thin Plate. The moment of inertia J about a point O of a small object whose mass is m is defined in Physics to be the product of the mass times the square of the distance from O to the object : / = mr'^. Given now a thin plate of metal of uniform density and thickness, whose boundary C is a given curve, let us divide the plate into small squares by lines equally spaced parallel to two rectangular axes through 0. Let P be a point in any one of these squares and let 0P = r = Vx^ ■+ yK Then the mass of the square is k • Ay Ax where k denotes the constant surface density (i.e. the mass per square unit) ; and the moment of inertia of this square about is, approximately, k Hence the moment of inertia / of the entire plate about is : 1, ^"^ ■^ / ' ^I- i / V A/ ' - '?;'; \ / , / J \ / y y-^ ; __ ^ / V - -Ax^ X Fig. 51 r- Ay Ax. lim i=^>^XX^''^-'^y^Si fjf- + y-) dy dx. * If any part of this work is memorized, it should be at most the figure drawn above. 214 GENERALIZED INTEGRALS [VII, § 116 where proper limits of integration are to be inserted to cover the area en. closed by C. If G is an oval, as shov?n in the figure, the limits of y are the values of y along the lower half oval and the upper half ; these must be given in the problem as functions of x. The limits for x are the ex- treme values of x on the two ends of the oval. Thus the moment of inertia of a plate bounded by the two curves y = (1 — x'^) and 2/ = (a:2 — 1), about the origin (draw the figure) is : I=k r^^^ Cy-='^-^' .2 ^ 2/2)d2/ dx = k f'^^Yx^?/ + ^ T"^~"'dx Ji=-1 Jy=x2-1 Ji=-1 L 3 _\y=x2-l = l^C=^\i-x^)ax = mx-^r^' = ^-k, 3 Jx=-i ^ ^ 3 1 7jx=-i 7 ' where k is the surface density. [D] Moment of Inertia in Polar Coordinates. Using the figure drawn for [B], it is easy to see that the moment of inertia of a thin plate of the shape of the area in [B] is : / = lim k . y'-' y'^'''' p^ApM = k. r' r'''' p' ^p ^^' A8^ where A; is the surface density {i.e. mass per unit area) as in [C]. Thus for a circle whose center is O, p=f{e) = a, the radius. Hence, the moment of inertia of a circular disk about its center is : i=k.r'''[^T'^cie=k-r'''-cio=.k^=.^, Je=o L4jp=o Jfl=o 4 2 2 where k is the surface density, and M = kira^ is the mass of the disk. EXERCISES XL VI.— DOUBLE INTEGRALS 1. Find the volume under the surface z = x'^ + y'^ between the xz- plane, the planes x = and x = 1, and the cylinder whose base is the curve y — x^. 2. Find the volume between the xy-plane and each of the following surfaces cut off by the planes and surfaces mentioned in each case : (a) z = x + y cut off by ?/ — 0, X = 0, X = 1, 2/ = Vx. (&) z = x^ + y cut off by ?/ = 0, X == 1, X — 3, 2/ = x^. (c) z = xy cut off by 2/ = 0, X = 2, X = 4, ?/ = x^ -}- 1. VII, § 116] DOUBLE INTEGRALS 215 cut off by J/ = 0, a; = 1, a; = 5, y =r a^. cut ofl by y = 0, X = 0, X = 1, y = x^. cut off by X = 0, y = 1, y = 4, y'^ = x. cut off by a; = 0, y = 2, 2/ = 5, y = X. cut off by y = and y = 1 — x^. cut off by y = x'^ and y = 1. cut off by y = x^ and y = x. cut off by y = x- and y = 2 — x^. 3. Find the volume of the portion of the paraboloid z = 1 — x^ — 4 y^ which lies in the first octant. 4. If two plane cuts are made to the same point in the center of a sircular cylindrical log, one perpendicular to the axis and the other mak- ing an angle of 45° with it, what is the volume of the wedge cut out ? 5. Show that the volume common to two equal cylinders of radius a which intersect centrally at right angles is 16 a^/3. 6. Show that the volume of the ellipsoid x716 + yV9 + 274 = 1 is 32 tt. 7. What part of the ellipsoid in Ex. 6 lies within a cube whose center is at the origin and whose edges are 6 units long and parallel to the 3oordinate axes ? 8. AVhere should a plane perpendicular to the x-axis be drawn so as to divide the volume of the ellipsoid in Ex. 6 in the ratio 2:1? 9. Calculate by double integration the areas bounded by the following Burves : (a) y = X- and y = Vx. (e) x = 0, y = sinx, and y = cos x. • (5) y = x2 and y = x^. (/) y = 0, y2 — x., and x^ — y- = 2. (c) y = x2 and - x^ + y2 = 2. (g) y = 2x, y = 0, and y = 1 — x. (d) x2 + y'^ = 12 and y = x'^. (A) y2 = x, and y = 1 — x. 10. Calculate the moment of inertia of a thin plate bounded by the curves y = x^, y = 2 — x^^ about the origin. 11. Calculate the moment of inertia of a thin plate about the origin, in each of the cases in which the shape of the plate is the area bounded by the curves in one of the parts of Ex. 9. 216 GENERALIZED INTEGRALS [VII, § 116 12. Find the moment of inertia of each of the following shapes of thin plate : (a) A square about a diagonal. About a corner. (6) A right triangle about a side. About the vertex of the right angle. (c) A circle about its center. (d) An ellipse about either axis. About the center. (e) A circle about a diameter. (/) A trapezoid about a line parallel to its parallel sides. 13. Find the moment of inertia of a thin spoke of a wheel about the center of the wheel. 14. Determine the entire area, or the specified portion of the area, bounded by each of the following curves, whose equations are given in polar coordinates : (a) p — 2 cos 6. Ans. ir. (b) One loop of p = sin 2 0. A7is. ir/8. (c) One loop of p = sin 3 0. Ans. ir/l2. (d) The cardioid p =: 1 — cos ^. Ans. 3 ir/2. (e) The lemniscate p^ = cos 2 0. Ans. 1. (/) The spiral p = from ^ = to tt. Ans. ir^/6. (g) The spiral p^ = 1 from = ir/i to ir/2. Ans. l/ir. " (A) p = 1 + 2 cos 6 from tf = to tt. Ans. 3 7r/2. (i) p = tan d from » = to 45°. Ans. 1/2 - tt/S. (j) The area between the nth and (n + l)th turns of each of the spirals in Exs. 14(/), 14 (gr). 15. Calculate the moment of inertia of a thin plate about the origin, for each of the shapes defined by the areas mentioned in Exs. 14 (a)-(O. 16. Calculate the following moments of inertia : (a) A thin circular plate, about its center. (6) A thin circular plate, about a point on the circumference. (c) A thin plate bounded by two concentric circles, about the center. (d) An equilateral triangle, about its center. (e) An equilateral triangle, about one vertex. 17. The square of the radius of gyration p„ of a body, about any point, is its moment of inertia about that point divided by its mass : Pg^ = T^ M. Find the radius of gyration for the example solved in [C], § 110 ; in [Z)], § 116. 18. Find the radius of gyration for each of the thin plates described in Exs. 9, 10, 12, 14, 16. Vll, § 117] MULTIPLE INTEGRALS 217 19. n/(r, y) is any function of x and ij, its average over a region is Average of f{jc, y) = j j/Xx, y)dxdy -e- j j dxdy. Show that the square of the radius of gyration about the origin of a ihin plate is the average value of 7^ = x- + y'^ over the surface of the plate. 20. Find the average value of x over the area described in [C], § 116. p. 214. Find the average value of y over the same area. [Note. The point whose coordinates are the averages of values oVx and / over an area is called the center of gravity or centroid of that area.] 21. Find the centroids of each of the areas mentioned in Exs. 9 and 14. 22. Find, for the area mentioned in [C], § 116, p. 214, the average ^alue of each of the following functions : (a) xy. (b) a;2 + 4 y^. (c) x + y. (d) x^ - y^. 117. Triple and Multiple Integrals. There is no difficulty .n extending the ideas of §§ 113-llG to threefold integrations )r to integrations of any order. Following the same reason- .ng, it is possible to show that, it w — F{x, y, z) lim V'^-' ^y y=d yAr=« ^^ ^^ ^^ -- f"" f^' f^ F{x, y,z)dxdy clz, x/z=e */y=c »Jx=a svhere the three integrations are to be carried out in succes- sion, where the limits for x may depend on y and z, and where }lie limits for y may depend on z : but the limits for z are, of 30urse, constants. Thus it is readily seen that the volume mentioned in (^1), \ 116, may be computed by dividing up the entire volume by ;hree sets of equally spaced planes parallel to the three coor- linate planes. Then the total volume is, approximately, the sum of a large number of cubes, the volume of each of which 218 GENERALIZED INTEGRALS [VII, § 117 is Aa: iyy Az; and its exact value is Az=y) Xx=5 /•.v=/(x) /*z=F(x,xj) I I ds; dy dec, wliiph reduces to the result of \_A], § 116, if we note that X^=F(x, y) '~\'=Fi,x, y) Likewise the moment of inertia / (see § 116, [C]) of the same volume with respect to the origin is approximately the sum of terms of the sort h^x^ -\- y^ -\- z^) Ax A?/ Ag where k is the density (mass per unit volume) ; whence the exact value of / is * Xx=6 /».v=/(x) /^z=F{=c,y) I I {j? + y' + z')dzdy =a %/y=0 t/z=0 dx. EXERCISES XLVII.— MULTIPLE INTEGRALS 1. Determine the volume bounded by the surface z = {x + yy, the coordinate planes, and the plane x ■\- y -{- z = \. 2. Write each of the volumes mentioned in Ex. 2, and in Exs. 3-6, List XLVI, as a triple integral ; show that one integration reduces the triple integral to the double integral used before, in each instance. 3. Find the volume of the sphere by triple integration. 4. Write down the moment of inertia about the origin of each of the solids bounded by the surfaces mentioned in Ex. 2, and in Exs. 3-6, List XLVI. Actually carry out each of these integrations. 5. Write down the moment of inertia of a right cylinder of height I whose base is any one of the areas mentioned in Ex. 9, List XLVI, about an axis through the orighi parallel to the elements of the cylinder. Show that one integration reduces the integral essentially to the double integral used in List XLVI, in each instance. * It is well to urge that such formulas should not be remembered, but obtained in each exercise by the simple reasoning used above. ^11, § 118] AVERAGES — CENTERS OF GRAVITY 219 6. The square of radius of gyration of a solid about a point (or about I line) is the moment of inertia divided by the total mass. Find its value :or each of the solids mentioned in Ex. 4, above ; for each of the figures nentioned in Ex. 12, List XLVI. 7. Find the average value : Average of /(x, y, z) = i \ j/(x, y, z) dxdydz ^ j j idxdydz, )t each of the following functions, over the region mentioned in Ex. 1 : (a) /(«, y, 3) = X. (d) f(x, y, z) = xyz. (g) /(.r, y, z) = x^ + z^. (b) f(x,y,z)=y. (e) f{x, y, z) = xy. (h) f(x,y,z)= x'^ + y^. (^c) f{x,y,z)=z. (/) /(x,y,3) = a;2+2/2. (i) f{x,y, z) = x + y -]-z. [Note. The point whose coordinates are the three values given by (a), 6), (c), is called the center of gravity, or centroid of the volume.] 8. Find the centroid of the solid mentioned in Ex. 3, List XLVI. 9. Show that, in spherical coordinates (p, 6, 0), the volume of a solid 3 given by an integral of the form | | \ p^ sin 6 dd d

Jo (■?) ji VTT^ (a;+l)Vi^^ d) j'_";cos2.xsinxdx.(^.) r^^d». (O ^l"^^^- -'' Vm — 2 e) I a^logx-dx. ^,v p <^^ (s) I xcosxcte. *^' ^ -^ J« at- + 2bt + c' ^" 7) |;; (cos 2 x)2 «^30o V5 - 2 sin2 a; Jo Vl^^ VT^^ 4. Show that n dx 1 p dx Jo 1 + 2 X cos <^ + x2 2 Jo 1 + 2 X COS (^ + x2 2 sin ' and explain what occurs when 0=0, and when ^ = ir/2. 5. Integrate the following general integrals; where /'(x) denotes the derivative of /(x). (a) j'/(x)/'(x)dx. id) jr(2x)dx. (&) j"e*Ax)/'^x)dx. («) ij^^^' C ' ff\ ( f'{x)dx (c) J/'(cosx)sinxdx. U) J i + [/(x)]2* 6. "Verify the result of integrating sin^ x (?x by comijaring it with the integral of cos^ u du by means of the substitution u — 7r/2 — x. 7. Evaluate each of the following integrals : (a) I i &\n{u-\- v)dudv. (6) t I se'dtds. 8. Calculate the area ^, between the x-axis and the curve y = x^ — 9 x^ + 23 X — 15, from x = 1 to x = 3, by direct integration and also by Simp- son's Kule. Find the centroid (x, y) of the same area. VII, § 118] GENERAL EXERCISES 223 9. Proceed as in Ex. 8 for each of the following curves, between the limits stated below : (a) y = 1 + X — X- + x^ ; x = to a; = 2. (b) y = a(l — x-/b-); 1st quadrant. Ans. A = 2b/S; x = S 6/8, y = 2 a/5. (c) y = x/(l + x^); X = to a; = 1. Ans. ^ = (1/2) loge 2; x = 0.6192, y = 0.2059. (d) y = (e« + e-''x)/2 a = (1/a) cosh ax ; x = to x = k: (e) The sine curve ; one arch. Ans. A = 2 ; x = 7r/2, y = tt/S. (/) The cycloid ; one arch. Ans. A = 3ira'^; x = air, y = 5 a/0. (g) a;2/3 + j/2/3 - a2/3[or x = a cos^ t, y = a sin^ t]; first quadrant. ^«s. 3 7raV32 ; x = y = 256 a/(315 n). Qi) x = a sin i + ft tan t, y = a cos < ; < = to « = ir/i. Ans. A = a\v + 2)/8 + ab log tan (.Stt/S). (i) X = 2 a sin2 0, j/ = 2 o sin^ (/> tan (f> ; between the curve and its asymptote. Ans. A =3 ncfi ; x = b a/3 ; j/ = 0. 10. Find the areas bounded by each of the following curves, or the part specified : (a) p = ad^; one turn. (ft) p = a cos d + b. (c) p = a sin cos ^/(sin^ ^ + cos^ d) [folium] ; the loop. Ans. a^JQ. {d) ±x + Va2 - y- = a log [(a + Va- - y-)/?/] [tractrix]; above y = 0. (e) y"^(a — x) = x^ [cissoid] ; to its asymptote z = a. 11. Find the volume generated by revolving each of the following curves about the line specified : (a) y = 5x/(2 + 3x); about y = 0; X = tox=l. Ans. 1.5558 ■••. (ft) 2 x2 + 5 y2 = 8 ; about y = ; total solid. Ans. 64 7r/15. (c) y"* = ax" ; about y = ; (0, 0) to (x, y). Atis. mny^x/{2 n + m). (d) y = b sin (x/a); about y = 0;x = Otox = ir. (e) y = a cosh (x/a); about y = 0; x = to x = a. (/) (■'■ — «)^ + 2/'" = '''^ > about X = ; total solid. ((j) The cycloid ; about base ; one arch. Ans. bir^a^. {h) The cycloid ; about tangent at maximum ; one arch. Ans. wa^. (i) The tractrix ; about asymptote ; total. Ans. 2 ira^/S. ( j) X = a sin « + ft tan t, y = a cos t ; about y = 0; 1 = to t = t. Ans. 7r[a'(sin t - (1/3) sin^ t) + a^ftf]. {k) y"(a — x) = x^ ; about asymptote ; total solid. Ans. 2 v^a^. {I) X = a cos* t, y = a sin* t ; about y=0 ; total solid. Ans. 32 wayiOb. 224 INTEGRATION [VII, § 118 12. Obtain a formula for the volume of a spherical segment of height Ji. 13. Show that the volume of an ellipsoid of three unequal semiaxes, a, 6, c, is 4 7ra6c/3. 14. Show that the volume bounded by the cylinder a;2 + y2 _ ^^x, the paraboloid x^ -\-y^= bz, and the xy-plane is (3/S2)('ira*/b). 15. Find the volume common to a sphere and a cone whose vertex lies on the surface and whose axis coincides with a diameter of the sphere. 16. Describe the solid whose volume is given by each of the following integrals ; and calculate the volume : (a) C C^"'-^' C^^'dzdydx. (c) f"" C Cdzdydx. Jo Jo Jo Jo Jo Jy ("> £1^"^'^' '''""•'■ ("> i'i/^ £"'■""■■■ 17. Show that the a;-coordinate of the center of gravity, or centroid, of any frustum of any solid is : «= ( x\ \ \dydz\dx-^V= \ As-xdx-^ \ Asdx, if As is the area of a section perpendicular to the x-axis, Fis the total volume, and x = a and x = b .are the truncating planes. State similar formulas for y and z. [Note. The integral ( ( ixdxdydz is often called the moment (or the first moment) of the solid about the yz plane.] 18. Find the centroid of each of the following frusta : * (a) Of the paraboloid x^ + y^ = 4caz by the plane z = c. Ans. i = 2 c/3. (6) Of a hemisphere. Ans. z = S r/8. (c) Of the upper half of the ellipsoid of revolution 4x- + 4y^ + 9z^ = 36. (d) Of the upper half of the ellipsoid x^ + 4y'^ + 9z- = 36. (e) Of the solid of revolution formed by revolving half of one arch of a cycloid about its base. Ans. x = ira/2 + 64 a/ (45 tt). 19. Show that, if Ag is any quadratic function of x, in Ex. 17, the moment of the volume about the yz plane is x-V= [aB+bT+2{b-a)31^ (6 - «)/6, where B, T, M denote, respectively, the areas of cross sections by a; = «, x~b, x = (b — a)/2. [Compare § 71, p. 126, and Ex. 3, p. 128.] VII, § 118] GENERAL EXERCISES 225 20. Find the lengths of the arcs of each of the following curves, between the points specified : (rt) y^logx; x-atox = b. Ans. f vT+x"-— log {( vT+x2 + l)/x}l . (6) e* cosx = 1 ; a: = to X = X. -'" (c) x^t^, y = 2at (or y- = 4 a%) ; t = tito t = t2. Ans. [tVa^ + t~ + a2 log (« + Va^ + f^)V^ . (d) One arch of a cycloid. Ans. 8 a. (e) p = a (1 + cos 6) [cardioid] ; total length. Ans. 8 a. 21. Find the moment of inertia and the radius of gyration (Ex. 6, List XLVII) of each of the areas mentioned in Ex. 9, about the origin. 22. Calculate the moment of inertia / for a right circular cone about its axis. Ans. (3/10) mass • square of radius. 23. Calculate the moment of inertia and the radius of gyration for the rim of a flywheel about its axis, the inner and outer radii being Bi, i?2- Ans. Mass (iiJi^ + i?2-)/2, y/iB{^ + Bi^)/2. 24. The moment of inertia of an ellipsoid about any one of its axes is (1/5) (mass) (sum of the squares of the other two semi-axes). 25. Calculate the moment of inertia for a spherical segment about the axis of the segment. 26. Show that, for any body, 2 /q = /- + /„ + 7^, where /o, 4, /„, I^ denote respectively its moments of inertia about a point and three rectan- gular axes through that point. 27. Show that for any figure in the xy-plane, I^—Ii + Iy, where /j,, 7„, [^ denote its moments of inertia about the three coordinate axes respectively. 28. Show that the total pressure on a rectangle of height h feet and width h feet immersed vertically in water so that its upper edge is a feet below the surface and parallel to it, is G2.4 hh(a + h/2). Show that the depth of the center of pressure is at (6 a- + 6 ah + 2 h^)/{Q a + 3h). 29. Show that the total pressure on a circle of radius r, immersed vertically in water so that its center is at a depth a+r, is 62.4 irr'^{a+r). Show that the depth of the center of pressure is a + r + r^/ {i r + 4 a). 30. Show that the total pressure on a semicircle, immersed vertically in water with its bounding diameter in the surface, is 41.6 r^. Show that the depth of the center of pressure' is 3 7rj'/16. 31. Show that if a triangle is immersed in a liquid with its plane verti- cal and one side in the surface, the center of pressure is at the middle of the median drawn to the lowest vertex. Q 226 INTEGRATION [VII, § 118 32. Show that if a triangle is immersed in a liquid with its plane verti- cal and one vertex in the surface, the opposite side being parallel to the surface, the center of pressure divides the median drawn from the highest vertex in the ratio 3:1. 33. Calculate the mean ordinate of one arch of a sine-curve. The mean square ordinate. [^Effective E. M. F. in an alternating electric current.] 34. Calculate the average distance of the points of a square from one corner. 35. What is the average distance of the points of a semicircular arc from the bounding diameter ? 36. When a liquid flows through a pipe of radius E, the speed of flow at a distance r from the center is proportional to B^ — r^. What is the average speed over a cross section ? What is the quantity of flow per unit time across any section ? 37. The kinetic energy E ot a, moving mass is lim ]^Am • v^/2, where Am is the element of mass moving with speed v. Show that for a disk rotat- ing with angular speed w, E = w~I/2. Calculate E for a solid car wheel of steel, 30 in. in diameter and 4 in. thick when the car is going 20 m./hr. 38. Show that the kinetic energy J? of a sphere rotating about a diame- ter with angular speed w is (l/5)(mass)r2w2. 39. Calculate the kinetic energy in foot-pounds of the rim of a flywheel whose inner diameter is 3 ft,, cross section a square 6 in. on a side, if its angular speed is 100 R. P. M. and its density is 7. 40. The x-component of the attraction between two particles m and m'. separated by a distance r, is (k ■ m ■ m'/r^) cos (r, x) where cos (r, x) de- notes the cosine of the angle between r and the x-axis. Hence the x-com- ponent of the attraction between two elementary parts of two solids M and M' is (^- • AM- AM'/f') cos (r, x). Show thUt the total attraction between the two solids is expressible by a six-fold integral. 41. A uniform rod attracts an external particle m. Calculate the com- ponents of the attraction parallel and perpendicular to the rod ; the re- sultant attraction and its direction. [Hint. Let A3/ be an element of the rod ; then AF= kAM ■ m/r"^ is the force due to AM acting on m, r being the distance from AM to m ; then the components of AF are AX = AFcos « and AY — Ai^'sin a, where a is the angle between r and the rod. Hence X = J^cosa, and r = j*^sina.] 42. A force at attracts a particle at P proportionally to the nth power of the distance OP. What \d the average force from Pi to P2 ? CHAPTER VIII METHODS OF APPROXIMATION PART I. EMPIRICAL CURVES INCREMENTS INTEGRATINCx DEVICES 119. Empirical Curves. Some of the methods used in science to draw the curves which represent simultaneous values of two related quantities and to obtain an equation which represents that relation approximately are given in Analytic Geometry. Usually the pairs of corresponding values are plotted on squared paper first ; in all that follows it is assumed that this has been done in each case. 120. Polynomial Approximations. It is advantageous to have equations which are as simple as possible. From experi- mental results, it is not to be expected that absolutely precise equations can be found, and the attempt is made to get an equa- tion of simple form which approximately represents the facts, in so far as the facts themselves are known. One simple kind of function which often does approximately express the facts is a polynomial : (1) y = a-\-'bx + ca? + cW-{- h kx". 121. Review of Elementary Methods. If the points lie reasonably close to some straight line, it is usual to assume 71 = 1 in (1), § 120, whence y = a-\-hx\ then h (the slope) and a (the ^/-intercept) may be found by direct measurements in the figure, or by one of the more general methods which follow. 227 228 APPROXIMATION [VIII, § 121 If the curve has the typical form of a parabola, it is advan- tageous to assume that the equation is of the form (2a) y = a-\-hx + cx\ or (26) (ji - B)= C(x - Af and then apply the methods of Analytic Geometry to find a, b, c, or A, B, C. One of the methods most often used is to find a, b, c, by assuming that the curve actually passes through three given points (see Ex. 2, p. 230). Another method that can be be used whenever the vertex of the parabola is clearly indicated, is based on the fact that (A, B) are precisely the coordinates of the vertex, and can therefore be measured directly. The value of C, which is all that remains to be found, can be obtained approximately by a variety of methods : one may lay over the experimental figure a sheet of transparent (tracing) paper on which the curves ?/= kx^ have been drawn for a large number of values of Ti-. or one may proceed as in § 122 ; or, finally, as in § 124, below.* In general, the equation (1) contains ?i + 1 unknown coeffi- cients. To obtain these values, it is possible to use any n-\-l points on the experimental curve, as in Analytic Geometry. In doing so, it is preferable to take, not the precise figures given by the experiment, but rather pairs of coordinates of points on a free-hand curve sketched into the figure. General formulas for the values of the coefficients have been worked out, and are given in the Tables, II, I, 17, under the name Lagrange's Interpolation Formula. In the theory of probabilities, formulas are derived (which are to be found in any large set of mathematical tables) for the most probable values of the coefficients a, b, c, d, etc. These formulas can be applied by any person even before studying the theory. See Tables, II, D, 4. * In any method, judgment on the part of the exi^erimenter is the final means of decidin^'whether the equation obtained will approximately repre- sent the facts. The amount of error which may exist in the experimental measurements is, of course, fundamentally important. VIII, § 122] EMPIRICAL CURVES 229 A few simple problems have been solved already by one of the methods of probabilities : in Exs. 18-23, p. 69, we assumed a formula of the type y = kx, and found A; by the requirement that the sum of the squares of the errors should be a minimum. This method is called the method of least squares; see also Example 2, § 105. 122. Logarithmic Plotting. The preceding forms of equa- tions may not represent the facts very well unless a large num- ber of terms of (1), § 120, are used. If the first graph resembles one of the curves y = x-, y = a^, y = X*, etc., ov y = o^'-, y = o^'^, etc., or y = l/a*, y = 1/x^, etc., it is advantageous to plot the common logarithms of the quantities measured instead of the actual values of those quantities. If X and y represent the quantities measured, and u = logioX, v = logioT/ are their common logarithms, the values of u and v may lie very nearly on a straight line, (1) V = a-\- bu, where a and h are found as in § 121. Then from (1), since u = logio X, V = logio y, (2) logio y = a + h logio X = logio ^' + logio ^ = logio (^"a;^), where logio ^' = « ; hence (3) y = kx^. This form of equation is very convenient for computation and is used in practice very extensively wherever the logarithmic graph is approximately a straight line.* This work applies equally well for negative and fractional values of h. * To avoid the trouble of looking up the logarithms, a special paper usually described in Analytic Geometry may be purchased which is ruled with loga- rithmic intervals. No particular explanation of this paper is necessary e.Kcept to say that it is so made tliat if the vahies of x and ?/ are plotted directly, the graph is identical with that described above. To secure this result the 230 APPROXIMATION [VIII, § 123 In many cases where the process just described fails, it is sometimes advantageous to assume that the equation has the form (y—B) = k{x—A)" which evidently has a horizontal tangent at the point (A, B) ii n>l, or a vertical tangent if n < 1. If the first graph (in x and y) shows such a vertical or horizontal tangent, that point {A, B) may be selected as a new origin, and the values x' = x — A and y' — x— B should be used ; thus we would plot the values of u = logio x' = logio (x-A), v = logio y' = logio (.y - B), in the manner described above. The values of A and B are found from the first graph (in x and y) ; the values of k and n are found from the log- arithmic graph as above, 123. Semi-logarithmic Plotting. Variations of this process of § 122 are described in Exercises XLIX below. In par- ticular, if the quantities are supposed to follow a compound in- terest law, y = fce*"^, it is advantageous to take logarithms of both sides : log^^ y ^ log^^ k + bx log^ e, and then plot m = x, v = logm?/ ; if the facts are approximately- represented by any compound interest law, the experimental graph (in u and v) should coincide (approximately) with the straight line v = A-\- Bu, where A = logio ^ and 5 = 6 logjo e. After A and B have been measured, k and b[=B log^ 10 = 2.303 B] can be found. EXERCISES XLIX. —EMPIRICAL CURVES: ELEMENTARY METHODS 1. Find the equation of a straight line through the points (—1, 3) and (2, 5); through (2, - 3) and (4, 5). 2. Determine a parabola whose axis is vertical, through the three points (0,3), (2,-1), (5, 8). [Hint : Assume the equation in each of the forms y = ax'^ + bx + c, y — B = C (x — A)'^ ; check the answers by comparing them.] successive rulings are drawn at distances proportional to logl(=^0), log 2, log 3, •• from one corner, both horizontally and vertically. Explanations and numerous figures are to be found in many books ; see, e.g., Kent, "Mechanical Engineers' Pocket Book " (Wiley, 1910) , p. 8.5 ; Traut- wine, "Civil Engineers' Pocket Book" (Wiley), (Chapter on Hydraulics). VIII, § 123] EMPIRICAL CURVES 231 3. Determine a cubic function of x which takes on the values — 10, — 2, 6, 20, respectively, when a; = — 1, 0, 1,2. 4. Determine n and c so that the curve y = ex" passes through the two points (1, 2) and (3, 54); through (1, 3), (4, 6); through (1, 3), (8, 12). 5. Plot the data of Ex. 18, List XIV, p. 69 ; draw a straight line as closely as possible through the points without giving a preference to any one ; determine the equation from this graph ; compare it with the result obtained in List XIV. 6. Proceed as in Ex. 5 for each of the cases in Exs. 19-23, List XIV. 7. Assuming the data of Ex. 1, § 124, p. 235, find graphically the equation connecting / and w and compare it with the result found in § 124. 8. Assuming the data of Ex. 2, § 124, sketch a parabola whose axis is parallel to the axis of ; determine its equation ; compare the result with that of § 124. 9. Find a parabolic curve of the second degree which coincides vrith y — sin X at the points where a; = 0, x =: ir/2, x = ir. Compare the areas under the two curves. 10. Proceed as in Ex. 9 for each of the following curves, taking the values of x specified in each case : (a) y = logio X, X = 1, x = 5, x = 10. (b)y = &', x=— 1, x = 0, x = +l, (c) y = tan x, x = 0, x = ir/S, x = ir/G. {d) y -x^ — 7 x + 2, x-0, x = 2, x = 4. 11. Find a parabolic curve of the third degree through four points taken at equal horizontal intervals on the curve y = sin x, between x = and X = ir/2. Compare the areas under the two curves. 12. Find a parabolic curve of the second degree which coincides with y — sin X at X = and x = 7r/2, and which has the same slope as y = sin x at X = 0. 13. Find a polynomial of second degree which, together with its first and second derivative, coincides with cos x at x = 0. 14. Proceed as in Ex. 12 for the curve y = e'. 15. Find a cubic which, together with its first three derivatives, coin- cides with each of the following functions when x = : (a) sinx, (6) tanx, (c) e*, C^^) 1/(1 + x). 232 APPROXIMATION [VIII, § 123 16. Plot each of the following curves logarithmically, — either by plot- ting logio X and logio ?/, or else by using logarithmic paper : (a) y = 2 x^. (c) y = A x^-^. (e) y = 5.7 x«. (6) 2/ =3x1/2. (d)y = 3x-^. (/) 2/ = - 1.4x2-4. 17. In each of the following tables, the quantities are the results of actual experiments; the two variables are supposed theoretically to be connected by an equation of the form y = kx": Draw a logarithmic graph and determine k and n, approximately : (a) [Steam pressure ; w = volume, p = pressure.] [Saxelby]. V 2 4 6 8 10 p 68.7 31.3 19.8 14.3 11.3 (6) [Gas engine mixture ; notation as above.] [Gibson.] 3.54 4.13 4.73 5.35 5.94 6.55 7.14 7.73 8.04 141. 115 95 81.4 71.2 54.( 50.7 45 (c) [Head of water ft, and time t of discharge of a given amount.] [Gibson.] h 0.043 0.057 0.077 0.095 0.100 t 1260 540 275 170 138 (d) [Heat conduction, asbestos; ^ = temperature (F.), C = coefficient of conductivity.] [Kent.] e 32° 212° 392° 572° 7.52° 1112° c 1.048 1.346 1.451 1.499 1.548 1.644 VIII, § 124] EMPIRICAL CURVES 233 (e) [Track records : d = distance, t = record time (intercollegiate).] d 100 yd. 220 yd. 440 yd. 880 yd. Imi. 2 mi. t 0:09f 0:211 0:48| l;56 4:17i 9:27| [Note. See KenneUy, Fatigue, etc., Proc. Amer. Acad. Sc. XLII, No. 15, Dec. 1906 ; and Popular Science Monthly, Nov. 1908.] 18. Plot the following curves, using logarithmic values of one quantity and natural values of the other : («) y (6) y = 10e3«. 19. Discover sets of data : a formula of the type y = ^•e<" for each of the fol (a) {;: .2 1.6 .4 2.2 .6 3.3 .8 6.0 1.0 7.4 if>) ' X: y- .6 3.0 1.2 4.4 1.8 6.6 2.4 10.0 3.0 14.8 (c) \7: .31 1.22 .63 1.49 .94 1.82 1.26 2.23 1.57 2.72 ((0 X: .2 .82 .8 .45 2.0 .13 4.0 .02. (e) X : y- .63 2.01 1.26 1.35 2.51 .60 3.77 .27 5.03 .12 (/) ' x: y- 1 1.63 2 1.34 3 1.08 4 .90 5 .73 20. A is the amplitude of vibration of a long pendulum, t is the time .since it was set swinging. Show that they are connected by a law of the form A = ke-~^. Ain.= 10 4.97 2.47 1.22 .61 .30 .14 fmin.=: 12 3 4 5 6 124. Method of Increments. A method which is often better in practice than those in § 121 is as follows. If the curve is supposed to be a parabola, (1) y = a + bx + CX-, 234 APPROXIMATION [VIII, § 124 and if we take two pairs of values of x and y, say (x, y) and {x + dkX, y + A^/) given by experiment, we should have (2) y = a + hx-{- ex-, y + d^y = a-\-h{x + ^x) -\- c{x-\- Axy, whence (3) Ay = b Aa; + 2 ex Ax + c Ace". If Ao; is constant, i.e. if points are selected at equal intervals on the crudely sketched curve drawn through the experimental points, we might write (4) Y=Ay=(bh-\-ch^)-\-2ch-x = A + Bx where h = Aa;. If we should actually plot this equation, Y=A-\-Bx, we would get (approximately) a straight line. Now Ay = Yis the difference of two values oi y; it can be found for each of the values of x selected above, and the (approximate) straight line can be drawn, so that A and B can be measured as in § 121. We may repeat the preceding process ; from (4) we obtain, as above, (5) AY=BAx = 2 ch% (h = Ax), whence AF is constant if h was taken constant. Now AF is the difference between two values of Y; that is, AF is the difference between two values of Ay : AY=A{Ay) = A% and for that reason is called a second difference, or a second increment. If the second differences are reasonably constant, we conclude that an equation of the form (1) will reason- ably represent the facts and we find c directly by solving equation (5). VIII, § 124] EMPIRICAL CURVES 235 Example 1 . With a certain crane it is found that the forces / measured in pounds which will just overcome a weight w are / 8.5 12.8 17.0 21.4 25.6 29.9 34.2 38.5 w 100 200 300 400 500 600 700 800 What is the law connecting power with the weight that it just overcomes ? [Pkrry.] Plotting the values of /and ip, it appears that the points are very nearly on a straight line f — a -\- bio. If they were on a straight line, Af/Aw would be constant and equal to df/dw = b. As a matter of fact, for each increase of weight, Af/Aw varies only from .042 to .044, its average value being 30/700 = .0429. Taking this value for b, one gets for the equation of the line, and hence for the relation between power and weight : /= 4.21 + .0429 10, 4.21 = 8.5 - 100 x .0429. Here 4.21 appears to be the power needed to start the crane if no load were to be lifted. Example 2. If 9 is the melting point (Centigrade) of an alloy of lead and zinc containing x % of lead, it is found that X = % lead 40 50 60 70 80 90 6 = melting point 186 205 226 250 276 304 Plotting the points (a;, d) will show them not to lie in a straight line as ds also shown by the differences A^. But A (A^) or A-^ does run uni- formly. Therefore one tries a quadratic function of x for 6, that is It is evident that and ^ = ffl + 6x + cx^. = 10?^-f c(20a; + 100), A:^e - 200 c. The average value of A2fl is 2.25. Hence c = .01125. If we subtract ca;2 from 6, we find 6 — cx!^ = a + bx. These values can be calculated from the data and from c = .01125 ; they will be found to lie on a straight line ; 2.5 3.0 2.1 2.8. .12 .14 .136 .163 236 APPROXIMATION [VIII, § 124 hence a and b can be found by any one of several preceding methods. The student will readily obtain, approximately, e= 133 + . 875 x + . 01125x2, a formula which represents reasonably the melting point of any zinc-lead alloy. [Saxelby.] EXERCISES L. — EMPIRICAL CURVES BY INCREMENTS 1. Express /(x) as a quadratic function of x, when x: 0.5 1.0 1.5 2.0 /(x): 2.5 1.9 1.6 1.5 1.7 2. Express /(x) as a cubic function of x, when x: .02 .04 .06 .08 .10 /(x): .020 .042 .064 .087 .111 3. Express ^Cw) as a cubic in tji, when to: .01 .02 .03 .04 .05 .06 .07 .08 <{>{m): .00010 .00041 .00093 .00166 .00260 .00385 .00530 .00690. 4. The specific heat 8 of water, at 6° C, is 6: 5 10 15 20 25 30 S: 1.0066 1.0038 1.0015 1.0000 0.9995 1.0000 1.002. Express S in terms of d. 5. Determine a relation between the vapor pressure P of mercury, and the temperature 5 C, from the data below : ^: 60 90 120 150 180 210 240 P: .03 .16 .78 2.93 9.23 25.12 58.8. 6. The resistance R^ in ohms per 1000 feet, of copper wire of diame- ter D mils, is * D: 289 182 102 57 32 18 10 B: .126 .317 1.010 3.234 10.26 32.8 105.1. Find a relation between B and D. 7. The Brown and Sharpe gauge numbers N of wire of diameter D mils, are iV: 1 5 10 15 20 25 30 D: 289 182 102 67 32 18 10. Express D in terms of N. VIII, § 124] EMPIRICAL CURVES 237 8. Find a relation between tlie speed a, I ydx\<\\ f{x)dx\ + \ I Mdx\ \Jx-a \ I «-'i=a I I \Jx=a \ ^\£_^J{x)dx\+Mi^-ay, that is, the error in the value of the integral calculated by using the approximation formula y =f(x) is not greater than M(b - a). * The pair of vertical lines | I indicate, as before (see pp. 16, 171), the posi- tive numerical value (or absolute value) of the quantity inclosed. VIII, § 127] APPROXBIATE INTEGRATION 241 The same result applies in cases in which a function to be Integrated has been replaced, for convenience, by a simpler function. Thus _J_z=l + x + a;2 + -^. 1 - X 1 — X If we replace 1/(1 — x), for convenience, by 1 + a; + x'^, the error E made in doing so is : l-x which, for values of x numerically less than 1/10, is numerically less than (.l)V-> < -0012 ; hence if we write the en-or £made in the value of the integral is less than .1 ■ .0012 = .00012. The exact value of the original integral is - log (1 - x)J^=^ - log (.9) = - logio (.9) loge 10 = .045757 • 2.30258 = .10536. In general, as in the example, the final error may be very much le.ss than the estimated upper limit of the error calculated above. 127. Derived and Integral Curves. In § 49, p. 77, we drew the derived curves by finding the derivatives and plotting their values. If the original curve was drawn from values found by some experi- ment, and if its equation is unknown, the derived curve can be drawn mechanically. To do so, draw, according to your be.st judgment, the tangents at each of a large number of points (xo, yo), (Xit Vi), (xo, 1/2), •••» («ni Vn), noting about how much uncertainty there seems to be in each case. Find the slope m,- of the tangent at each point (x,-, ?/,■) by measuring its rise per horizontal unit. Plot the points (m,-, x,), indicat- ing the estimated uncertainty in each value of m. Draw a smooth curve which passes near each of these points, allowing the most variation at the points where the values of ??i seemed to be most uncertain. Check by comparing the slope of the original curve and the ordinate of the derived curve for various other values of x. This process may not be very reliable, and every possible check must be used. (See § 143(d).) 242 APPROXIMATION [VIII, § 127 Likewise, if any function y =f(x) is given, the integral curve : . I = jf(x)clx = (x) + 0, which, represents the area under y =f(x) from some fixed left- hand boundary to the ordinate x = x can be drawn.* But if the equation of the curve is not known, this can still be done by the methods of § 125 ; or by simply estimating the area from some left-hand vertical line up to various points a^, iCj, • ••, x„ and marking at each value of x, as a new ordinate, the value of the area up to that point. The result is surprisingly accurate if the curve is drawn on millimeter paper and the area obtained by actually counting the squares. The accuracy of this process as compared with the uncertainty of mechanical construction of the derived curves, is a consequence of § 126. EXERCISES LI.— APPROXIMATE EVALUATION OF INTEGRALS 1. rind the area iinder the curve y = 1/(1 — x) from a- = to a: = .1 by use of the prismoid formula, and show that the result is accurate to five decimal places. 2. Draw the curve y= 1/(1 —x) and construct the integral curve from X = to any value of x less than 1, first by actually counting the squares on the cross section paper, second by actually integrating betvi^een the limits x = and x = x. 3. rind the area under the curve ?/ = l/x^ between x = l and x = 2, approximately, first by using the prismoid formula, then by using Simp- son's rule with three intermediate points of division. Compare the results with the precise answer obtained by integration, ^ 4. Find the error made in computing the value of the area of one arch of the curve y = sin x if the approximating parabola of Ex. 9, List XLIX, p. 231, is used instead of the sine curve. 5. Proceed as in Ex. 4, for each of the curves and their approximating parabolas mentioned in Ex. 10, List XLIX, taking the extreme values of x mentioned there as limits of integration. * Different values of C give, of course, different integral curves, all con- gruent, obtained from any one of them by a stiff vertical motion. VIII, § 12S] APPROXIMATE INTEGRATION 243 6. Show that a;2.5 ijes between x- and x'^ from x = to .r = 1. Hence show that f a;2.5 c^^ lies between 1/3 and 1/4. Find the exact vahie of the integral. 7. Show that 1/Vl - x* lies between 1/Vl-x- and l/\/2(l - x^) between x = and x = 1 ; hence find extreme limits between which £ [l/Vl-x*]dxlies. 8. Compute the value of the integral T [1/(1 + x2)](Zx (a) by the prismoid rule ; (b) by the trapezoid rule, with two intermediate points of division ; (c) by Simpson's rule, with three intermediate points of division ; (rf) precisely by direct integration. Compare the results for accuracy. 9. Show by long division that 1/(1 + ,r-) = 1 — x~ + x* - x^/(l + X'^). Hence show that the area under y — 1/(1 + x'-) from x = to x = .5 dif- fers from that under y = 1 — x^ + x* by less than 1/128. Actually com- pute both areas, and show that this estimate of the error is far larger than the actual error. 10. Draw the curve y = 1/(1 + x'-) and construct its integral curve, starting from the initial point x = 0. Verify by direct integration. 11. Draw the curve y = e'~^ and construct its integral curve. Find the value of the integral from x=0 to x = l, approximately, [a) from this inte- gral curve ; (6) by the prismoid formula ; (c) by the trapezoid rule, with one intermediate point ; (d) by Simpson's rule, with one intermediate point. 12. In each of the exercises of Ex. 17, List XLIX, p. 232, estimate from the figure an upper limit of the difference between the given data and the values represented by the empirical formula obtained. Hence find an upper limit of the total error which would be made in using the empirical formula to find the area underneath the curve. 13. Given a function /(x) defined by the following set of data : X .1 .2 .3 .4 * .5 .(5 .7 .8 .9 1 /(x) 1 .9 .7 .4 -.4 -.7 -.9 -1. -.0 -1 find approximately the derivative of /(x) at each of the points x = .2, a; = .3, x = .7. Find approximately the value of the integral of /(x) from X = to each of the preceding values of x. 128. Integrating Devices. It is important in many prac- tical cases to know approximately the areas of given closed curves. Thus the volume of a ship is found by finding the areas of cross sections at small intervals. 244 APPROXIMATION [VIII, § 128 Besides the methods described above, the following devices are employed : A. Counting squares on cross section paper. B. Weighing the figures cut from a heavy cardboard of uniform known weight per square inch. C. Integraphs. These are machines which draw the inte- gral curve mechanically; from it values of the area may be read off as heights. The simplest such machine is that invented by Abdank-Abakanowicz. A heavy carriage CDEF on large rough rollers, R, B' is placed on the paper so that CE is par- iinii fM allel to the y-axis. Two sliders S and S' move on the parallel sides DF and CE ; to S is attached a pointer P which follows the curve y = f(x) . A grooved rod AB slides over a pivot at A, which lies on the .r-axis, and is fastened by pivot B to the slider S. A parallelogram mechan- ism forces a sharp wheel IV attached to the slider S' to remain parallel to AB. A marker Q draws a new curve i = (p{x), which obviously has a tangent parallel to W, that is, to AB. If AB makes an angle a with Ox, tan « is the slope of the new curve ; but tan « is the height of S divided by the fixed horizontal distance h between A and B : hei^htoTS^fJx). h h ' whence -'"=li7j'^''^^''' where a is the value of x at P when the macbine starts, and /„ denotes the vertical height of the new curve at the corresponding point. VIII, § 128] APPROXBIATE INTEGRATION 245 D. Polar Planimeters. — There are machines which read off the area directly (for any smooth closed curve of simple shape) on a dial attached to a rollini? wheel. The simplest such machine is that invented by Amsler. Let us first suppose that a moving rod ab of length I always remains per- pendicular to the path described by its center C. The path of C may be- regarded as the limit of an inscribed polygon, and the area swept over by the rod may be thought of as the limit of the sum of small quadrilaterals, the area A^ of each of which is lAp, approximately, where Ap is the length of the corresponding side of the polygon inscribed in the path of C. Hence the total area A swept over by the rod is evidently Ip^ where p is the total length of the path of C. But if the rod does not remain perpendicular to the path of C during the motion, and if ^ is the angle between the rod and that path, the area AA becomes Z sin Z': > V(l x-^)(l tiZ^) Vl — .64 sin2 9 (0 J^'e-'x-erfx. 250 POLYNOMIAL APPROXIMATIONS [VIII, § 129 11. Reduce each of the following integrals to standard forms to be found in the tables by means of the substitutions indicated, and then evaluate them : a) C"' Vl - .09 cos-^edd; dd V'l-.49sin2 2^ dx '^ X logx' i) \ e idx; dx \/(4-a;2)(4- .36x2) put e = 90° — xp. put 2 ^ = i^. put log X = u. put - = M. PART II. POLYNOMIAL APPROXIMATIONS SERIES TAYLOR'S THEOREM 130. Rolle's Theorem. Let us consider a curve where f(x) is single-valued and continuous, and where the curve has at every point a tangent that is not ver- tical. If such a curve cuts the a;-axis twice, at a; = a and x = b, it surely either has a maximum or a mini- mum at at least one point x = c between a and b. It was shown in § 39, p. 64, that the derivative at c is zero : [A] If /(a) =/(&)= 0, tJien \ *^f(-^U = o, (a)-f(a) Fig. 59 given in § 130. Another similar state- ment, which is true under the same restrictions and is equally obvious geo- metrically, is : An arc of a sinqjle smooth curve cut off by any secant has at least one tangent iKirallel to that secant. If the curve is y =f(x), and if the secant yiS cuts it at points P: [rt, /(tt)] and Q: [b, f{b)'], the slope of S is A^-Ax=[/(6)-/(a)]-(&-a). The slope of the tangent CT at a; = c is equal to this : {a(x)/dx = Oatx = c, {a, R, and h are all to be taken in the same units.) If D is tabulated for values of h at inter- vals of one foot, what is the tabular difference at the place where A = 60 ? 254 POLYNOMIAL APPROXIMATIONS [VIII, § 132 14. If the boiling point of water at height H ft. above sea level is 2', if = 517 (212° - T) - (212° - T)^, T being the boiling temperature in degrees F. Discuss the uncertainty in H, if T can be measured to 1°. If H be tabulated with argument T at intervals of 1°, what is the tabular entry and the tabular difference when T = 200° ? 15. When a pendulum of length I (feet) swings through a small angle a (radians), the time (seconds) of one swing is T = tt VT/g (1 + a^/16). What is the effect on T of a change in a, say from 5° to 6° ? Of a change in I from 36 in. to 37 in. ? Of a change in g from 32.16 to 32.2 ? 16. The viscosity of water at 6° C. is P = 1/(1 + -0337 + .00022 ff^). Discuss the change in P due to a small change in d. What is the average value of P from ^ = 20° to ^ = 30° ? 17. The quantity of heat (measured in calories) required to raise one kgm. of water from 0° C. to 0° C.is H = 94.21 (365 - 6i)0-3i25 + j^. How much heat is required to raise the temperature of one kgm. of water 1° C. when e = 10°? 20°? 30°? 70°? To find A;, observe that lf=:0 when ^=0. 18. The coefficient of friction of water flowing through a pipe of diameter D (inches) with a speed V (ft. /sec.) is /= .0126 + (.0315 — .06 D)/\/V. What is the effect on / of a small change in V? in D? 19. If the values of \ VI — .2sin^xfto were tabulated with x as Jo argument, for every degree, what would be the tabular difference at the place in the table where x = 30° ? See Tables, V, E. 133. Limit of Error. In using the formula [B] the uncer- tainty in the value of c is troublesome. If the value of dy/dx at a; = a is used in place of its value at a? = c, the error made in finding Ay by [JB] can be expressed in terms of the second derivative d'^y/dx'. We shall use the convenient notation mentioned in Ex. 33, p. 57, and Ex. 5, p. 222, for the derivatives of f(;x) : f(x) = M^) = ^ (the slope of 2/ = f(x)). cix ctx y .(^) = ^m =tE^^ . \ ^J^ \l^ 1^ .. L^^-^ I ^^ M ti^. t .zn zB^n^ r _, ... \ w ^^/ 1 -4-.7a:\-^^l^ 7 --S\w V y 1 TTji/^ A_/ //T /17 " .m y "± i/j^f/ / Aj¥^ / -'^cr^^ L -. JZXIJ?'^ I - ;? ?^ j?^ i 0^0 X 0\ \ /L... 1 (4) dx \ or since /'(a) and 3/2 are constants and df(x)/dx=f'{x), 256 POLYNOMIAL APPROXIMATIONS [VIII, § 133 whence, substituting the limits in the usual manner, which holds for all values of x between x = a and x='b. This formula may be written even if a; < a : ^ [C*] /(ic) = /(a) + /'(a)(£c— a) +^2, where I ^2 1 ^ -Ms^^^^^, and En is the error made in using f'(a) in place of /'(c) in formula [C] ; for (x — a)^ =\x — a\'-. It should be noticed that E2 is exactly the error made in substituting the tangent at a; = a for the curve, i.e. it is the difference between A?/ [=/(.«) —/(a)] and d?/[=/'(a)(a;— a)] mentioned in § 31, p. 50, and shown in Fig. 12. The formula [-B*] is exactly analogous to [C*] ; since A// =f(x) — /(«) if Ax = a; — a, [-B*] may be written [5*] f(x)=f{a)^E„ \E,\^M,.\x-a\. Example 1. In Ex. 1, p. 252, we found for L = logio N, AL = -^ (nearly). Applying [C*], with f(N) = logio iV, a = N, x = J^ + AJST, x—a = AN = .001, we find AL=AN+AN)-fm=-^ + E.,, \E.J<-^.M,, where I/2 is the maximum value of |/"(2V) | = (logio e)/N'^ between iV = 1 and iV= 10. Hence E^ < .00000022. The value of AL found before x, was therefore quite accurate, — absolutely accurate as far as a five-place table is concerned. Example 2. Apply [C*] to the function f(x) = sin x, with a = 0, and show how nearly correct the values are for x < ir/90 = 2°. Since /(x) = sin x, and a = 0, [ C*] becomes sin X = sin (0) + cos (0) • {x - 0) + Eo = x + E2, {Eol^M^j, where J/o is the maximum of | / "(x) | = | — sin x | between and tt/QO, that is M-z = sin (tt/OO) = sin 2° = .0.349. Hence Eo < .0175x2. Since VIII, § 134] ERROR — TAYLOR'S THEOREM 257 a;<7r/90, x2 <7rV8 100 < .0013 ; hence ^2 < .000023, and sin a; = a; is correct up to x ^ 7r/90 within .000023. Similarly, for a = 7r/4, we have, by [C*], where J/o < 1- If (a; - 7r/4) < tt/OO, | -F, ; < (7r/90)2 h- 2 = .0007. 134. Extended Law of the Mean. Taylor's Theorem. The formula [C*] can be extended very readily. Let f\x), f^\x), /'"(x), ••• p"^x denote the first n successive derivatives oi fix) : -' ^ ' dx" dx ' and let the maximum of the numerical value of /^"^(x) from jc = a to ic = 6 be denoted by J/„. Then |/<")(.r) I ^ Jf„, and I f^f'Xx) dx | ^ I C^~' M„ dx\, or |/("-i)(.r) _/"-i)(a) I ^ I M,,{x-a)\ for all values of x between a and h. Integrating again, we obtain, as in § 133 : |/(»-2)(^.) _/(n-2)(a) -fi^^\a)(x - a) I < 1 3f/-^Hjzi^' | ; and, continuing this process by integrations until Ave reach f{x), we find : [I>] \f(oc)-f(a)-f'(a)(oc-a)-^'^p-(x-ay-:. (M — 1)1 I Ml 258 POLYNOMIAL APPROXIMATIONS [VIII, § 134 or, where i«ni*]' /(x) =/(0) +/'(0)x+^^:r2^ -. /(-i)(0) , where | ^„ 1 < iJ/"„ | .t" | /n ! This special case of Taylor's Theo- rem is often called Maclaurin's Theorem. The formula [D*] replaces /(.«) by a polynomial of the nth degree, with an error E^. These polynomials are represented graphically by curves, which are usually close to the curve which represents f(^x) near x = a. See Tables, III, K. Since the expression for £"„ above contains n ! in the denomi-^ nator, and since n ! grows astoundingly large as n grows larger, there is every prospect that E^ will become smaller for larger n ; hence, usually, the polynomial curves come closer and closer to f{x) as n increases, and the approximations are reasonably good farther and farther away from x = a. But it is never safe to trust to chance in this matter, and it is usually possible to see what does happen to E^ as n grows, without excessive work. VIII, §134] ERROR — TAYLOR'S THEOREM 259 Example 1. Find an approximating polynomial of the third degree to replace sin x near x = 0, and determine the error in using it up to X = 7r/18 = 10^ Since f(x) = sin x and a = 0, we have f'(x) = cos x, f"{r) = — sin x, /"'(.t) = — cos x,/i^ (.»•) = + sinx, whence /(O) = 0, /'(O) = l,/"(0) = 0, /'"(O) = - 1 ; and [Max. \f"{x)\ ] = [Max. | sin .r | ] = sin 10° = .1736, between x = and x = tt/IS = 10°. Hence sin X = + 1 . (X - 0) + + (- 1) . ^-—^ + .^4 = a^ - 1^ -!- ^4, where | ^4 |< (.1736) • .rV4 ! < (.1736) (tt/IS)* - 4 ! < .000007, when x lies between and tt/IS. In general, the approximation gi'ows better as n grows larger, for |/<"'(.v)| is always either | sin .r | or [ cos x |; hence M^ < 1, and \ E„\-^ x"/n ! which diminishes very rapidly as n increases, especially if x < 1 = 57°.3. For n = 7, the formula gives, for x > 0, smx = x-$+'^ + E„ \E^\ 0, Ay > 0, and f(x) is a minimum at a; = a ; if / "(a) < 0, Ay < 0, and /(a;) is a maximum at x = a. (See § 47, p. 75.) If f"(a) = 0, the question is not decided.* But in that case, by [b*] : Ay = f(x) -/(a) = + + ^^^ (x - af +-^ (x - «)"+ E„ where | ^5 1 < 3/5 1 x — a 1 75 !, J/,^ |/^ (x) \ . From this we see that if/'" (o) 9!= there is neither a maximum nor a minimum, for (.X — o)^ changes sign near x = a. But if f"'{a) =0, then /'''(a) determines the sign of Ay, as in the case of /"(o) above. In general, if /^*X") i^ *^^^ fi^"^*- °"® °^ *^^® successive deriva- tives, /'(a), /"(a), •••, which is not zero at x = a, then there is : no extreme if k is odd ; a maximum if k is even and /(^'^Ca) < ; a minimum if A- is even and f^^^{fi) > 0. Example 1. Find the extremes for y — x*. Since f(x) = :c*, f'{x)= 4x3 ; hence the critical values are solutions of the equation 4 a:^ = 0, and therefore x = is the only such critical value. Since /"(x)=12x2, /"'(x) = 24x, /i^(a:) = 24, the first derivative which does not vanish at x=:0 is /"'(a:), and it is positive ( = 24). It follows that /(x) is a minimum when x = 0; this is borne out by the familiar graph of the given curve. EXERCISES LV. — EXTREMES 1. Study the extremes in the following functions : (a) x6. (0 (.c + 3)5. (0 x2sinx. (6) (x-2)8. {f) x\2x-iy. (j) x^cosx. (c)4x8-3x<. {(j) sinx8. (A;) x8 tan x. (d)x8(l + x)8. (;t)x-sinx. (0 e-^^'^ * Tlie methods which follow are logically sound and can always be carried out when the derivatives can be found. But if several derivatives vanish (or, what is worse, fail to exist) , the method of § 40, p. 64, is better iu practice. 262 POLYNOMIAL APPROXIMATIONS Wm, § 135 2. Discuss the extremes of the curves y = x", for all positive integral values of n. 3. Solve the problem of Ex. 18, List XIV, p. 69, by the method of § 135. 4. If a set of observed values of a quantity y which depends upon another quantity x are yo, 2/i, 2/2, •■■■, yni when x has the values .co, cci, X2, •■■, Xni and if y is connected with x by means of an equation of the form y = kx, the sum of the squares of the differences between the observed and the computed values of ?/ is : S=iyo- kxo)' + (2/1 - ^•Xl)2 + (1/-2 - kxo^ + ••• + (?/„ - ^x„)2. Show that the sum S, as a function of k, is least when 2 xoiyo - kxa) +'2xi (yi - kxi) + ••• + 2 x„ (2/„ - A;x„) = 0, or A;=2^^^.-2/.^2^,^-'^r. [Note. Under the assumption of Ex. 18, p. 69, this value of k is the best compromise, or the most probable value.] 5. Using the result of Ex. 4, recompute the value of each of the con- stants of proportionality k in Exs. 18-23, p. 69. 6. An open tank is to be constructed with square base and vertical sides so as to contain 10 cu. ft. of water. Find the dimensions so that the least possible quantity of material will be needed. 7. Show that the greatest rectangle that can be inscribed in a given circle is a square. [See Ex. 25, p. 70. Other examples from List XIV may be resolved by the process of § 135.] 8. What is the maximum contents of a cone that can be folded from a filter paper of 8 in. diameter ? 9. A gutter whose cross section is an arc of a circle is to be made by bending into shape a strip of copper. If the width of the strip is «, show that the radius of the cross section when the carrying capacity is a maxi- mum is u/tt. [Osgood.] 10. A battery of internal resistance r and E. M. F. e sends a current through an external resistance E. The power given to the external circuit is „ 2 W= —^ (B + ry If e = 3.3 and r = 1.5, with what value of R will the greatest power be given to the external circuit '? [Saxelbv.] VIII, § 136] INDETERMINATE FORMS 263 11. Find the shortest distance from the origin to the curve y = a'; show that it is measured along a straight line from the origin to the intersection of the given curve with the curve x = — y^ log a. 12. Show that the maximum and the minimum distances from a point (rt, b) to the curve y = x- join (a, h) to the intersections oi y = :r- with x(y- b + i) =a. 136. Indeterminate Forms. The quotient of two functions is not defined at a point where the divisor is zero. Such quotients f(x)-r-(a)=0, are called indeterminate forms.* We may note that the graph of (1) ^ = :r7^' (/(a.) = <^(a) = 0), may be quite regular near x = a; hence it is natural to make the definition : (2) If we apply [D *], we obtain, , f(x) 0+f'(a)(x-a)+E,' •d where ^ <^(x) + \a){x-a)+E2"' \E,'\^ M.: (x - ay/2 1, 1 E," \ ^ MJ' (x - ay/2\, and J/2' ^ I /" (x) I, M2" > I "(x) I, near x = a. Hence q = VH = ^ *U 4>(a)=0 but f(a) ^ the quotient q evidently becomes infinite ; in that case the graph of (1) shows a vertical asymptote. 264 POLYNOMIAL APPROXIMATIONS [VIII, § 136 where p' and p" are numbers between — 1 and + 1. It follows that (3) hm g = lim ^;-( = ^yf^ unless '(a) = 0. But if ^' (a) = 0, g becomes infinite, and the graph of (1) has a vertical asymptote Sit x = a unless /' (a) = also. If both f (a) and 4>' (a) are zero, it follows in precisely the same manner as above, that _f{x) ^ ^ '^^ *^' (A: + l)! ^^^) <^-(a) + y'3/Ui;^^ where either /(*Xa) or >^^^\a) is not zero, but all preceding de- rivatives of both fix) and ^{x) are zero at x = a\ and where ■^I+i ^ 1/'"+'^^;) I, iWlVi ^ |'(ic) I near x = a and where _p' and ^" are numbers between — 1 and + 1. It follows that lim g = lim •-—— = -TTtVTT' provided all previous derivatives of both f(x) and <^ (x) are zero at X = a, and provided <^(*> (a) =^ 0. If <^<*' (a) = 0, /<*^ (a) ^ 0, then g becomes infinite and the graph of (1) has a vertical asymptote at x = a. It should be noted that (3) is only a repetition of Rule [VII], p. 36. For if u -f{x) and^; = (p{x), since /(a) = 0(a) =0, _ /(a:) ^ /(■r)-/(ffl) ^ 4 m ^ A« _^ Au ^~{x)-ia) ^v Ax ■ Ax' where Ax = x — a ; and therefore ,. A?( ,. Atj rd?< dv-^ _r/!Ml --^ S '^ = i^o :^ " i"o ^x-\Tx- TA^ - L 0'(x) J^. - 0' provided (x) = x; /(O) = 0(0) = ; hence y^^ tan X _ f'(0) ^ [sec2 xl^^ ^ ^ x^ X 0'(O) 1 Draw the graph 5=(tana;) — x and notice that this value (7 = 1 fits ex- actly where x = 0. This limit can be found directly as follows : lim ^^" ^ ^ lim ^^^ ^-^ + ^^ ~ ^''" '^^^ = ^ ^^" ^- 1 =sec2x1 =1 h^ h b^ (0 + /i)-(0) dx Jx=o Jx=o Compare the work done in § 96, p. 167, for lim (cos A6 — 1)/A^. Example 2. To find lim (i _ cos x)/x2. Here /(x) = 1 - cos x, 0(x) = x2 ; /(O; = <^(0) = ; /'(O) = sin (0) =0 and ^'(0) = ; /"(x) = cos x, 0"(x) = 2 ; hence lim h 2 Jx=o~2" Draw the graph of g=(l — cosx)/x2, and note that (x = 0, q = 1/2) fits it well. 137. Infinitesimals of Higher Order. When the quotient (1) 0=^ approaches a finite number not zero when x is infinitesimal : (2) \imq = \im^^ = ]c=^0, ^ ^ z = x = X" then f(x) is said to be an infinitesimal of order n with respect to X. An infinitesimal Avhose order is greater than 1 is called an infinitesimal of higher order. The equation (2) may be reduced to the form (3) lim[/(aj)-^u-"]=0, or (4) fix) = (k^E)x'', 266 POLYNOMIAL APPROXIMATIONS [VIII, § 137 where lim E =0. The quantity A;x" is called the principal part of the infinitesimal /(x). The difference / (x) — kx" = Ex"" is evidently an infinitesimal whose order is greater than «, for lim (Ex'' H- x'^) = lim ^ = 0. Thus by Example 2, p. 265, 1 — cos x is an infinitesimal of the 2d order with respect to x ; its principal part is x^/2. Note that 1 — cos X = x'^/2 + px^/3 !, by [D*], where — 1 ^ p ^ + 1 ; the principal part is the first term of Taylor's Theorem that does not vanish. In general, if /(0) = /'(0) = /''(0)= - =/(^-i)(0)=0, hutfW(0)^0, the formula [D *] gives, for a = 0, /(a:) =/W(0) . xk/k \+pM„+iX^-+y(k + 1) ! where M^+i ^|/<*^+i*(a:)| near x = 0, and — l°» X — sinx C9) ° • (0 (u) (s) 2smx-^ I (^) sin 6 X I „/6 Va- VIII, § 138] INDETERMINATE FORMS 267 2. Determine the order of each of the quantities below when the vari- able X is the standard infinitesimal : (rt) X - sin X. (e) e - e^'n-^. (i) sin 2 x — 2 sin x. (6) t'—e-'. (/) a'—\. (j) log cos X. (c) x2 sinx2. {g) log[(a+x)/(«-x)]. (^•) log(l + e-Vx). ^d) log (1 + x) — X. (ft) X cos X — sin x. (?) tan-i x — sin-i x. (w) log cos X— sin^ X. («) 2x— e*+e~*. (o) cos-i(l— x) — \/2x — x^. 3. Show that Ex. 1 (a) can be expressed as the derivative of sin x at X = 0, as in Example 1, p. 265. . 4. Show that Exs. 1 (e), (/), (/i), (j) can be expressed as the deriva- tives of the numerators, for x = 0. 5. Show that Ex. 1 (rZ) can be expressed as the derivative of its numer- ator divided by the derivative of its denominator, at x = 1. 6. Find the limit of the ratio of the surface of a sphere to its volume, as the radius approaches zero. 7. Find the limit of the ratio of a chord of a circle to the distance along a radius perpendicular to the chord from the chord to the circle. 8. Given two quantities u and v which vary with the time «, so that u =f(t) and V = 0(0, show that lim ^'=rilmf^M-| lim"^ flim ^'1 9. Show that the slope of the path of a moving body is the ratio of its vertical speed to its horizontal speed. 138. Double Law of Mean. Let y =/(x) and y z={b)-4>{a) '{€)' To prove this, consider the parallel curves y =fi^) -f(a) and J/ = (x) - (a), which both go through the same left-hand point (a, 0) and the ratio of the slopes of whose secants is, as above, [/(6) — /(a)]/[0 (&) —

{a) by this ratio, we have the new curve {a)/4> (b) shall be as small as we please. For however close to A one takes a, /(a) and 0(a) are finite. Taking now b between a and x4, we can give f{b) any value above /(a). Therefore the first of the preceding ratios (and in like manner the second) can by proper choice of 6, after any choice of a, made as small as one pleases. Notice that a and b must be on the sa side of the vertical asymptote. Fig. 62 be VIII, § 140] INDETERMINATE FORMS 269 Let the choice of a and b be made as just indicated. The theorem of § 138 still holds : 0'(c) 0(6)-,^ (a) fi '^(^)lrf,(M (a If lim s^ = S exists, the series is called convergent ; if S does not exist, the series is called divergent; if the series formed by ticking the numerical (or absolute) values of the terms of (1) converges^ then (1) is called absolutely convergent. Infi- nite series which converge absolutely are most convenient in actual practice, for extreme precaution is necessary in dealing with other series. (See § 143, p. 276.) Example 1. The series 1 + r + r- + ••• + r" + ••• is called a geometric series ; the number r is called the .common ratio. A geometric series converges absolutely for any value of r numerically less than 1 ; for s„ = 1 + r + r2 + - + r»-i = -J— - -^^ , 1 -^ ;• 1 — r hence liml— ^i s„| = lim |-^| = 0, if | r| < 1, ; since r" decreases below any number we might name as 7i becomes in- finite. It follows that the sum ^S" of the infinite series is 8 = lim s„ = -i— , if I r |< 1 ; and it is easy to see that the series still converges if r is negative, when r is replaced by its numerical value \r\. Example 2. Any series ao + «i + as +••• + «« + • -of positive num- bers can be compared with the geometric series of Ex. 1. Let (r„ = ao + «! + ao + ••• + a„-i ; VIII, § 142] TAYLOR SERIES 273 then it is evident that (7-„ increases with n. Comparing with the geometric series a^{\ + r + r- + ••• + r» + •••), it is clear that if ■where s„ = 1 + r + ••• + »""~^. Hence o-„ approaches a limit if s„ does, i.e. ?/ < ?• < 1. It follows that the given series converges if a value of r < 1 can be found for which «„ ^ agr", that is, for which a„ -r- a„_i < r <1. There are, however, some convergent series for which this test can- not be applied satisfactorily. It may be applied in testing any series for absolute convergence ; or in testing any series of positive terms. For example, consider the series 1!2!3! 71 ! ' here a„ = 1/n !, a„_i = l/(n — l)!i and therefore a„/an-i = (n — 1) \/n I = 1/n. Hence «„/"«-! < V^ when re ^ 2, = 1 + s„_i, where s„_i = 1 + r + ••• + r''-^, r = 1/2. It follows that the given series converges and that it^ sum is less than 1+2 = 3. [Compare § 143, p. 278 ; it results that e < 3. Compare Ex. 2, p. 275.] 142. Taylor Series. General Convergence Test. Series which resemble the geometric series except for the insertion of con- stant coefficients of the powers of r, (1) A + Br-hCr + D)^-\--, arise through application of Taylor's Theorem [Z)*](§ 134, p. 258); such series are called Taylor series or power series. The prop- erties of a Taylor series are, like those of a geometric series, comparatively simple. Comparing (1) with [i5*], we see that r takes the place of (x — a), while A, B, C, D, "• have the values : A=m, B^fM, c^-CM, D^qf-. 274 POLYNOMIAL APPROXIMATIONS [VIII, § 142 If we consider the sum of n such terms : we see by [Z)*], that f(x)=s^ + E^, where |^„|< lf„ 1^-=-^, M„>\p-\x)\', or s„ =f{x) - E^. It follows that if En approaches zero as n becomes infinite, the infinite Taylor Series converges, and its sum is ,S = lim s„ = fix).* This is certainly true, for example, whenever |/^"'(cc) | remains, for all values of n, less than some constant C, however large, for all values of x between x= a and x= b. For in that case lim|^Ji=oo ~- n^=« n ! * This result is forecasted in § 134, p. 258. t This results from the fact that n eventually exceeds (a; — a) numerically; afterwards an increase in n diminishes the value of En more and more rapidly as n grows. VIII, § 142] TAYLOR SERIES 275 therefore the infinite series [Z)**] for a = is sina; = + x + O -~a;3 + + — x^ + ... ; 3 ! 5 ! this series certainly converges and its sum is sin x for all valurs of x, since lim | £"„ | = 0. Example 2. Derive an infinite series for e* in powers of (x — 2). Since /(x) = c*, we have f'(x)-e', ••-, /(")(x) = e* ; hence /(2) = e^, /'(2) = e2, •■•,/'»>(2) = e2, and If^^^x) | ^ e'> where b is the largest value of X we shall considei*. Then the series e» = e2 + e2(x-2) + ^(x-2)2+ ... +-^(x-2)»+ ••• 2 1 n\ = e2[l+(x-2) + ±(x-2)2+...+-l(x-2)»4--] converges and its sum is e', for all values of x less than b ; for lim \E„\< lim «'"]■'''- -]" = q. Since b is any number we please, the series is convergent and its sum is e* for all values of x. EXERCISES LVm. — TAYLOR SERIES 1. Obtain the infinite Taylor series for cos x in powers of x. Show that lim | ^„ | = 0. 2. Derive the following series, and account, when possible, for the fact that lim I ^„ I = : (c) e' = l + x + xV2 ! + xV3 !+•••; (all x). (b) e-' = l-x + x2/2 ! - xV3 ! + •••; (all x), (c) tan X = X + xV3 + 2x5/15 + 17x7315 + .•• ; ( | x |< 7r/2). (d) log (1 + x) = X - x2/2 + xV3 - xV4 + ••• ; ( | x |< 1). (e) sinh x = (e' - e-')/2 = x + xV3 ! + x^/5 ! + •••; (all x). (/) cosh X = (e^ + e-')/2 = 1 + x^l'2 ! + xV4 ! + •••; (all x). (jr) tanh X =sinhx/coshx =x— xV3 + 2xV15-17xV3l5+ •••; (all x). 3. Show that the series of Ex. 2 (e) can be obtained from those of Exs. 2 (a) and 2 (6) if the terms are combined separately. 4. Show that the series of Ex. 2 (6) results from the series of Ex. 2 (a) if X is replaced by — x. 276 POLYNOMIAL APPROXIMATIONS [VIII, § 142 5. Obtain the series for sin x in powers of (x — 7r/4). 6. Obtain the series for e^ in terms of powers of (x — 1). 7. Obtain the series for log x in powers of (x — 1). Compare it with the series of Ex. 2 (d) . 8. Obtain the series for log ( 1 — x) in powers of x, directly ; also by replacing x by — x in Ex. 2(d). 9. Using the fact that log [(1+x) /(I -x)] = log (1+x) - log (1-x), obtain the series for log [(1 + x)/(l — x)] by combining the separate terms of the two series of Ex. 8 and of Ex. 2 (d). This series is actually used for computing logarithms. 10. Show that the terms of the expansion of (a + x)" in powers of x are precisely those of the usual binomial theorem. 11. Show that the series for e°'+' in powers of x is the same as the series for e* all multiplied by e". 12. Show that the series for 10"^ is the same as the series for e* with X replaced by x/3/, where 3/ = 2.30 •••. 143. Precautions about Infinite Series. There are several popular misconceptions concerning infinite series which yield to very commonplace arguments. (a) Infinite series are never used in computation. Contrary to a popular belief, infinite series are never used in computation, and can never be used. This is because no one can possibly add all the terms of an infinite sei'ies. What is actually done is to use a few terms (that is, a polynomial) for actual computation ; one may or may not consider how much error is made in doing this, with an obvious effect on the trustworthiness of the result. Thus we may write ■> ^3 ^5 r2*+l smx=: X- — + - ± — T ••• (forever); 3! 5! (2^ + 1)! ^ ^' but in practical computation, we decide to use a few terms, say sin x = x — x'y3 ! +x^/5 !. The error in doing this can be estimated by § 134, p. 257. It is I ^7 1 <|x"/7 1 1 . For reasonably small values of x [say | x | < H'^ < 1/-4 (radians)], | £"7 1 is exceedingly small. Many of the more useful series are so rapid in their convergence that it is really quite safe to use them without estimating the error made ; but if one proceeds without any idea of how much the error amounts to, one usu- VIII, §143] INFINITE SERIES — PRECAUTIONS 277 ally computes more terms than necessary. Thus if it were required to calcu- late sin 14° to eight decimal places,*most persons would suppose it necessary to use quite a few terms of the preceding series, if they had not estimated E^. (b) No faith can be placed in the fact that the terms are becom- ing smaller. The instinctive feeling that if the terms become quite small, one can reasonably stop and suppose the error small, is unfortunately not justitied.* Thus the series 1111 1 _L 4. _L + _L + J- + ... + _i_ + ... 10 20 30 40 10 ?i has terms which become small i-ather rapidly ; one instinctively feels that if about one hundred terms were computed, the rest would not affect the result very much, because the next term is .001 and the succeeding ones are still smaller. This expectation is violently wrong. As a matter of fact this series diverges ; we can pass any conceivable amount by continuing the term-adding process. For ^ + T*o + - + xb > 8 • xk = irs, and so on ; groups of terms which total more than 1/20 continue to appear forever ; twenty such groups would total over 1 ; 200 such groups would total over 10 ; and so on. The preceding series is therefore very decep- tive ; practically it is useless for computation, though it might appear quite promising to one who still trusted the instinctive feeling mentioned above. (c) If the terms are alternatehj ^iositive and negative, and if the terms are numerically decreasing loith zero as their limit, the in- stinctive feeling just mentioned in (b) is actually correct : the series a^ — Oj -f ag — ag -j — converges if a„ approaches zero ; the error made in stopping with a„ is less than a„+i.t For, the sum s„ = Oo — cfj + ••• ± a„_i evidently alternates * This fallacious instinctive feeling is doubtless actually uxed, and it is re- sponsible for more errors than any other single fallac-y. The example here mentioned is certainly neither an unusual nor an artificial example. tOne must, however, make quite sure that the terms actually approach zero, not merely that they become rather small ; the addition of .0000001 to each term would often have no appreciable effect on the appearance of the first few terms, but it would make any convergent series diverge. 278 POLYNOMIAL APPROXIMATIONS [VIII, § 143 between an increase and a decrease as n increases, and this alter, nate swinging forward and then backward dies out as n increases, since a„ is precisely the amount of the nth swing. On each swing s„ passes a point S which it again repasses on the return swing ; and its distance from that point is never more tlian the next swing, — never more than a„+i. Since a„ approaches zero, s„ approaches S, as n becomes infinite. Thus the series for sin x is particularly easy to use in calculation : the error made in using x — x^/fi ! in place of sin x is certainly less than x^/b !. The test of § 134 shows, in fact, that the error | E5\ 0) is less than M5 x^/b !, where 3/5 is the maximum of p{x) — e^ between x = and x = x, i.e. e' ; hence E5 1 f or x > 0. Another means of convincing oneself that the preceding series converges is by comparison with a geometric series with a ratio x/2, as in Example 2, p. 272. But this method would require the computation of a vast number of terms, to make sure that the error is small. (d) A consistently small error in the values of a function may make an enormous error in the values of its derivative. Thus the function y = x— .00001 sin (100000 x) is very well approxi- mated by the single term y = x, — in fact the graphs drawn accurately on any ordinary scale will not show the slightest trace of difference be- tween the two curves. Yet the slope of y = x is always 1, while the slope oiy = x — .00001 sin (100000 x) varies from to 2 with extreme rapidity. Draw the curves, and find dy/dx for the given function. One advantage in Taylor series and Taylor approximating polynomials is the known fact — proved in advanced texts — that differentiation as ivell as integration is quite reliable on any valid Taylor aiyproximation* * See, e.g., Goursat-Hedrick, Mathematical Analysis, Vol. I, p. 380. VIII, §143] INFINITE SERIES — PRECAUTIONS 279 Thus an attempt to expand the function y = x — .00001 sin (100000 x) In Taylor form e;lves ^ L a! 5! J' which would never be mistaken for y —xhy any one ; the series indeed converges and represents y for every value of x, but a very hasty exam- ination is sufficient to show that an enormous number of terms would have to be taken to get a reasonable approximation, and no one would try to get the derivative by differentiating a single term. If the relation expressed by the given equation was obtained by ex- periment, however, no reliance can be placed in a formal differentiation, even though Taylor approximations are used, for minute experimental errors may cause large errors in the derivative. Attention is called to the fact that the preceding example is not an unnatural one, — precisely such rapid minute vibrations as it contains occur very frequently in nature. EXERCISES LIX. — INFINITE SERIES 1. Show that the series obtained by long division for 1 -?- (1 + x) is the same as that given by Taylor's Series. 2. Obtain the series for log (1 -f- a*) (see Ex. 2 (d), List LVIII), by integrating the terms of the series found in Ex. 1 separately. 3. Find the first four terms of the series for sin-i.c in powers of x di rectly ; then also by integration of the separate terms of the series for l/Vl— x^. 4. Proceed as in Ex. 3 for the functions tan-i x and 1/(1 + x-). 5. Show that the series for cos x in powers of x is obtained by differ- entiating separately the terms of the series for sin x. 6. Show that repeated differentiation or integration of the separate terms of the series for e^ always results in the same series as the original one, 7. From the series for tan-ix compute ir by using the identity ir/4 = i tan-1 (1/5) - tan-i (1/239). 8. Six)^f;isinu/u)au = x-lf^ + lf-:.. Show that S(.l) = .0999+ ; 8(1) = .94G1 ; ^(3) = 1.8487. 9. The Gudermannian of x is gd(x) = 2 tan-'e' - ir/2 ; expand in powers of x ; calculate gd(.l) = 5° 43', and gd(.l) = 37° 11'. 10. The Fresnel integrals are C(.) =-^ f --^d.; S(z)=-^r'^dz. V2irJo y/z y/2irJo y/z 280 POLYNOMIAL APPROXIMATIONS [VIII, § 143 Obtain power series in z for C{z) and Si^z). Calculate C(.l) = .2521, C(l) = .7217, (7(3) = .5610 ; {x), is given, z reduces by simple substitution to a function of one variable : z=f(x, y), y = (^) gives z =f(x, <}>(x)). Now any change Aar in x forces a change Ay in y; hence y cannot remain constant (unless, indeed, (f>(x) = const.). Hence the change Az in the value of z is due both to the direct change Ax in x and also to the forced change Ay in y. We i^shall call Az= the total change in z = f(x+ Ax, y + Ay) —f(x, y), A^z= the partial change due to Ax directly = f{x+Ax,y)-f(x,y), A/=the partial change /orced by the forced change Ay = Az-A^, =f{x + Ax, y + Ay) -fix + Ax, y). • If z is negative, of course the lower end is the one to take. 286 SEVERAL VARIABLES [IX, § 148 It follows that ,-,s dz T Az (1) — = lim — dx Ax=y) Ax {' ^)}' ^lim ^-' + V ajs£=o Aa; I Ay^ \ AX I /•(y + Aa-. y + Ay) - / («• + Aa-. y) Ay N 1 Ay ^"^ / j ' whence, if the partial derivatives exist and are continuous,* (2) dz dx Ax=oL Ax Ay Ax J dz . dz dy dx dydx* or, multiplying both sides by dx{=Ax) (3) dz = ^dx-\-^ dy, since dy = ^ dx, Sy dx dx v^ Qo ^ V=0(a:) Fig. 64 PS = AB = Ax ST= C2) = Ay SR = A«3 = TM= R„R - PgP MQ = A^»lx=^+Ax = QoQ - ^off A 3 = ^0^ - PoP= TQ^SS+MQ = V+\^].=. +Ai where % = ^'(.^') dx. Since <^(a;) is any function whatever, dy is really perfectly arbitrary. Hence (3) holds for any arbitrary values of dx and dy whatever, where dz=(dz/dx) dx is defined by (2) ; dz is called the total differential of z. These quantities are all repre-^ sented in the figure geometrically: thus Az = A^z + AyZ is represented by the geometrical equation TQ = SR +MQ. It should be noticed that dt is the height of the plane drawn tangent to the surface at P, since * For a more detailed proof using the law of the mean, see Goursat-Hed* rick, Mathematical Analysis, I, pp. 38-42. IX, § 149] PARTIAL DERIVATIVES 287 dz/dx and dz/dy are the slopes of the sections of the surface by y =yp and x= Xp, respectively. [See also § 1G4, p. 321.] If the curve PqQq in the a;j/-plane is given in parameter form, x=(t), y = i}/(t), we may divide both sides of (3) by dt and write / . . dz _ dz dx dz dy ^ ^ Tt~^Yt TyW since dx h- dt = dx/dt, dy -=- dt = dy/dt. 149. Elementary Use. In elementary cases, many of which have been dealt Avith successfully before § 148, the use of the formulas (2), (3), and (4) of § 148 is quite self-evident. Example 1. The area of a cylindrical cup with no top is (1) ^ = 2 7rr/i + 7rr2, where h is the height, and r is the radius of the base. If the volume of the cup, irr-h, is known in advance, say irr^A — 10 (cubic inches), we ac- tually do know a relation between h and r : whence (3) ^:^2 7rr^+7rr2=^ + 7rr2 irr- r from which dA/dr can be found. We did precisely the same work in Ex. 7, p. 68. In fact even then we might have used (1) instead of (3), ind we might have written U) ^ = 2 irr— -I- 2 tt/H- 2 Trr, or dA = 2 nr dh + (2 irh + 2 irr)dr, dr dr where dh/dr is to be found from (2). This is precisely what formula (2), § 148, does for us ; for (5) ?ii = 2 7rft + 2,rr, ^ = 27rr, — =(2 7r;i + 2irr)+(2 7r?-)— , or dA =(2 7rh + 2irr)dr + 2 rrrdh. dr dr We used just such equations as (4) to get the critical values in finding extremes for dA/dr = at a critical point. We may now use (2), § 148, to find dA/dr ; and the work is considerably shortened in some cases. 288 SEVERAL VARIABLES [IX, § 149 Example 2. The derivative dy/dx can be found from (2), § 148, if we know that z is constant. Thus in § 26, p. 44, we had the equation (1) a;2 + 2/2 = 1, and we wrote : (2) ^(^i±l!i = 2x + 22/^ = ^ffi = 0, dx dx dx whence we found (3) x + ,f^=:0, or^ = -5. dx dx y This work may be thought of as follows : Let = a;2 + j^ ; then dz ^ d{x'' + y^) ^dz ^ ^zdy^^^ I 2^*^^; dx dx dx dy dx dx' but ;? = 1 by (1) above ; hence dz/dx = 0, and 2x + 2y^ = 0, or ^ = -^. dx dx y Thus the use of the formulas of § 148 is essentially not at all new ; the preceding exercises and the work we have done in §§ 26, 34, etc., really employ the same principle. But the same facts appear in a new light by means of § 148 ; and the new formulas are a real assistance in many examples. 150. Small Errors. Partial Differentials. Another applica- tion closely allied to the work of § 132, p. 252, is found in the estimation of small errors. Example 1. The angle vl of a right triangle ABG (C = 90^), may be computed by the formula tan^=^, or ^ = tan-i ?, where a, 5, c are the sides opposite A, B, C. If an error is made in measuring a or b, the computed value of A is of course false. We may estimate the error in A caused by an error in measuring a, supposing tem- porarily that b is correct, by § 132 ; this gives approximately IX, § 150] PARTIAL DERIVATIVES 289 1 A.A = ^Aa = —^Aa = — ^ Aa, da 1 + ^ «'" + b- where d is used in place of cZ of § 132, since ^1 really depends on b also, and we have simply supposed b constant temporarily. Likewise the error in A caused by an error in b is approximately, _ a AU = Ma6 = —^,Ab = -F^A6. cb I 4-11 «■" + b'^ If errors are possible in both measurements, the total error in A is, approximately, the sum of these two partial errors : \AA\^\A^A\ + \A,A\=^-^^^1+^^1^^- a'^ + b- The methods of § 133, p. 256, give a means of finding how nearly cor- rect these estimates of AaA, A^A, and A.-l are ; in practice, such values as those just found serve as a guide, since it is usually desired only to give a general idea of the amounts of such errors. This method is perfectly general. The differences in the value of a function z =f{x, y) of two variables, x and y, which are caused by differences in the value of x alone, or of y alone, are denoted by ^^z, ^^z, respectively. The total difference in z caused by a change in both x and y is Az =f(x + Ax, y + Ay) -fix, y) = lf{x + Ax,y-\-Ay)-f{x + Ax,y)-]-\-lf{x+Ax,y)-f{x,y)-] as in § 148. The differences A^z and A^z are, approximately,* -1 * More precisely, these errors are A:,Z = ^ . AZ + £'2, A,Z = ^ I • A?/ + E' where | E'^ \ and 1 £"2 1 are less than the maximum M^ of the values of all of the second derivatives of 2 near (r, ?/) multiplied by Ax^, or Ay^, respectively (see §133). And since dz/dy is itself supposed to be continuous, we may AZ = — AX + — A?/ + T'-i, ?j- ru where \ E-i\ is less than 3/2( I Az I + I A^ I )2. [Law of the Mean. Compare §133.] V 290 SEVERAL VARIABLES [IX, § 150 A^z = ■ — Act-, A„2! = — i dx dy whence, approximately, Az = Az + A^z = —Ax- + — Ay. ax dy The products {dz/dx)dx and {dz/dy)dy are often called the partial differentials of z, and are denoted by d^z = —-dx, d z = — dy, whence dz = d^z + d z, ox dy where dx = Ax and dy ={dy/dx)Ax= Ay, approximately. We have therefore, approximately, Az = d^z + dyZ, within an amount which can be estimated as in § 133 and in the preceding footnote. Similar formulas give an estimate of the values of the changes in a function ri =f(x, y, z) of the variables ,r, y, z\ we have, approximately, A^?f = —Ax, ^yU = —Ay, A,u = —Az, dx dy dz Am = A^M + AyU + A,u = —Ax + —Ay + —Az, dx dy dz within an amount vphich can be estimated as in the preceding footnote. The generalization to the case of more than three variables is obvious. EXERCISES LXI. — TOTAL DERIVATIVES AND DIFFERENTIALS 1. Express the total surface area A of a cylindrical can vyith a bottom but no top, in terms of the height h and the radius of the base r. If the volume of the can is given, say 100 cu. in., find a relation between h and r ; and find dA/dr. 2. Find the most economical dimensions for the can described in Ex, 1. 3. Find the most economical dimensions for a funnel made in the form of a right cone, neglecting the outlet hole. IX, § 150] PARTIAL DERIVATIVES 291 4. The pressure p, the volume v, and tlie temperature 6 of any gas are connected by the relation pv = kd, when k is a constant. When no heat escapes or enters it is found by experiment that;^ = c • v-^-^i for air. Ex- press 6 in terms of v alone and find dd/dv. Find the same result directly by § 149. 5. Find dz when z is given in terms of x and y, and y is given in terms of x, by one of the following sets of equations : (a) z = x-^ + y^ 2/ = 2x + 3. (d) z = xy, y = V2x + 3. (6) z = x- - 2/2, y = x3/2. (e) z = sin (r + y), y = x. (c) z-x^ — y"-, y = z. (/) z=Vx^ + y\ y = l/x. 6. Find dz/dt when z = xy, and x = sin f, y = cos t by expressing 2 in terms of t ; without expressing z in terms of t. Interpret this result geometrically. 7. Find dy/dx in each of the following implicit equations by method of Ex. 2, § 149 : (a) x2 + 4 y2 = 1. (c) x3 + 2/3 _ 3 .r2/ = 0. (6) 4x2-92/2 = 36. (rZ) 2/-(2 a - x) = x'. 8. li A, B, C denote the angles, and a, b, c the sides opposite them, respectively, in a plane triangle, and if a, A, B are known by measure- ments, ft = ffl sin B/sin A. Show that the error in the computed value of 6 due to an error da in measuring a is, approximately, dab = sin B esc A da. Likewise show that ?Aft = — a sin B CSC A ctn A dA, and csb = a cos B esc A dB ; and the total error is, approximately, db = dab + d^b + dsb. Note that A and B are expressed in radian measure. 9. The measured parts of a triangle and their probable errors are a = 100 ± .01 ft., A = 100° ±1', i? = 40'^ ± 1'. Show that the partial errors in the side ft are dab = ± .007 ft., dib = ± .003 ft., deb = ± .023 ft. If these should all combine with like signs, the maximum total error would be db=± .033 ft. 10. If a = 100 ft., B := 40°, A - 100°, and each is subject to an error of 1 %, find the per cent of error in 6. 292 SEVERAL VARIABLES [IX, § 150 11. Find the partial and total errors in angle B, when a = 100 ± .01 ft., 6 = 159 ± .01 ft., 4 = 30^ ± 1'. 12. The radius of the base and the altitude of a right circular cone being measured to 1%, what is the possible percent of error in the volume? Ans. 3%. 13. The formula for index of refraction is m = sin i/sin r, i being the angle of incidence and r the angle of refraction. If i = 50" and r = 40°, each subject to an error of 1%, what is m, and what its actual and its percentage error ? 14. Water is flowing through a pipe of length L ft., and diameter Bit., under a head of ^ft. The flow, in cubic feet per minute, is Q = 2356 J ^^^ — liL = 1000, Z> = 2, and fi- = 100, determine the change in Q due to an increase of 1% in ^; in i; in D. Compare the partial differentials with the partial increments. 15. If the coordinates (x, y) are changed to polar coordinates (p, ^), find z in terms of p and ^ if 2 = 4 x^ + y- ; find dz/dp and dz/dd. 16. Find dz/dp and dz/dd it z = xy - ^ y"^. 17. Find dz/dx and dz/dy if 2 = p2 — 2 p cos 6, where (p, 6) are the polar coordinates of the point (x, y) . 18. Find dz/dd if ;s = x^ — 4 ?/2, where x = a tan 6, y = a sec 0, by ex- pressing z in terms of 6 ; without expressing z in terms of d. 19. Find dz/dt ifz- e^'+J'' sin (x^+y) , where x = l + 2t + t-,y = te-*. 151. Significance of Partial and Total Derivatives. The <. formulas of § 148 become of vital importance in scientific and mathematical problems. The methods employed are illustrated by the following typical examples. Example 1. Expansion of a Gas at Constant Temperature.* Thus in the case of a gas under pressure p, we have (1) pv = ke, * Often called isothermal expansion. IX, § 151] PARTIAL DERIVATIVES 293 where v is the volume and d is the absolute temperature, that is, e — C + 273'^ where C is the temperature (C). In general, we have f9\ dB _dB d6 dp _p . v dp dv dv dp dv k k dv If the temperature is constant during a change in volume, the pressure must change, for dd/dv = 0, and therefore (3) ^(JP ^^ = 0, or ^ = _^^Z^ = _£, dp dv dv ' dv dd/dp v where 30/ dv = p/k is the rate of change of temperature which would occur if V alone were changed. Here again the bare fact that dp/dv = — p/v can be obtained with- out using § 148. For since ^ is constant, pt)= const., hence pdv + i; dp = and dp/dv = - p/v. (See Ex. 26, p. 90.) The new fact discovered by § 148 is the second equation in (3), which says that the rate of change of pressure with respect to volume in expan- sion at constant temperature, is equal to the negative of the ratio of the rate of change of temperature when the volume alone changes to the rate of change of temperature when the pressure alone changes. This fact remains strictly true even when (1) is not strictly true ; for if the tem- perature can be expressed as any function of the volume and the pressure, the first equation under (3) remains true. (See Ex. 28, p. 57.) Example 2. Expansion when no Heat escapes or enters* The funda- mental equation j)v = kd of Ex. 1 and equation (2) hold true in any case for periect gases. If no heat escapes from nor enters the gas, its temperature is bound to rise or fall if the product pv of the pressure and the volume does not remain constant ; for dd/dv = if and only if dp/dv = — p/v. For any gas the application of sudden mechanical pressure — such as that of the piston of an air compressor — results in some actual decrease in volume, but the rate dp/dv depends on the nature of the particular gas. For air, it is found experimentally that pv^*^ = const, (nearly) ; whence, from (1), e^m^'^jn^, ^lP^-lA\c.v-^-», ^ = -OAl'-v-^*K k k dv dv k Using (2), we might have written de^p v {x) be any curve through a point P ; if we follow the variations in temperature along that curve, (2) e=f{x,y),y = {x), ^cdh -op IX, § 151] PARTIAL DERIVATIVES 295 we have, by (3), § 148, (3) de = ^dx + ^ dy. ^ ' ex cy Let s be the length of the arc of the curve ; then ds"^ = dx^ + dy^, dx/ds = cos a, dy/ds = sin a, where a-AxDT.hj § 62, p. 107 ; hence, dividing both sides of (3) by ds, we have ^ = L^^ + e^^ = £?cos« + i^sin«. ^ ^ ds dx ds dy ds ex dy This equation shows that the rate of change of the temperature along the curve depends only on the angle a ; all curves tangent to DPT at P give the same result for dd/ds. The equation (4) evidently holds for any function e=f(x, y) what- ever ; often the derivative d9/ds is called a directional derivative, that is, a derivative (or rate of change) of in the direction FT. Example 5. Flow of Water in Fipes. Farticle Derivative. When water is flowing in a pipe, the speed of the water may be considered as follows : . (a) Fixing our attention ^ ;2^ A^ upon a particular point ^ ni y | . ^ Jl l \ the pipe, — say its mouth, — ( — we may consider the speed pj^, gg of various water particles which pass that point. This speed Sa may change as time goes on, it the water pressure varies from any cause. (6) Fixing our attention on a particular tcater particle P, as it moves through the pipe, that particular particle has a speed, Sp, which may change even when the flow through the pipe is perfectly constant ; for if F moves from a wide part of the pipe to a comparatively narrow part (as in the nozzle of a hose) the speed Sp increases. It is clear that Sp = Sa, when F is at A. (c) Let us suppose the pressure is constant. Then Sa depends only on the (fixed) position of the point A ; but Sp depends upon the time t, since the position of F changes with the time : ct ct {d) If the xcater pressure changes, both Sp and S^ change ; that is, S^ and Sp both depend on the pressure p : Sa = ^ function of p alone ; Sp=& function of p and of t. 296 SEVERAL VARIABLES [IX, § 151 If t is assigned a fixed value, sA = Sa, where A is the position pf P at the time « = A;. We have -^'^* dSp^dSp ^ dSpdp^ dt ct dp dt where dSp/dp is the rate of change in Sp with respect to p which would occur if t alone could be kept constant, i.e. the rate of change of Sa with respect to p, or dS^/dp ; and dSp/dt is the rate of change of Sp with respect to t which would exist if p alone were constant.* The equation therefore shows that the actual rate of change of Sp with respect to t is equal to the rate at which Sp would change if p were constant plus the rate at which Sa changes with respect to p times the rate of change of p with respect to t. All of these concepts can be illustrated by the speeds of water particles near the nozzle of an ordinaiy garden hose as the water is turned on or off. EXERCISES LXII. — APPLICATIONS OF TOTAL DERIVATIVES 1. Find dz when z = x^ + 4 y^. Hence find dy/dx for the point x = 1, y = 2 on the curve x2 + 4 2/2 — 17. Find dy/dx for that curve of the family a;^ + 4 2/2 = ^ which passes through x = 2, y = 1, at that point. 2. Find dy/dx for that curve of the family xy = k which passes through the point x = 2, y = 3, at that point. 3. Find dy/dx for that curve of the family x^ -\- y^ — Zxy = ]c which passes through the point (1, 1) at that point. 4. For steam, it is found by experiment that pv^'^"^^ = const., for adiabatic expansion. Find dp/dv and dd/dv, where d, v, p denote the temperature, volume, and pressure, respectively, and pv = kd. 5. The strength of a beam is proportional to bd?/l, where b is the breadth, d the depth, and I the length of the beam. Discuss the effect ^ upon the strength of changes in each dimension separately ; the effect of simultaneous changes in 6 and d when I is constant. 6. In the beam of Ex. 5 if & and d are changed while 1 is constant, find a relation connecting b and d if the strength remains unchanged. Find the rate of change of 6 with respect to d under these circumstances. * For this reason, the partial derivative dSp/dt is often called a "particle " derivative : it is in this case the fictitious rate at which Sp would change if the flow were steady, as P moves along the pipe. The other derivative dSp/dp may be replaced by dSJdp. IX, § 151] PARTIAL DERIVATIVES 29? 7. The amount of deflection Z) of a rectangular beam under a load is proportional to l^/bd^, in the notation of Ex, 5. Find the rates of change of D with respect to each dimension separately. Find a relation between I and d for which D is constant while b is constant ; and find dl/dd in this case. Find a relation between I and b for which D and d are constant and find db/dl in this case. 8. For the beam of Ex. 7 show that if b is constant, dD = (3 l''/bd^)[_dl - il/d)dd]. 9. The collapsing pressure of a boiler tube is given by Fairburn as pro- portional to t^/ld where t, 1, d, respectively, denote the thickness of the material, the length, and the diameter, of the tube. Show how the col- lapsing pressure changes with respect to changes in t and d. 10. The resistance J?, due to water friction for a boat in still water, is proportional to S-D-^^, where S is the speed and D is the displacement. Show how the resistance changes when S changes ; when D changes. 11. If the boat of Ex. 10 is loaded more heavily, D increases ; but S is usually decreased. Find dS/dD if i? is kept constant. 12. The temperature at points of a certain square plate OABC varies inversely as 1 -|- r-, where r is the distance from O. The temperature at O is 100°, Find the rate of change of the temperature (a) along the diagonal OB, (ft) along AC at the center of the plate; (c) along a verti- cal line through the center of the plate. 13. If M is a function of three variables, such as the density or the temperature at points of a solid, the rates of change of u in the directions of the coordinate axes are, respectively, du/cx, du/cy, cu/cz. If s is a variable distance along a line making angles a, /3, y with the coordinate axes, show that the rate of variation of « along this line is du du dx , du dii , du dz du „^„ ^ , du „„^ „ , ^m „„„ — = 1 2-1 — = — cos a -^ cos p -\ cos 7. ds ex ds dy ds cz ds dx dy dz ^ 14. In a spherical shell of inner radius 5 and outer radius 10, the tem- perature decreases uniformly from 100° at the inner surface to 0° at the outer. Show that the rate of variation of the temperature along a radius, at right angles to a radius, along a line inclined 45° to a radius at their point of intersection are, respectively, — 20, 0, — IOV'2. 15. From the value of dy/dx found in Example 3, § 151, show that the equation of the tangent to a plane curve whose equation is given in im- plicit form, f(x, y) = 0, is (x - Xp) (df/dx)p+ (y - yp) (cf/cy)p = 0, where (xp, yp) is the point of tangency, and where the values of the derivatives are to be taken at that point. 298 SEVERAL VARIABLES [IX, § 151 PART II. APPLICATIONS TO PLANE GEOMETRY 152. Envelopes. The straight line (1) y — lex — l?, where k is a constant to which various values may be assigned, has a different position for each value of k. All the straight lines which (1) represents may be tangents to some one curve. If they are, the point P^, (a-, y) at which (1) is tan- gent to the curve, evidently depends on the value of k : (2) x = (k), y=xl^(k); these equations may be con- sidered to be the parameter equations of the required curve. The motive is to find the functions {k) and ij/(k) if possible. Since P^. lies on (1) and on (2), we may substitute from (2) in (1) to obtain : Fig. 67 (3) xl;{k) = k{k) -k', which must hold for all values of k. Moreover, since (1) is tangent to (2) at P^, the values of cly/clx found from (1) and x. from (2) must coincide : from (1) dx_\ from (2) (k) and i{/(k) from the two equations (3) and (4), it is evident that it is expedient to differentiate both sides of (3) with respect to k : (3*) ^p'Qc) = H'(k) + <|>{k)-2k^, (^) k = '-^1 = —1 ^ ■' clx} from (1) dx_\ or kcj>'{k) = yl,'(k). IX, §152] GEOMETRY — ENVELOPES 299 this equation reduces by means of (4) to the form (5) = + (^•) -2k, or <^{k) = 2k, and then (3) gives (6) 4,(Jc) = k(2k)-k^ = k\ Hence the parameter equations (2) of the desired curve are (7) x = 2k, y = k\ and the equation in usual form results by elimination of k : (8) y = f. It is easy to show that the tangents to (8) are precisely the straight lines (1) The preceding method is perfectly general. Given any set of curves (1)' F{x, y, k) = 0, where k may have various values, a curve to which they are all tangent is called their envelope ; its equations may be written (2)' x = {k),y = f{k-); whence by substitution in (1)', (3)' Fi{k)-k-]-^l.i>'{k)-ll * The reason for this differentiation is brought out in the example of § 152. Notice that the equations here are numbered to correspond exactly to the equations of § 152. IX, § 153] GEOMETRY — EVOLUTES 301 Moreover, since (1) is tangent to (2) whence (3*) reduces to (5) 0-2A: = A_[^(fc)_A;]-J^(0-l). Solving for ^(^•), we find : (6) «^ (fc) = ^^ + 2 ^-2^ - 2 A; - ^^ = - 4 A;3. [t follows from (3) that [7) ^ (A;) = Ar' - i^ [<^ {k)-k^ = 3 Tc^^ 1/2, whence the equations of the new curve are [8) a; = - 4 Ar', y = 3 A;- + 1/2. We might proceed to eliminate A:, as in the example of § 152, in order to express the equation of the new curve in usual form ; but when the elimination is at all diificult, as it is here, it is best to keep the equation in the parameter form (8). The 'raph may be plotted from these equations as usual. The iccurate construction of a few normals to the given curve is a jreat assistance in drawing this graph. The new curve is called the evolute of the given curve; the *iven curve is called an involute of the new one : the evolute to xny curve is the envelope of its normals. (See § 154 and Ex. 4, p. 172.) The method used above is perfectly general, and may be used in any problem. Thus the normal to a curve y = / (jc) at a point x — k is [ly y-f(k)=-^{x-k). If these normals, for all values of k, are tangent to the new curve [2)' x = ik), y = Hk), 302 SEVERAL VARIABLES [IX, § 153 direct substitution, followed by differentiation with respect to k, gives (3)' HJc) -f(k)^-y^[ whence (3*)' reduces to (5)' 0-f'{k)=+ f"'^^^ \_{k)-k'\ ^[0-11; w J \ ) [f'{k)Y f'{ky -* an equation which might have been found directly from (5)' of § 162, Solving (5)' for (t>{k), we find : * (6)' "^^ ^ ^ /"(/fc) L ■' ^ ^ /'(A:)J whence, from (3)' (7)' ^(k) =f(k) -^lci>(k)-k] =f(k) + '^ + lf'[\^^' . Denoting /(fc) by y^, f'(k) by m* (the slope at x=k), f"(k) by bk (the flexion at x = k), the equations of the evolute may be written in the form : CSV , »Wfc(l + mfc2) 1+mjl w x = k V , and 2/ = 2/fc H j ^ "k "k These equations may be used to write down the equations of the evo- lute directly ; but it is strongly recommended that the direct solution, as above, be practiced. Frequently the elimination of k between the two equations (8)' is rather difficult, as in the example given above ; hence the equations are very often left in the parameter form (8)'. * Notice that the work breaks down at this point iif"{k) =0. The advan- tage of this direct solution is that such special cases are not so troublesome as when the final formulas alone are used. IX, §153] GEOMETRY — E VOLUTES 303 EXERCISES LXm.— ENVELOPES EVOLUTES 1. Show that the envelope of the set of straight lines y = 3 kx — k^ is 2. Find the envelopes of each of the following families of curves : (a) 2/ = 4 kx — k*. Ans. y^ = 27 x*. (6) y'^ = kx- A;2. Ans. y=±\x. (c) y = kx± Vl + k\ Ans. x- + j/2 = 1. (d) r' = ^•-.r-2^•. Ans. xy^ = -l. (e) (x - A-)- + 2/- = 2 A. .4«s. «/2 = 2 x + 1. (/) 4 a;2 + (y _ A;)2 = 1 _ fc2. ^hs. t/2 + g x2 = 2. (j^) X cos ^ + 2/ sin 5 = 10. ^/ts. x2 + y- = 100. 3. Show that the normal to the curve y = x^ a,t any point (k, F) on it, is y — k^ = — (x — k)/S k'-. Hence show that the evolute oi y = x^ is given by the parameter equations x = (^• — 9 k^)/2, y ={lb k* + \)/{Q k). 4. Taking the equations of an ellipse of semiaxes a and b in the form X — a cos e, y = b sin 0, show that the equation of the normal at any point is by = ax tan 6 + {b- — a-) sin 6. Hence show that the evolute of the ellipse is given by the parameter equations ax=(a2— 62) cos* $, by = (62 _ a2) sin3 0. 5. Find the evolute of the curve y^ = x*. 6. Find the evolute of the curve y = e*. 7. Show that the envelope of a family of circles through the origin with their centers on the parabola 2/2 = 2 x is y'^(x + 1) + x^ = 0. 8. Show that the envelope of the family of straight lines ax + by = I where a + 6 = a6, is the parabola x^/^ + yV'^i = 1. 9. Show that the envelope of the family of parabolas y = xUna- «»x2 sec2 a is 2/ = 1/(4 w) — tox2. [Note. If m = g/(2 vd^), the given equation represents the path of a projectile fired from the origin with initial speed vo at an angle of eleva- tion a.] 10. Find the evolute of the curve y — (e^ + e-')/2. 304 SEVERAL VARIABLES [IX, § 154 154. Properties of Evolutes. In general, taking the equa- tions (8)' of § 153, the equation of the evolute of a given curve y=f{x) may be written: (1) x-x^= ^^- ^, y-yk= \ S where Xi,=k, y^=f(k), mk = f{k) (the slope at x=^'k), h^=if"{lk) (the flexion at x='k). The point (a;^, y,) lies on the given curve y = f(x); the point (a;, y) lies on the evolute; the normal at (x^, y^) to the given curve is tangent to the evolute at (a;, y). Hence the distance B, measured along the normal, from the given curve to the point of tangency on the evolute, is given by the equation : zy = (^ - .,)' + (y - y,y = «\+ <>' + ^^ '^k ^k it follows that D is precisely the radius of curvature (see § 97, and Ex. 4, p. 172) : ^k Hence the radius of curvature of a curve is shown graphically when the evolute is drawn : in Fig. 68, p. 300, for example, the radii of curvature at A, B, C, D, are the lengths AA', BB', CC, DD\ 00', respectively. Notice particularly that the change in the radius of curvature can be followed by the eye very clearly by means of the evolute, as the point on the given curve moves.* 155. Center of Curvature. The point at which the normal to the given curve is tangent to the evolute is at a distance R from the given curve (along the normal) ; this point is called * This is of importance in laying out railroad curves, etc., where the change in the radius is of great moment ; in particular the minimum value of the radius is often important. IX, §156] GEOMETRY -CURVATURE 305 the center of curvature ; its coordinates are precisely the x and y of equations (1), § 154. (See Ex. 2, p. 171.) The angle y8 which the normal makes with the a>axis is shown in Fig. 68, p. 300 ; from the right triangle CAC we have Z ACC = 13, CC = E, CA = x - x,, AC'=>/- y„ and therefore (1) X — x\ = R cos /3, i/ — y^ = R sin ^ ; moreover (2) .V^. = tan^ = -1 = -^V=* x—x^ W/,. dy^./(Lx\ ax since tan fi is the slope of the normal (= — l/'»^/fc) of the given curve, and also the slope of the tangent of the evolute. The circle whose radius is E (the radius of curvature) and whose center is the point (x, y) at which the normal is tan- gent to the evolute, is called the circle of curvature ; its equa- tion is {X — cc)^ + ( Y— yY = E^, where {x, y) is the fixed point on the evolute and (X, Y) is the variable point on the circle of curvature. 156. Rate of Change of R- The rate at which R changes, which was mentioned m § 154, can be obtained as follows. Since B^ = (x- Xk)^ +(y- yk)\ we have (2) RdR = (x- Tk) (dx - dXk) + (y - y*) (dy - *a-). or (3) dH = Cx-Xt)c^x+(2/-y*)(?y ^ Vix-x^y^+iy-yk)-^ ' since ^x - Xu) dXk + (y - yk) dy^ = 0, by (2), § 165. But since (y - yt)/(x - x^) = dy/dx, by (2), § 155, (4) dR = _^^*, = ^=== = Vdx:^ + dy^. and since Vdx:^ + dy^ = ds, where s is the length of arc of the evolute, X 306 SEVERAL VARIABLES [IX, § 156 (5) dR = ds, or r ~ 'di? = ("'"'"rfs, or ^2 - ^i = Sa - Si ; that is : the rate of growth of the radius of curvature is equal to the rate of growth of the arc of, the evolute ; and the difference between two radii of curvature is the same as the length of the arc of the evolute which separates them. This fact gives rise to an interesting method of drawing the original curve (the involute) from the evolute : Imagine a string wound along the convex portion of the evolute, fastened at some point (say D', Fig. 68, p. 300) and then stretched taut. If a pencil is inserted at any point (say C, Fig. 68) in the string, the pencil will traverse the involute as the string, still held taut, is unwound from the evolute. 157. Illustrative Examples. Example 1. The evolute of the curve y = x"^ was found in § 153 to be x = -4F, ?/ = 3A;2+l/2. The radius of curvature of the given curve at the point (x^ = k, yk = k^) is therefore B = V(x - ky +(y- i-2)2 = V(- 4 A;3 - ky + (2 ^2 + 1/2)2 = H4 A;2 + 1)3/2. The rate of change of B with respect to k is — = 6A;(4A;2 + 1)1/2; dk and JB is a maximum or a minimum only where this rate is zero, i.e. where A; = 0. Since dB/dk is negative when ^ < 0, and positive when ^>0, it follows that i? is a minimum when k = 0, i.e. at the point in Fig. 68, p. 300. Tiie value of B at this point is Bo = 1/2 ; this is also evident in the figure. Example 2. To find the evolute and radius of curvature of the cycloid ; ,,v (x = a(t — sin t), ^ [y = ail -cost). The slope m^ at a point (x*., y^) where « = A; is dyk ^ ^ dyk^ dt ^ asint ^ smt ^^^^i. * dx^ dxk a(l — cos<) 1 — cos« 2 dt IX, § 157] GEOMETRY — CURVATURE and the second derivative b^ = cPy,,/dx^k is 307 dm^ cos < ( 1 — cos t) — sin'' t _ dnik _ dt * ~ dx^ ~ dx^ (1 -cosO^ dt a{\-co&t) a{l-cos,tY Aasm\t/2) It follows that the evolute is given by the equations : X 1 + mr . .Tfc - "'*C^ + ""■■'^) = rt(< - sin t)+2a sin t = a{t + sin t). y = Vk+- hk a{\ — cos «) — 2 rt(l — cos t) = — a{\ — cos t) which is another cycloid of the same shape and size, with its vertices at the points y =—2a, x — wa, 3 ira, etc., as shown in Fig. 69. y <\ Inv A ^ — *^ A' / ^^' """'-■-^ X Fig. 69 The radius of curvature is given by the equation ii;2 = (x - a-4)2 + (2/ - yj2 ^ (2flsin02 + (- 2 a(l - cosi))^ - 8 a2(l - cos t). whence i?=4a^Lzi|2ii = 4a^/8in2(|)=4«sinQ) The value of i? at is given by « = : J?o = ; the value of II at C is given by « = tt : 7?c = ^ «• Since the arc OC^ of tlie evolute is equal to the difference of these values of 7?, we have arc OC = 4 a; hence the length of a whole arch of the cycloid is 2 00' = 8a (Ex. 16, p. 155). 308 SEVERAL VARIABLES [IX, § 157 EXERCISES LXI v. — PROPERTIES OF E VOLUTES 1. Find the general equation of the circle of curvature for the curve y = x3. Draw it for the points (1, 1), (2, 8), (1/2, 1/8). 2. Find the radius of curvature, the circle of curvature, and the evolute, for any point on the curve a; = 4 cos 6, y = sin d. 3. Find the radius of curvature, the circle of curvature, and the evolute (in parameter form) for each of the following curves at any point : (a) (^ = si^^. (fZ) (^ = «' [y = 2 cos d. \y = cos t. ,,. fx = sec e, . . jx = cost + t sin t, \y = tan 0. [y = sin t — t cos t. 4. Find the minimum value of the radius of curvature for the curve y = x^ Ans. E = (3/5) ) of (3) are the coordinates of a singular point of the contour line of the surface z=f(x, y) cut out by the plane z = c, where /(a, b) = c. Usually a due amount of care in plotting the curve near tlie singular point will indicate its nature. A detailed discussion is 310 SEVERAL VARIABLES [IX, § 158 given in advanced texts on Calculus.* The points of (3) for which df/dy alone vanishes should also be inspected carefully 159. Illustrative Examples. Example 1. Examine the curve x^ + y^—Sxy = Ofor singular points. In this example /(x, y) = x^ + y^ — S xy = 0. In § 27, p. 45 we found dy/dx for this curve by a process which amounts to the same thing as writing dx dydx dx This equation can be solved for dy/dx [or for dx/dyl imless x^ — y = 0, y^ — x = 0. These equations have the two pairs of solutions (a; = 0, y = 0) and (x = 1, 2/ = 1). The point (0, 0) lies on the curve, and is therefore a singular point. A careful figure, drawn as in Ex. 12, p. 63, shows that the curve crosses itself at this point, and has no single tangent ; such a point is called a double point. (See Tables, III, I5.) The point (1, 1) does not lie on the given curve since /(I, 1) =1 + 1 — 3 = — 1. But it does lie on the contour line of the surface z = x^ + y^—Sxy cut out by the plane z = — 1. A careful graph of this contour line x^ + y^ — Sxy = — I reveals the fact that there is no other point on the curve near (1, 1), although there is another portion of the curve some distance away ; such a point is called an isolated point. The plotting of the figures is facilitated by first rotating the x?/-axes through 45°. Example 2. Examine the curve y^ = x^ for singular points. Here/(a;, y) = y^ — x- = 0, and we write : ^£ + ^f.^y=^2x + 3y^^ = o, dx dy dx dx an equation which determines dy/dx [or dx/dy"] except when x = y = 0. This point (0, 0) lies on the given curve ; hence it is a singular point. Careful plotting (see Tables, III, A) near the point indicates that the curve has a sharp corner at this point. At any other point dy/dx = 2x/(Sy-) =2/(3x1/3). As x approaches zero from either side, this quantity becomes infinite. Hence the tangent approaches a vertical posi- tion as X approaches zero from either side. A corner is called a cusp if the two branches of the curve which meet there have, as here, a conmion tangent line. * See, e.g., Goursat-Hedrick, Mathematical Analysis, Vol. I, p. 110. IX, § 160] ASYMPTOTES 311 Example 3. Examine the cycloid X = a{t — sin <), y = a{\ — cos i) for singular points. The value of dy/dx [or dx/dy"] is given, as above, by the equation dy _ dy/dt _ a sin < _ sin t dx dx/dt a(l— cos 1 — cos «' unless sin « = and 1 — cos t — 0; these equations are both satisfied when « = 0, ± 2 tt, etc., (not at t — ir). Hence the points where t — 0, [i.e. (x = 0, 2/ = 0)], « = 27r \_i.e. (x = 2 7ra, ?/=0)], etc., are singular points. It results from the rules for indeterminate forms (§ 136, p. 263) that lim ^ _ lim 1 - cos ^ ^ lim sin < _ ^ . t^ dy hence dx/dy approaches zero as t approaches zero [i.e. as (x, y) approaches (0, 0)] ; therefore the tangent becomes more and more nearly vertical as we approach the singular point (0, 0) from either side ; the singular points of a cycloid are therefore cusps. When the equations of a curve are given in parameter form, the singu- lar points can be located as in this example, by finding the common solu- tions of the equations dx/dt = 0, dy/dt = ; and it is usually possible to determine, by the rules for indeterminate forms, what happens to the tangent as that point is approached. 160. Asymptotes. The search for the asymptotes of a curve is often facilitated by our knowledge of the Calculus. Vertical or horizontal asymptotes are usually best found by the purely algebraic methods of analytic geometry. Thus if f(x) is a fraction, the curve y =f{x) has a vertical asymptote ic = fc if a factor of the denominator vanishes w^hen x='k. (See, however, § 139, p. 268). If f{x) has a factor tan x or log x or sec X, •■■, y = f(x) may have a vertical asymptote at any point where that factor becomes infinite. Useful rules for horizontal asymptotes result by interchange of x and y. If the asymptote is neither horizontal nor vertical, these elementary means are insufficient. If the tangent to a given curve : (1) y-yp = '>np{x-xj), 312 SEVERAL VARIABLES [IX, § 160 at the point P, (xp, yp), approaches a fixed limiting position (2) y = ax + b as the distance OP from the origin to P becomes infinite, the line (2) is called an asymptote. This will be true if and only if lim 7np = a, and lim (yp — mpXp) = b, where a and b are constants. The value of m^ can be com- puted by any of our usual methods and then lim m^ can be found if it exists. In this work it is useful to notice that the ratio [y/x']p also ajyproaches a if there is actually an asymptote (2) which is not vertical. Example 1. Examine curve x^ + y^ — 3 xy = for asymptotes. The method used in Ex. 1, p. 310, gives hence mp = m =^"1 ; dx]p x — y^Ap lim trip = lim p = lim JL since 1/xp approaches zero, and {_y/x']p approaches a if a exists. Since lim mp = a, we have a —— 1/a^, if a exists, whence a = — 1. The equation of the given curve may be written in the form ;ir= 1 + 3 '?A1 whence it is evident that y/x does approach — 1 as x becomes infinite. Finally the expression yp — mpXp becomes 2xy — x^ — y'' -y^ l _ -xy -| _ y/x 1 . ^ Jp x-y'Up (y/xy^-l/xJp' hence lim (yp — nipXp) = lim IX, § 161] CURVE TRACING 313 since lim [y/x']p =—1 and lim [1/a;]/. = 0. The values of a and b are therefore a =— I, b = — I, and the line ?/ =— a; — 1 is an asymptote. The knowledge of this fact assists materially in drawing an accurate figure. In general, an equation of the form f(x, y) = gives m = - (cf/cx) - (cf/dy). If f(x, y) is algebraic, the value of m can be arranged as above in powers of (y/x) and (1/x) [or of (x/y) and (l/y)] ; and the equation f{x, y) = can also be written in terms of (y/x) and (l/x). The work in any case is similar to that of the preceding example. 161. Curve Tracing. In order to draw a curve whose equa- tion is given, it is often desirable to find whether there are any asymptotes or any singular points before an attempt is made to draw the curve. It is also useful to know the posi- tions of any maxima and minima (§§ 37, 47, 13o) and of any points of inflexion (§ 46, p. 75). The actual construction of a few tangents is often useful, particularly at points of inflexion. Elementary methods should not be abandoned ruthlessly. Building up a graph by adding, multiplying, or dividing the ordinates of two simpler curves ; moving a curve vertically or horizontally; increase or decrease of scale on one axis at a time ; plotting from equations in parameter form ; in some rare instances, rotation of axes ; in all cases, inspection of the given equation for possible simplijications : these elementary methods are even more fundamental and vital than the newer ideas explained above. EXERCISES LXV. — SINGULAR POINTS, ASYMPTOTES, CURVE TRACING 1. Find the asymptotes A, and the singular points .S' for each of the following curves ; then trace each curve. Use elementary methods whenever possible, and use the points of inflexion and the extremes, if any exist. In every case, try to build up the curve from simpler ones ; in most of these exercises, this can be done. 314 SEVERAL VARIABLES [IX, § 161 (a) y = ; A:x = a; no S. a — X (b) y^ = ; A: z = a; no S. X— a (c) 2/2 = x^ — a;* ; no A ; S : (0, 0) , double point. (d) yi = 2x^-x^; A:y=-x+2/3; S:(0, 0), cusp. (e) X = y(x - a)'^ ; A : x = a, y = ; no S. (/) y\2 a-x) = x3; A:x = 2a; s : (0, 0), cusp. (g) x^y = 4 a2(2 a-y); A:y = 0; no /S". (h) y^ = 9 x^ + x^ ; A : y = x + 3 ; S : {0, 0), cusp. (i) ?/2(x2 + 1) = x2(a;2 - 1) ; A:y = ±z; S:(0,0), isolated. U) 2/^(-« - 2) == x3 - 1 ; .4 : a; = 2, y = ± (x 4- 1) ; no ,5. (*) y = e^ ; ^ : ?/ = ; no *9. (I) y = (e^ + e-0/2 ; no ^ ; no S. (»n) y = e-" ; J. : y = ; no /S. (n) ?/ = sec X ; ^ : y = n7r/2, n any odd integer ; no 8. 2. Show that the curve y = 2/(e^ + e^^) = sech x is asymptotic to the X-axis, by building up its graph from that of 1 (l). 3. Show that each of the curves y = xe-^, y = x^e-^, y = xV"*, •••, y = x"e-*, is asymptotic to the x-axis (See Exs. 3, 5, p. 271). 4. Show that the curve 1 (a) has no area, in the sense of § 111, be- tween x = a and x =a +1, nor from x = a + ltox = Go. 5. Show that the curve 1 (6) has an area between x = a and x = a + 1, but not from x = a + ltox = oo. 6. Show that the curve of Ex. 1 (e) has no area between x = a and X = a + 1 , and has no area from x = a-|-ltox = co. 7. Show that the curve y = xlogx ends abruptly at the origin, by building up its graph. [See § 140, p. 269.] 8. Show that the curve y = e~^ sin x is asymptotic to the x-axis. 9. Show that y ■= sin (1/x) has an infinite number of maxima and minima near the origin ; and that it is asymptotic to the x-axis. 10. Build up the graph oiy =x sin(l/x) from Ex. 9. 11. Show that the curves y = (e* -f- e-*)/2=cosh x and y = (e* — e-'^)/2 = sinh X are asymptotic to each other, and to the curve y = e''/2 as x becomes infinite. See Tables, III, E. 12. Build up the graph of y = e-^/*' ; show that it is asymptotic to the line y = 1. [IX, § 162] GEOMETRY OF SPACE 315 PART ni. gp:ometry of space extremes 162. Resume of Formulas of Solid Analytic Geometry. (rt) Distance between two Points (.t-j, ^/i, Zi) and (xo, y.^, z^ : (1) V(a^2 - x,y + (2/2- y,y + (z, - z^\ (Jb) Distance from Origin to {x, y, z) : r = V.«- -f- y- -f z^. (c) Direction Cosines. If «, /3, y denote the angles that a given line makes with the positive directions of the x, y, z axes respectively, then cos a, cos ji, cos y are the direction cosines of the given lines ; and we always have (2) cos- a + cos- p + cos2 7 = 1. If the direction cosines are proportional to three numbers a, b, c, their actual values are cosa = — - , cos/? = /gx Va^ + ft^ + c^ Va2 + 62^c2' cos y = Va^ +b- + c^ If we indicate the direction cosines by single letters, say (4) I = cos «, m = cos 13, n = cos y, we speak of the direction (Z, m, n), (rf) Angle between Two Directions. The angle 6 between the directions (I, m, n) and (V, m', n') is given by (5) cos d = ll' + mm' + nn'. The directions are parallel, if (6) W + ??m' + ?in' = 1. They are perpendicular, if (7) IV + mm' + «n' = 0. 316 SEVERAL VARIABLES [IX, § 162 (e) The Plane. If p is the length of the perpendicular from the origin upon a plane and (I, m, n) is its direction, the plane is denoted by {I, m, n ; p), and its equation is (8) Ix + my + nz =p, or x cos a + y cos /3 + z cos y =p. Since the distance d from the plane (I, m, n; p) to the point {^x, yi, Zi) is (9) d = Ix^ + myi + nzi — p ; the form (8) is called the distarice form, or the normal form, of the equation of a plane. If the axial intercepts of a plane are a, b, c, its equation is (10) ^ ■ ^ ■ ' ^ + f + -=l. a c (11) = 0. The plane through the points (x^, y^, %), (^2, 2/2; ^2), (^3, Vz, ^3) ^^ a;, ?/, 2, 1 ^•1, Vi, ^1, 1 ^2j y2i ^2) 1 ^3J ^SJ ^3> 1 The general equation of the plane is the general equation of the first degree, namely : (12) Ax + By-i-Cz + D = 0. If the direction of the normal to the plane from the origin is {I, m, n), and its distance from the origin is p, we have A B I =■ (13) ^A' + B'+C C p = ^A' + B'+C' -D -JW+W+C' VA' + B'+C' The angle between two planes is the angle between their * The definition of a determinant is given in the Tables, II, C, 5. IX, § 162] GEOMETRY OF SPACE normals ; it is given by (14) cos 317 AA' + BB' + CO' V(^2 + B^ + C'){A'^ + B'' + C"2) The planes are ijarallel, if A/A' = B/B'-^C/C') they are perpendicular, if AA' + BB'+CC' = 0. The distance from the plane ^.r + jB?/ 4- Cfe + Z) = to the point (xi, ?/i, Zi) is ^A^^W+U' (15) (/) The Straight Line. In general, a straight line is repre- sented by the intersection of two planes : (16) Ax-\-By+Cz + D = 0, A'x + B'l/ + C'z + D' = 0. The direction cosines of the line are given by the proportion (17) l:m:n = BC B'C CA • CA' AB A'B' together with the principle (3). The equations of the straight line joining the points (xi, y^, z^) and (o:.2, y-2, z^) are (18) X — Xi _y — Vi _ z —Zi X., -Xi ?/2 - ?/i Z, - 2i The line through the point (x^, y^, Zj) in the direction (J, m, n), is (19) x — Xt _ _ y — ?/i ^ z — Zi I m n 318 SEVERAL VARIABLES [IX, § 163 (gr) Cluadric Surfaces. Equations of the Second Degree. Spheres, center (a, h, c), radius r : (20) (^x-ay + (y-by+(z-cy^r^. Cones, vertices at origin : (Imaginary, if all signs are alike ; otherwise real, and sections parallel to one of the reference planes elliptic.) Ellipsoids and hyperboloids, centers at origin (Tables, III, N) : (22) ±^^±f,±'~ = l. a~ ¥ r All signs on the left +, ellii^soid. One sign on the left — , hyperholoid of one sheet Two signs on the left — , hyperholoid of two sheets. Three signs on the left — , imaginary. Paraboloids, vertices at the origin (Tables, III, ^4,5) : (23) ±S4 = '^- Like signs, elliptic paraboloid; unlike, hyperbolic paraboloid. 163. Loci of One or More Equations in Three Variables. A single equation in three variables, (1) J^Ccc, 2/, s)=0, represents, in general, a curved surface in space. If z is given a series of constant values a^, a^, a^, ••• successively, the coor- dinates X, y will satisfy the equations of the curves (2) F(x,y,a^)=0, F(x,y,a,)=0, F(z, y, a,) = 0, ••' in which the planes (3) z = a•^, z = a2, z = a3, .•• IX, § 163] GEOMETRY OF SPACE 319 cut the surface. These curves are, in fact, contour lines on the surface, aud the totality of them, for all possible values of z, makes vip the surface. Two independent simultaneous equations: (4) f{x,y,z)=(i, ^(U7, t/, 2) = 0, are satisfied, in general, by the intersection of two surfaces, and therefore represent a curve in space. TJiree independent simultaneous equations, (5) f{x,y,z)=0, (x,y,z)=-.0, ^pifc, y, z) =0, are true, in general, only at certain isolated points; those, namely, in which the curve represented by two of the equations cuts the surface represented by the third. A single equation from tohich one of the coordinates is missing is a cylinder ivith axis parallel to the axis of the missing coordi- nate. Thus (6) /(a?, 2/)=0, interpreted in space, is a cylinder parallel to the z-axis. Its trace on the a;i/-plane is the plane curve, (7) f{x,y) = Q,z = Q. EXERCISES LXVL— RESUME OF SOLID GEOMETRY 1. Find a straight line tlirough each of the following pairs of points ; find its direction cosines. (a) (0, 1, 0) and (2, 3, 5). (c) (4, 1, - 5) and (2, 1, -3). (b) ( - 1, 2, - 3) and (2, - 1, 0). {d) (5, 3, 7) and (5, - 2, 7). 2. Find the direction cosines of each of the following planes : (a) 2 X - 3 y + 4 = 5. {c) y-Zz^2. iP) X -\-y -\-z = 0. {d) z = 2x -y + i. 3. Find the equations of a line formed by the intersection of the planes 2 (a) and 2 (6), in the form (19), and find its direction cosines. 4. Proceed as in Ex. 2 for each of the combinations formed by two of the planes mentioned in Ex. 2. 320 SEVERAL VARIABLES [IX, § 163 5. Reduce each of the equations in Ex. 2 to normal form ; find the distance from each of these planes to the origin. 6. Find the equation of a plane through the origin which (a) also passes through the two points of Ex, 1 (a); or (b) is parallel to the plane 2 (a); or (c) is perpendicular to each of the planes 2 (a) and 2 (c). 7. Find the angle between each pair of planes in Ex. 2. 8. Find the angle between the direction specified by Ex. 1 (a) and that specified by Ex. 1 (b) ; between the directions specified by each pair of lines mentioned in Ex. 1. 9. Find the center and the radius of each of the following spheres : (a) x2 + 2/2 + 2:2 + 2 X - 4 2/ + 6 5! = 2. (6) x^ + y'^ + z^+12x — y — 4z + 40 = 0. 10. Find the equation of a sphere (a) whose center is (2, — 1, 4) and whose radius is 3 ; (b) one of whose diameters joins (2, 4, — 1) and (3, 1, 6); (c) whose center is (1, 0, 5) and which passes through (3, 1, — 2). 11. Reduce to standard form and identify each of the following sur- faces : (a) a;2 + 41/2 + ^2 _6x + 2^ = 6. (d) 9x^-y^ + iz^-\-6x + 10y--25. (b) x2 - 4 j/2 _ 6 a; + 2 s = 6. (e) 4x^ - y- — 4:X + 6y = 15. (c) 9x2-2/2+4s24.6x+10y = 10. {/) z^ + 9 x"^ - 2 z + iy ^ 0. 12. Represent each of the following equations or groups of equations geometrically in space of 3 dimensions ; find the trace, if any exists, on each coordinate plane, and on each of a series of parallel planes : (a) z = xy. (b)x^ = y^ + z^. (c) x^ + y^ + z"^ = 1. (d)y = smx. (e) xyz = \. (/) x2 + 2/2 = sin z. (g) x + y = e'. (h) z = e'+y. (i) x^ + y'^ + z^ = i,x + y = 0. (j) z- - x^ + 2/2, ^ = 1 - x. {k) X = cos z, y= sin x. {I) x + y =: e', y = 2 x. (m) y = z^, X = 2/2. (n) x- — z — y, x + y = 0, z + y=0. (o) x2 — y2 —4 2, x — 2/ = 4, X + 2/ = 7. (p) X + y = z, y + z = x, z + x-y. IX, § 164] TANGENT PLANE — EXTREMES 321 164. Tangent Plane to a Surface. Let Pq be the point (xo, rjo, Zo) on the surface z =f(x, y). Let P„7\ be the tangent line at Pq to the curve cut from the surface by the plane y = yo and PqT^ the tangent line to the curve cut from the surface by the plane x = X(t. The plane containing these two lines is the tangent plane to the surface at Pq. Since this plane goes through Pi), its equation can be thrown into the form (1) z-Zf^ = A(x - .To) -\-B(y- ?/n). If we set y = yo we find the equation of PqT^ in the form : (2) But, from the form : (3) Hence Z — Zn = A(X — Xn). 33, p. 58, the equation of PqT^ may be written in =^1 dxjo (x - Xo). A=^\; likewise B=^']- Thus the equation of the tangent plane is ^o)! or, what is the same thing, (6) z -Zo= -^ (x-Xo) + ~\ (y- yo)- 322 SEVERAL VARIABLES [IX, § 164 It is important to notice the great similarity between this equation and the equation of § 148. In fact (7) expresses the fact that if dx, dy are measured parallel to the x and y axes from the point of tan- gency {x^, y^, Zq), dz represents the height of the tangent plane above (xq, yo, Zq). Equation (7) furnishes a good means of re- membering (6). 165. Extremes on a Surface. If a function z=f(x, y) is represented geometrically by a surface, it is evident that the extreme values of z are represented by the points on the surface which are the highest, or the lowest, points in their neighborhood : (1) f{x^, yo) >f{xo + h,ijo + k), if / {xo, yo) is a maximum, (2) /(.To, yo) = 9, ioo = bO), (p.^^lo, 703 = 90). A.ssuming that p = aw + /3, find the values of a and /3 which make the sum S of the squares of the errors least. (Compare Ex. 18, p. 69, and § 121, p. 229.) Computing p by the formula aio+p, the three values are p'i = 20a + p, p'2 = 50 a + /3, p'3 — 90 a + p. Hence the sum of the squares of the 324 SEVERAL VARIABLES [IX, § 165 errors is S={p'i- Pi) 2 + (p'2 - P2Y + (P'z - PzY = (20 a + ^ - 5)2 + (50 a + iS - 9)2 4- (90 a + /3 - 15)2. In order that ^S be a minimum, we must have 1 :^' = 20 (20 « + iS -5) + 50 (50 a + /3 - 9) + 90(90 a + /3- 15) = 0. 1 £i? = (20 « + ^ - 5) + (50 a + ^ -9) + (90 « + ^ - 15) = 0. that is, after reduction, 1100 a + 16 ^ - 190 = 0, « = lU = .143, whence 160 « + 3 ^ - 29 = 0, /3 = -V?if = 2-03. If the usual grapli of the values of p and w is drawn, it will be seen that p = aw + /3 represents these values very well for a = .143, j3 = 2.03 and it is evident from the geometry of the figure that these values render S a minimum, S = .0545 ; for any considerable mcrease in either « or /3 veiy evidently makes >S' increase. Since this is the only critical point, it surely corresponds to a minimum, for the function S has no singularities. This conclusion can also be reached by thinking of ^S" as represented by the heights of a surface over an a^ plane, and considering the section of that surface by the tangent plane at the point just found as in Ex. 3 below ; but in this problem the preceding argument is simpler. It IS customary to assume that the values of a and /3 which make S a Ininimum are the best compromise, or the "most probable values " ; hence the most probable formula for p is p — .143 lo -\- 2.03. The work based on more than three trials is quite similar ; the only change being that S has n terms instead of 3 if n trials are made. Example 3. Find the most economical dimensions for a rectangular bin with an open top which is to hold 500 cu. ft. of grain. Let X, ?/, h represent the width, length, and height of the bin, respec- tively. Then the volume is xyh ; hence xyh = 500 ; and the total area z of the sides and bottom is (a) z = xy + 2ky + 2hx^xy + '-^+m. X y If this area (which represents the amount of material used) is to be a minimum, we must have ^ ^ dx ^ x^ ' dy y2 The plane 2 = 300 —15— Fig. 71 IX, §166] TANGENT PLANE — EXTREMES 325 Substituting from the first of these the value y = 1000/x- in the second, we find r^ y (c) X — 0, whence x =: 0, or x = 10. 1000 ,„. The value a; = is obviously not worthy of any consideration ; the value x — 10 gives y = 1000/x- — q- = 10 and h = bOO/(xy) = o. The value of z when x = 10, y = 10 is 300. If the equation (a) is represented graphically by a surface, the values of z being drawn vertical, the section of the surface by the plane z = 300 is represented by the equation (d) xy + i^ + ^^ = 300, or x^ - 300 xy + 1000(x + y)=0. X y This equation is of course satisfied by x = 10, y = 10. If we attempt to plot the curve near (10, 10), — for example, if we set ?/ = 10 + A; and try to solve for x in the resulting equation : (10 + kyx"- - (300 k + 2000)x + 1000(10 + k) = 0, the usual rule for imaginary roots of any quadratic ax^ + 6x + c = shows that 62 - 4 ac =- 1000 k^-iAk-^s- 30] <0 for all values of k greater than — 7.5. Hence it is impossible to find any other point on the curve near (10, 10). It follows that the horizontal tangent plane z — 300 cuts the surface in a single point ; hence the sur- face lies entirely on one side of that tangent plane. Trial of any one convenient pair of values of x and y near (10, 10) shows that z is greater near (10, 10) than at (10, 10) ; hence the area z is a minimum when X = 10, 2/ = 10, which gives h =<>. 166. Final Tests. Final tests to determine whether a func- tion fix, y) has a maximum or a minimum or neither, are somewhat difficult to obtain in reliable form. Comparatively simple and natural examples are known which escape all set rules of an elementary nature.* (See Example 1 below.) * For a detailed discussion, see Goursat-Hedrick, Mathematical Analysis, Vol. I, p. 118. 326 SEVERAL VARIABLES [IX, § 166 One elementary fact is often useful: if the surface has a maximum at (xq, y^), every vertical section through (xq, y^ has a maximum there. Thus any critical point (xq, y^) may be dis- carded if the section by the plane x = Xq has no extreme at that point, or if it has the opposite sort of extreme to the section made hy y = y^. The safest final test, and the one very easy to apply, is to actually draw the section of the surface made by the horizontal tangent plane, as in Ex. 3, § 165. Then a test of a few values quickly settles the matter. Example 1. The surface z = (y — x!^) (y — 2 x'^) has critical points where ¥.= -exy + 8x^ = 0. S^ = 22/-3x2 = 0; dx ' dy that is, the only critical point is (x = 0, y = 0). The tangent plane at that point is 2 = 0. This tangent plane cuts the surface where (y-x^){y-2x'')=0; that is, along the two parabolas y = x^,y-2 x"^. At a; = 0, 2/ = 1, the value of 2 is + 1 ; hence z is positive for points (x, y) inside the parabola y = 2 x^. At x = 1, y =0, the value of 2 is +2 ; hence z is positive for all points (x, y) outside the parabola y = x^. At the point x = 1, ?/ = 1.5, the value of is — .25 ; hence z is negative between the two parabolas. It is evident, therefore, that z has no extreme at x = 0, ?/ = 0. Fig. 72 A qualitative model of this extremely interesting surface can be made quickly by molding putty or plaster of paris in elevations in the unshaded regions indicated above, with a depression in the shaded portion. Another interesting fact is that every vertical section of this surface through (0, 0) has a minimum at (0, 0) ; this fact shows that the rule about vertical sections stated above cannot be reversed. Moreover, this surface eludes every other known elementary test except that used above. IX, § 166] TANGENT PLANE — EXTREMES 327 EXERCISES LXVIL — TANGENT PLANES EXTREMES 1. Find the equation of the tangent plane to eacli of the following surfaces at the point specified : (a) z = x-^ + dy^, (2, 1, 13). Ans. 3 = 4x + 18y-13. (b) e = 2x2 -4 2/2, (3,2, 2). Ans. z=12x-16y-2. (c) z = xy, (2, - 3, - 6). Ans. 3x~2y + z = 6. (d) z = (x + yy, (1, 1,4). Ans. 4x + 4y- z = 4. (e) z = 23-2/2 + 2/3^ (2, 0, 0). Ans.z^O. 2. The straight line perpendicular to the tangent plane at its point of tangency is called the normal to the surface. Find the normal to each of the surfaces in Ex. 1, at the point specified. 3. At what angle does the plane x + 2y — z + S =0 cut the parabo- loid X- + 2/- = 4 2 at the point (6, 8, 25) ? 4. Find the angle between the surfaces of P^xs. 1 (a) and 1 (6) at the point (VlT^, 1, 22). Find the angle between each pair of surfaces in Ex. 1, at some one of their points of intersection, if they intersect. 5. Find the tangent plane to the sphere o:- + ?/" + z'^ = 25 at the point (3, 4, 0) ; at (2, 4, V5). 6. At what angles does the line x = 2y = 3z cut the paraboloid y = x^ + z"'. 7. Find a point at which the tangent plane to the surface 1 (a) is horizontal. Draw the contour lines of the surface near that point and show whether the point is a minimum or a maximum or neither. 8. Proceed as in Ex. 7 for each of the surfaces of Ex. 1, and verify the following facts : (6) Horizontal tangent plane at (0, 0) ; no extreme. (c) Horizontal tangent plane at (0, 0) ; no extreme. (rf) Horizontal tangent plane at every point on the line x + y = ; weak minimum at each point. (e) Horizontal tangent plane at every point where 2/ = ; no extreme at any point. 328 SEVERAL VARIABLES [IX, § 166 9. Find the extremes, if any, on each of the following surfaces : (a) 3 = X"- + 4 2/2 - 4 x. (Minimum at (2, 0, — 4).) (&) z^x^-Zx-y'^. (See Tables, Fig. Ij.) (c) 2 = x3 - 3 X + y- (x - 4). (See Tables, Fig. Ij.) (d) z=[(x- ay + ?/-] [(x + a)2 + ?/2]. (Similar to TaftZes, Fig. I7. ) (e) 2 = x* — 6 X — 2/2. (Draw auxiliary curve as for Fig. Ii.) (/) = x^ — 4 2/2 + xy-. (Draw auxiliary curve as for Fig. I2.) (gr) 2 — x^ + 2/^ — 3 xy. (Draw by rotating xy-p\ane through 7r/4.) 10. Redetermine the values of a and /3 in Example 2, § 165, if the additional information (p = 23, ta = 135) is given. 11. Find the values of u and v for which the expression (aiu + biv — ci)2 + (aou + 62V — C2)2 + (asM + 63U — C3)2 becomes a minimum. (Compare Ex. 10. ) 12. Show that the most economical rectangular covered box is cubical. 13. Show that the rectangular parallelopiped of greatest volume that can be inscribed in a sphere is a cube, [Hint. The equation of the sphere is x"^ + y^ + z^ = 1 ; one corner of the parallelopiped is at (x, y, z); thenF = 8x2/2, where z =Vl— x^— 2/2 ] 14. Show that the greatest rectangular parallelopiped which can be inscribed in an ellipsoid x''/a2 + y'-/b'- + z'^/c'^ = 1 has a volume F= 8 a6c/ (3V3). 15. The points (2, 4), (6, 7), (10, 9) do not lie on a straight line. Under the assumptions of Ex. 2, § 165, show that the best compromise for a straight line which is experimentally determined by these values is 24 2/ = 15 X + 70. 16. The linear extension E (in inches) of a copper wire stretched by a load W (in pounds) was found by experiment (Gibson) to be ( PT = 10, E = .06), ( TT = 30, ^ = .17), ( >r = 60, ^ = .32). Find values of a and /3 in the formula E = aW + ^ under the assumptions of § 165. 17. The readings of a standard gas meter 8 and that of a meter T being tested were found to be ( T = 4300, S = 500), (T = 4390, S = 600), (T' = 4475, 6' = 700). Find the most probable values in the equation T = aS + )3 and explain the meaning of a and of /3. IX, § 167] TANGENTS AND NORMALS 329 18. The temperatures d° C. at a depth d in feet below the surface of the grouud iu a iiiiue were found to be d = 100 ft., 6 = 15^.7, d = 200 ft., — Iti'^.d, (1 — 300 ft., 6 = 17^.4. Find an expression for the temperature at any depth. 19. Redetermine, under the assumptions of § 166, the most probable values of the constants iu Exs. 1-5, p. 2ot). 20. The points (10, 3.1), (3.3, 1.6), (1.25, .7) lie very nearly on a curve of the form «/x + ^/y = 1. Use the reciprocals of the given values to find the most probable values of « and /3. 21. The sizes of boiler flues and pressures under which they collapsed were found by Clark to be {d = 30, p = 76), (d = 40, p = 45), (d = 50, p = 30). These values satisfy very nearly an equation of the form p z= k • d" or logp = n log d + log k, where d is the diameter in inches, and p is the pressure in pounds per square inch. Using the logarithms of the given numbers, find the most probable values for n and log k. 22. Recompute, under the assumptions of § 165, as in Ex. 21, the values of constants in Exs. 17, 19, pp. 232-233. 167. Tangent Planes. Implicit Forms. If the equation of a surface is given in imjilicit form, F{x, y, z) — 0, taking the total differential we find : /ix dF. , dF J , dF, -. (1) dx + -—dy + -~dz = 0. ox ay dz But, by virtue of F(x, y, z) = 0, any one of the variables, say z, is a function of the other two ; hence (2) dz=^dx + ^^dy. dx ay Putting this in the total differential above and rearranging: ^ ^ \dx dz dx) ^\dy^ dz dyj ^ But dx and dy are independent arbitrary increments of x and of y J and since the equation is to hold for all their possible 330 SEVERAL VARIABLES [IX, § 167 pairs of values, the coefficients of dx and dy must vanish separately. This gives .^. dz ^ dF/dx dz ^ dF/dy ^ ^ dx dF/dz' dy~ dFjdz Substituting these values in the equation of the tangent plane, and clearing of fractions, we obtain X (6) fl/^-^'^ + fl'^-^'^ + flo^^--^^"' the equation of the tangent plane at {x^, y^, Zq) to the surface F{x,y,z)=0. 168. Line Normal to a Surface. The direction cosines of the tangent plane to a surface whose equation is given in the explicit form z =f(x, y) are proportional (§ 164) to (1) dz/dx^, dz/dy\, and - 1. Hence the equations of the normal at {xq, y^, Zq) are (2) 2/o z-Zo dz/dx^o dz/dy^o — 1 The direction cosines of a surface whose equation is given in the implicit form F(x, y, z) =0 are proportional to (3) dF/dx-]o, dF/dy-],, dF/dz],, so that the equations of the normal to this surface are x-Xq y — yo ^-zo (4) dF/dx]o dF/dy]o dF/dz], 169. Parametric Forms of Equations. A surface S may also be represented by expressing the coordinates of any point on it in terms of two auxiliary variables or parameters : \_S] X =f(u, v), y = (f>(u, v), z = ij/{u, V). IX, § 169] TANGENTS AND NORMALS 331 If we eliminate u and v between these equations, we obtain the equation of the surface in the form F (x, y, z) = 0. Similarly a curve C may be represented by giving x, y, z in terms of a single auxiliary variable or parameter t : [C] x^fit), y=(t), z = ^(t). The elimination of t from each of two pairs of these equations gives the equations of two surfaces on each of which the curve lies, in the form (4), § 163. In particular, taking t = x gives the curve as the intersection of the projecting cylinders : [P] y = (x), z = ^{x). If, in the parametric equations of a surface, one parameter (say u) is kept fixed while the other varies, a space-curve is described which lies on the surface. Now if u varies, this curve varies as a whole and describes the surface. The curve on which u keeps the value k is called the curve u = k. Similarly, keeping v fixed while ?< varies gives a curve v = k' . The intersection of an u = k with an v = k' gives one or more points (k, k') on the surface. The pair of numbers (^•, k') are called the curvi- linear coordinates of points on the surface. Simple examples of such coordinates are the ordinaiy rectangular coordinate system and the polar coordinate system in a plane. Thus (2, 3) means the point at the intersection of the lines a: = 2, ?/ = 3 of the plane ; in polar coordinates, (5, 30°) means the point at the intersection of the circle r = 5 with the line = 30°. Example 1. The equations of the plane x + y -{- z = 1 may be written, in the parametric form : x= u, y = V, z z=l — u — v. Let the student draw a figure from these equations by inserting arbitrary values of u and v and finding associated values of x, y, z. Another set of parameter equations which represent the same plane is x = u + v, y = u — v, z = — 2u + I. Thus several different sets of parameter equations may represent the same surface. In the first form, put u = k. Then, as v varies, we obtain the straight line z = k, y = v, z — \ — k — V, 332 SEVERAL VARIABLES [IX, § 169 which lies in the given plane. As k varies this line varies ; its different positions map out the entire plane. Likewise, u = A;' is a line varying with k' and describing the plane. The intersection of two of these lines, one from each system, is point {k, k') of the plane. Example 2. The sphere x^ + t/^ + ^2 - cfl may be represented by the equations : X = a cos 9 cos 0, y = a cos ^ sin 0, z =a sin 6. Here the parameters 6 and (p are respectively the latitude and the longi- tude. Thus = k is a parallel of latitude ; <^ = ^•' is a meridian ; and their intersection (A-, k') is a point of latitude k and longitude k' . [If a is allowed to vary, the equations of this example define polar coordinates in space ; but the colatitude 90° — 6 is often used in place of ^.] Example 3. The equations X = a cos t, y = a sin t, z = bt, represent a space curve, namely a helix drawn on a cylinder of radius a with its axis along the ^-axis. The total rise of the curve during each revolution is 2 nb. If a is replaced by a variable parameter u, the helix varies with u, and describes the surface X = u cos t, y = u sin ^, z = bt, which is called a helicoid. The blade of a propeller screw is a piece of such a surface. 170. Tangent Planes and Normals. Parameter Forms. When a surface is given by means of parametric equations, (1) x=/(m, v), y = 4>(u,v), z = ^{u,v), the equation of the tangent plane is found as follows. Elimination of u and V would give the equation in the implicit form jP(x, ?/, z) - 0. If the parametric values of x, y, z are substituted in this equation the resulting equation is identically true, since it must hold for all values of the inde- pendent parameters ii, v; hence (2) ^ = 0, and ^ = 0, ^ du dv that is 4 dx du dy du dz du ' dx dv dy dv dz dv IX, § 170] TANGENTS AND NORMALS Solving these, we find : 333 (4) dx ' dy ' dz dy 8z dz dx dz dy_ du du du du du du dy dz dz dx dx dy dv cv dv dv dv dv hence the equation of the tangent plane is (x — Xo) dy^ dz^ du du dy dz dv dv + (y - 2/o) az dx du du dz dx dv dv + (^ dx dy ru du zo) dx dy dv dv while the equations of the normal are X — .To y — 2/0 di dz_ du du cz ex du du dx dy_ du du dy dz dv dv dz dx dv dv 1, dx dy_ dv dv = 0; EXERCISES LXVra. - EQUATIONS NOT IN EXPLICIT FORM 1. Determine the tangent plane and the normal to the ellipsoid x-^ + 4?/2 + 2- = 36 at the point (4, 2, 2), first by solving for z, by the methods of § 164 ; then, vfithout solving for z, by the methods of §§ 167-168. 2. Detei-mine the tangent planes and the normals to each of the fol- lowing surfaces, at the points specified : (a) x^+y'^ + z^ = a2 at (xq, t/o- 2^0). (6) x^ - 4 ?/2 + ^2 = 36 at (6, 1, 2). (c) x2 - 4y2 _ 9^2 = 36 at (7, 1, 1). (d) x2 +2/2 _ «2 = at (3, 4, 5). (e) a^ + a:2y - 2 22 = at (1, 1, - 1). (/) 22 = gx+y at (0, 0, 1). 3. Find the angle between the tangent planes to the ellipsoid 4 x2 4-9 j/2 + 36 ^2 = 36 at the points (2, 1, z^) and (- 1, - 1. 21). 4. At what angle does the 2-axis cut the surface z^ = e^+i' ? 334 SEVERAL VARIABLES [IX, § 170 5. Obtain the equation of the tangent plane to the helicoid X = u cos V, y = n sin V, z — v, at the point m = 1, u = 7r/4. 6. Taking the equations of a sphere in terms of the latitude and longi- tude (Example 2, § 169), find the equation of its tangent plane and the equations of the normal at a point where 6 = — 45° ; at a point where = 60°,

, y = S cos sin 4>, z = sin ^, at ^ = <^ = 7r/4. 10. The surfaces z = x'^ — 4 y^ and z = 6x intersect in a curve, whose equations are the two given equations. Find the tangent line to this curve at the point (8, 2, 48) by first finding the tangent planes to each of the surfaces at that point ; the line of intersection of these planes is the required line. 11. Find the tangent line to the curve defined by the two equations 16 x2 - 3 2/2 = 4 2 and 9 x2 + 3 2/2 _ 22 - 20 at (1, 2, 1). 171. Area of a Curved Surface. Let s he a portion of a curved surface and R its projection on the ,T?/-plane. In B take an element Ax Ay and on it erect a prism cutting an element AS out of S. At any point of AS, draw a tangent plane. The prism cuts from this an ele- ment AA. The smaller Ax Ay (and therefore A*S') becomes, the more nearly will the ratio AA/AS approach unity, since the limit of this ratio isl. Suppose now that the area B is all divided up into elements AxAy and IX, § 171] AREA OF A SURFACE 335 that on each a prism is erected. The area (S' will thus be divided up into elements AS and there will be cut from the tangent plane at a point of each an element A^. One thus gets (1) S: lim V A^. But if 7 is the acute angle that */ the normal to any AA makes with the 2-axis, we have (2) AA = sec y AxAy; hence (3) im V A^ = lim X (sec y AxAy) = \ \ sec y dx dy. : li Aj=0 ■ Ak:^) Of course sec y is a variable to be expressed in terms of x and y from the equation of the surface. The limits of integration to be inserted are the same as if the area of B were to be found by means of the integral lldxdy. If the surface doubles back on itself, so that the projecting prisms cut it more than once, it will usually be best to calculate each piece separately. When the equation of the "surface is given in the form z =f{x, y), the direction cosines of the normal are given by cos a : cos /3 : cos y = ■ 1. Taking cos y positive, that is y acute, we may write (4) gjnaD^'' and 4 -=nv(i)^+(i)'™»- The determination of sec 7, when the .surface is given in the form F(x, y, z) = 0, is performed by straightforward transformations similar to those used in §§ 167-170; they are left to the student. 336 . SEVERAL VARIABLES [IX, § 171 EXERCISES LXIX.-AREA OF A SURFACE 1. Calculate the area of a sphere by the preceding method. 2. A square hole is cut centrally through a sphere. How much of the spherical surface is removed? 3. A cylinder intersects a sphere so that an element of the cylinder coincides with a diameter of the sphere. If the diameter of the cylinder equals the radius of the sphere, what part of the spherical surface lies within the cylinder ? 4. How much of the surface z = xy lies within the cylinder x^+y'^=l? 5. How much of the conical surface z"^ = x"^ + ?/2 lies above a square in the x2/-plane whose center is the origin ? 6. Show that if the region B of § 171 be referred to ordinary polar coordinates, AA = rsecy ArAO, approximately. (See [B], p. 212.) 7. Using the result of Ex. 6, show that S = \ I »' sec y dr dd. 8. Show that, for a surface of revolution formed by revolving a curve whose equation is z =f(x) about the z axis, sec 7 = Vl + \_df{i')/dr']--, where r = Va;^ + y'^. 9. By means of Exs. 7, 8, show that the area of the surface of revo- lution mentioned in Ex. 8 is where a is the value of r at the end of the arc of the generating curve. (See Ex. 13, p. 129.) 10. Compute the area of a sphere by the method of Ex. 9. 11. Eind the area of the portion of the paraboloid of revolution formed by revolving the curve z^ = 2 mx about the x axis, from x = to x = k. 12. Show that the area of the surface of an ellipsoid of revolution is 2 7r& [?) +(a/e) sin-^e], where « and b are the semiaxes and e the eccentricity, of the generating ellipse. 13. Show that the area generated by revolving one arch of a cycloid about its base is G4 ira^/S. 14. Show that the area of the surface generated by revolving the curve a;2/3 -j- 2^2/3 _ (j2/3 about one of the axes is 12 way 5. IX, §172] TANGENTS TO CURVES — LENGTHS 337 172. Tangent to a Space Curve. Let the equation of the curve be given in paninietric form x =/(<), y = {p, q), obtain cu/dx and du/dy by a process analogous to that of Exs. 4, 5. 7. Proceed as in Ex. 3, by the methods of Exs. 4, 5, in each of the following cases : (a) u = r^- cos2 6, (b) u = rc.^^ (r) u = log r. 8. Find the volume of that portion of a sphere of radius 4 ft, which is bounded by two parallel planes at distances 2 ft. and 3 ft., respectively, from the center, on the same side of the center. 340 SEVERAL VARIABLES [IX, § 173 9. Determine the position of the center of mass of the solid described in Ex. 8. 10. What is the nature of the field of integration in the integral Show that the same integral may be written in the form lo"^' r -^(^' y^ 'y ''^ + i/vJo^"^-^^^' y^ '^ 'y- 11. Find the volume cut from the sphere x"^ + y"^ + z^ = a^ by the cylinder x^ + y^ — ax = 0. 12. Find the volume cut from the sphere x'^ + y'^ + z- = a^ by the cone (x - ay + 2/2 - 22 = 0. 13. Show that the surface of a zone of a sphere depends only upon the radius of the sphere and the height 6 — a of the zone, where the bounding planes are z = a and z = b. 14. Find the area of that part of the surface k-z = xy within the cylinder a;2 + 2/2 = k'^. 15. Find the center of gravity of the portion of the surface described in Ex, 14, when k = I. 16. Find the moment of inertia about its edge, of a wedge whose cross section, perpendicular to the edge, is a sector of a circle of radius 1 and angle 30°, if the length of the edge is 1, and the density is 1. 17. The thrust due to water flowing against an element of a surface is proportional to the area of the element and to the square of the com- ponent of the speed perpendicular to the element. Show that the total thrust on a cone whose axis lies in the direction of the flow is k7rr^vy(r^+}i^)i. 18. Calculate the total thrust due to water flowing against a segment of a paraboloid of revolution whose axis lies in the direction of the flow. (See Ex. 17.) 19. Show that the thrust due to water flowing against a sphere is 2 kirrH^/Z. Compare with the thrust due to the flow normally against a diametral plane of this sphere. IX, § 173] GENERAL EXERCISES 341 20.*t Given a function /(a;, y), consider the function (f>{t) =/(a + ht, b + kt), and show, by means of Maclaurin's series for 0(f) [see [D*]', § 134, p. 258,] that, upon inserting the special value f = 1, we obtain : f(a + h, b + k) =f(a, b) + \h^ + k^~\ + L ex C'yjx=a +i.[h^^^+2hk-^+k^m+... '" 2!L dx^ dxdy dy^Jx=a »=» (71 - 1) 1 L c^''-i ^ ^ ax»-2gy J^ !r=b where | ^„ ] < ilf (| A | + | A; |)" -j- ?( ! , and where M is the maximum of the absolute values of all the nth derivatives in a rectangle whose sides a.Te X = a, X = a + h, y = b,y = b + k. [Taylor's Theorem.] 21,t Assuming the truth of the formula of Ex. 20, show that the spe- cial values a = 0, b = 0, h = X, k = y, lead to the formula /(.,,)=/(0,0) + [.| + !,|]_^^^^+... + S.. 22. Expand each of the following functions by use of the formula of Ex. 21, in powers of x and y as far as terms of the second degree : (a) sin(x + j/). (6) e2«+3v. (c) cos(^x^ + y^. 23. Find the critical points, if any exist, for the surface z = x- + 2y^ — 4 X — 4 y + 10. Is the value of z an extreme at that point ? Draw the contour lines near the point. 24. Determine the greatest rectangular parallelepiped which can be inscribed in a sphere of radius a. 25. The volume of CO2 dissolved in a given amount of water at tem- perature ^ is \ e 6 10 15, [v 1.80 1.46 1.18 1.00. Determine the most probable relation of the form v = a + bd. 26. Determine the most probable relation of the form S — a + bF^ from the data : | P 550 650 750 850, \S 26 35 62 70. 27. Determine the most probable relation of the form y = ae^ from the data : f x 1 2 3 4, y .74 .27 .10 .04. 342 SEVERAL VARIABLES [IX, § 173 28. The barometric pressure P (inches) at height H (thousands feet) is P 30 28 26 24 22 20 18 16 , H 1.8 3.8 5.9 8.1 10.5 13.2 16.0. Determine the most probable values of the constants in each of the assumed relations : (a) H - a + hP ; (6) H=a + hP + cP^ ; (c) H = a + b log P or P = Ae^^. Which is the best approximation ? 29t. If the observed values of one quantity y are iiii, mo, ms, corre- sponding to values Zi, h, h of a quantity x on which y depends, and if y = ax + b, show that the sum S = (ah + b- mi)2 + (ah + b- jno)2 + (ah + b- ws)"^ is least when I h (ah -\-b — mi) + h (ah + b — m-i) + h (ah + b — ms) = 0, I (ah + b — Mil) + (ah + b - m^) + (ah +b — mg) = ; that is, when a-^h^ + b- 2^1- V^niZi^Oanda- VZi + 36- ^ mi h' where ^ indicates the sum of such terms as that which follows it. [Theory of Least Squares.] 30. Show that the equation of the tangent plane to 2z = x^ + y"^ at (xo, 2/o) isz +Zo = xxo + 2/2/0. 31. Determine the tangent plane and normal line to the hyperboloid a;2 _ 4 2/2 + 9 ^2 ^ 36 at the point (2, 1, 2). 32. Study the surface xyz = 1. Show that the volume included be- tween any tangent plane and the coordinate planes is constant. 33. Study the surface z = (x^ 4- y"^) (x^ + y^ -1). Determine the ex- tremes. 34. At what angle does a line through the origin and equally inclined to the positive axes cut the surface 2z = x:^ + y^? 35. Determine the tangent line and the normal plane at the point (1, 3/8, 5/8) on the curve of intersection of the surfaces x + y + z = 2 and x2 + 4 2/2 - 4 22 = 0. 36. Determine the tangent line and the normal plane to the curve x = 2 cost, y = 2 sin t, z = f^ a.t t = ir/2 and at t = v. IX, § 173] GENERAL EXERCISES 343 37. Find the length of one turn of the conical spiral x = t cos (a log t), y — tsm (a log t), z = bt, starting from t = t. 38. Determine the length of the curve x = a cos 6 cos <(>, y = asin 6 cos 0, 2 = a sin 0, from (p = by the equation d = k log cot (ir/4 — /2). (Loxodrome on the sphere.) 39.*t Show that the surfaces /(a:, y, z) = and (x, y, z) =0 cut each other at right angles if /x0x + /y0y + f^^z = 0. 40.* Show that the surfaces xV(a2 + \) + 2/"/(&- + ^) + z-/(c- + X) = 1, a>b>c>0, are always (i) ellipsoids if X> — c^, (i7) hyperboloids of one sheet if — b^<\ < — c^, ( Hi) hyperboloids of two sheets if — a"-^ < X < — b-. (CONFOCAL QUADRICS.) Show also that these surfaces cut each other mutually at right angles. 41.*t On the surface x =f(u, v), y =

= a da? dx 176. Determination of the Arbitrary Constants. The deter- mination of the arbitrary constants appeared in the very first examples. Thus, in § 54, p. 91, the rate at which water is being poured into a tank was considered. The total amount y was found to be y = r ■ t+C, where r is the rate per second, t is the time in seconds, and C is the amount already in the tank when t = 0. The arbitrary constant C is determined as soon as the value of y is given for some value of x. Thus in the problem of fall- ing bodies (§ 113, p. 206), from the fact that jj, == — const. = — g=z— 32.16 ft./sec./sec, we found that If the body is dropped from rest (v = 0), at a height s = 100 ft., we have s| =C2 = 100, vl =ci = 0, Je=0 J(=0 X, § 177] REVERSED RATE PROBLEMS 347 whence s=-|r/^- + + 100, in which the arbitrary constants have disappeared. Essentially the same process was used in determining the arbitrary constants in a compound interest law (§ 81, p, 143). Finally, in the case of direct integration, the arbitrary con- stant was disposed of by taking the difference between two values of y which correspond to two given values of x : 2/]'^'=J_7''^(^)c^-«; f=/(^), given; and the same scheme was used in motion problems (§ 59, p. 100) and in compound interest examples (§ 81, p. 142). 177. Vital Character of Inverse Problems. These problems are reversed or indirect only from a mathematical standpoint. From the standpoint of science, or of everyday life, many such problems are more direct than those which seem to be the original ones from a mathematical standpoint. Thus, from the standpoint of science, it is just as much a direct problem to find the distance passed over from a given acceleration, as to find the acceleration from the distance; as a matter of fact the former is usually the real scientific problem. , We found (§ 97, p. 1G9,) that the radius of curvature of any given curve is [1 + vi-Y'^/l', where m = chj/dXy b = d-y/dx^. If the curve is given, this formula indeed gives the radius of curvature. But it is more desirable in practice to find a curve whose radius of curvature behaves in a way we wish: given the radius of curvature R = \p (x), it is desired to find a curve y = f(x) which will actually have just this radius at each point : 348 DIFFERENTIAL EQUATIONS [X, § 177 We shall solve such differential equations later (§ 191, p. 371) j just now it is important to see that they actually arise in con- \ Crete direct scientific and mathematical problems. 178. Elementary Definitions. Ordinary Differential Equa- tions. Au ordinary differential equation is one involving only one independent variable. The derivatives in such an equa- tion are therefore ordinary derivatives. An ordinary differential equation may contain derivatives of various orders, and these derivatives may enter in various powers. The order of a differential equation is the order of' the highest derivative present in it. The degree of a differential equation is the exponent of the highest power of the highest derivative, the equation having been made rational and integral in the derivatives which occur in it. Thus, equation (5), § 175, is of the second order and first degree; (1), § 174; (1), § 175; (4), § 175, are of the first order and first degree ; and (1), § 177, when rationalized, is of the second order and second degree. 179. Elimination of Constants. Differential equations also arise in the elimination of arbitrary constants from an equation. Example 1. Thus, if A and B are arbitrary constants, the equation 1/ — Ax + B represents a straight line in the plane, and by a proper choice of A and B represents any line one pleases in the plane except a vertical line. One differentiation gives m = dy/dx = A, which represents all lines of slope A. A second differentiation gives (1) flexion = b = d-y/dx- = 0, which represents all non-vertical lines in the plane, since all these and! no other curves have a flexion identically zero. X, § 179] INTEGRAL CURVES 349 Example 2. Any circle whose radius is a given constant r is repre sented by the equation (2) (x - A)- + (y - B)-^ = f\ from which A and B may be eliminated as in the preceding example. Differentiating once, (3) x-A + {y-B)yi = 0, where y' = dy/dx. Differentiating again, (4) l + y'-2 + ^y_ B)y'i = 0, where y" = d-y/dx'^. Solving (3) and (4) iov x — A and y — B and sub- stituting these values into (2), A and B are eliminated, giving (6) (1 + 2/'2)3 = r2t/"2. This says that every one of these circles, regardless of the position of its center, has the curvature l/r, — a statement which absolutely character- izes these circles. In general, if (6) f{x,y,c^,c.,,—,c,) = (i is an equation involving x, y, and n independent arbitrary con- stants Ci, C2, •••, c„, n differentiations in succession with regard to X give <^> !=»' S=»' ••■' '£='■' these equations, together with (6), form a system of ?i-|-l equations from which the constants Ci, c.,, •••, c„ may be elimi- nated. The result is a differential equation of the nth order, free from arbitrary constants, and of the form (8) (x,y,y',y", '",?/"") = 0. Equation (6) is called the primitive or the general solution of (8). The term general sohition is used because it can be shown that all possible solutions of an ordinary differential equation of the nth order can be produced from any solution that involves n independent arbitrary constants, with the ex- ception of certain so-called " singular solutions " not derivable 350 DIFFERENTIAL EQUATIONS [X, § 179 from the one general solution (6) (see Ex. 20, List LXXV, p. 362). Thus, to solve an ordinary differential equation of the nth. order is understood to mean to find a relation between the variables and n arbitrary constants. These latter are called the constants of integration. If, in the general solution, particular values are assigned to the constants of integration, a particular solution of the dif- ferential equation is obtained. 180. Integral Curves. An ordinary diiferential equation of the first order, (1) <^(^, y, y') =0, or 2/' =f(x, y), where y' = dy/dx, has a general solution involving one arbi- trary constant c : (2) F{x,y,c)=^. This represents a singly infinite set or family of curves, there being in general one curve for each value of c. Any curve of the family can be singled out by as- signing to c the proper value. The differential equation deter- mines these curves by assigning, for each pair of values of x and y, that is, at each point of the plane, a value of the slope y'\_ = f(x, ?/)] of the particular curve going through that point. Thus the curves are outlined by the directions of their tangents in much the way that iron filings sprinkled over a glass plate arrange themselves in what seem to the eye to be curves when a magnet is placed beneath the glass. Straws on water in inotion create the same optical illusion. A differential equation of the second order: (x, y, y', y") = 0, or y" =f(x, y, y'), X, § 180] INTEGRAL CURVES 351 has a general solution involving two arbitrary constants, F(x, y, c„ Co) = 0. This represents a doubly infinite or tioo-parameter family oi curves; for each constant, independently of the other, can have any value whatever. The extension of these concepts to equations of higher order is obvious. The curves which constitute the solutions are called the integral curves of the differential equation. EXERCISES LXXII.— ELIMINATION INTEGRAL CURVES Find the differential equations wliose general solutions are the follow- ing, the c's denoting arbitrary constants : 1. x^ + y^ = c2. Ans. x + yy' = 0. 2. x- — y- = ex. Alls, a;- + y^ = 2 xyy'. 3. y = ce' — |(sin x + cosx). Ans. y' = y + sin x. 4. y = ex + c^. Ans. y = y'x + y''^. 5. y = cx+f(c). A71S. y = y'x+f(y'). 6. y = eie^ + e2e^. Ans. y" -5y' + 6y = 0. 1. y = CiC^ + e-z^'. Ans. y" — (a + h)y' + ahy = 0. 8. xy =^c + c-x. Ans. x*y'- — y'x + y. 9. y = {ci + x)e^'' + c^e'. Ans. y" - ■iy'+3y=2 e^. 10. y - Cie^ + Coe-^ + c^e^'. Ans. y'" - Q y" + \\y' -Qy = 0. 11. r = c sin d. Ans. rcosd = 7-' sin 6. 12. r=e'». Ans. r]ogr = r'0. 13. Assuming the differential equation found in Ex. 1, indicate the values of ?/'(= —x/y) at a large number of points (x, y) by short straight- line segments through each point in the correct direction. Continue doing this at points distributed over the plane until a set of curves is outlined. Are these the curves given in Ex. 1 ? 14. Proceed as in Ex. 13 for the equation ?/' = y/x. Do you recognize the set of curves ? Can you prove that your guess is correct ? 15. Draw a figure to illustrate the meaning of y' = x^. Find y. Gen- eralize the problem to the case y' = / (a;). 352 DIFFERENTIAL EQUATIONS [X, § 180 16. Find that curve of the set given in Ex. 1 which passes through (1,2). Find its slope (value of y') at that point. Do these three values of (x, y, y') satisfy the differential equation given as the ansvrer in No. 1 ? 17. Proceed as in Ex. 16 for the equation of Ex. 2. 18. Proceed as in Ex. 16 for the first equation of Ex. 15. 19. Find the differential equation of all circles having their centers at the origin. 20. Find the differential equation of all parabolas with given latus rectum and axes coincident with the x-axis. 21. Find the differential equation of all parabolas with axes falling in the X-axis. 22. Find the differential equation of a system of confocal ellipses. 23. Find the differential equation of a system of confocal hyperbolas. 24. Find the differential equation of the curves in which the sub- tangent equals the abscissa of the point of contact of the tangent. 25. A point is moving at each instant in a direction whose slope equals the abscissa of the point. Find the differential equation of all the possi- ble paths. 26. Write the differential equation of linear motion with constant acceleration ; of linear motion whose acceleration varies as the square of the displacement. The same for angular motion of rotation. 27. A bullet is fired from a gun. Write the differential equations which govern its motion, air resistance being neglected. How must these equations be modified, if air resistance is assumed proportional to velocity ? 181. General Statement. We shall now consider methods for solving differential equations. Since the most common properties of curves involve slope and curvature, and since in the theory of motion we deal constantly with speed and acceleration, the differential equations of the first and second orders are of prime importance. Ordinary differential equations of the first order and first degree have the form (1) M-\- N^ = 0, or Mdx + Ndy = 0, dx where M and N are functions of x and y. I X, § 182j SEPARATION OF VARIABLES 353 No general method is known for solving all such differential equations in terms of elementary functions. We proceed to give some standard methods of solution in special cases. 182. Type I. Separation of Variables. It may happen that Jf involves x only, and N involves y only. The variables are then said to be separated and the primitive is found by direct integration : ^Mdx+ ^ N-dy = C, C being an arbitrary constant. Example 1. Find the curves having a constant subnormal equal to k. The differential equation is subnormal = y • -^ = k. dx Separating the variables : y dy = k dx. Integrating both sides : ^y'~ = kx + c, or 2/2 = 2 kx + c', a family of parabolas. The constant c' is determined if the parabola is required to pass through some given point in the plane. Check this result by eliminating c again by the methods of § 179. Example 2. Given the relative rate of change (logarithmic derivative) of a function of x in terms of x, find the function : i.e. given (dy/dz) -4- 2/ = 0(x), to find y =f(x). The differential equation ^^y = (x) dx is of the type mentioned above ; separating variables and then integrating we find : -^ = (x)dx, whence log y = (

(x) = x, for example, y = ke^"/-; if also the value of y is given for some value of x, say y = .3, when x = 2, we liave 3 = ke^, whence fc = 3 e-2 and w = 3 e'-l'^-'^. Check this result. 354 DIFFERENTIAL EQUATIONS [X, § 183 183. Type II. Homogeneous Equations. When M and N are homogeneous * in x and y and of the same degree, the equa- tion is said to be homogeneous. If we write the equation in the form dy _ M (£~ N' and make the substitution dii , xdv ^ = '"''' dx^'^^'d^' we obtain a new equation in which the variables can be separated. Example 1. (1) {xy + y2) dx + {xy - x2) dy = 0, dy _ xy + y^ dx x^ — xy (2) Substituting as above : (3) ^^^(?l._«x2 + .2^2_^ + dx x'' — vxP' 1 — ■» ^dv ^ 2v'^ . dx 1 — t) ' separating variables, ^dv — — -Iv^ X Integrating : - log v = log a; + c. Keplacing v by y/x, A_llog^ = logx 2u 2 X or \ogxy = 2c; y hence (4) xy = e-'/'J-"^", or xy = Ae-^/J', where k = e-^^. * Polynomials are homogeneous in a: and y when each term is of the same degree. In general, /(x, y) is homogeneous if f{kx, ky)= ^"/(a;, y) for some one value of n and for all values of k. I X, § 183] SEPARATION OF VARIABLES 355 Check : Differentiating both sides of (4) with respect to a;, we find (5) ydx + xdy^A-e-^/v[- y^y^^y ] ; dividing the two sides of (5) by the corresponding sides of (4) respectively (6) [_ydx + xdy^^xy=-y^^^^=^', show that (6) agrees with (1). EXERCISES LXXm. — SEPARATION OF VARIABLES Solve the following exercises by separating the variables : 1. xdy + y dx = 0. Ans. xy = c. 2. X Vl + y- dx-y^\ +x'^dy = 0. Ans. Vl + x^ = Vl + y- + c. 3. sin tf (ir + r cos ^ (Z^ = 0. Ans. r&\ne — c. 4. xVl + y dx = yVl + xdy. Solve the following homogeneous equations : 5. (x + y) dx+ (^x — y) dy = 0. Ans. x"^ -\- 2 xy - y~ — c. 6. {x? + y')dx = 2xydy. Ans. x^ - y'^ = ex. 7. (3 x2 — y-) dy = 2xy dx. Ans. x- — y" = cy^. 8. (x2 4- 2 xy - 2/2) ax = {x^-2xy- 2/2) dy. Ans. x2 + y2 = c(x+j/). The following Ex. 9-18 are intended partially for practice in recogniz- ing types : 9. Vl — y-dx + y/l — x2 dy = 0. Ans. sin-' x + sin-J y = c. 10. x^ dx + (3 x2y + 2 y3) (^y = 0. Ans. x^ + 2y'^ = cy/di^ +yK 11. dy + y sin X dx = sin x dx. 12. rdd = tan 6 dr. 13. (2/ - 1) dx = (x + 1) dy. 14. ydx+ (x-y)dy = 0. 15. X (1 + 2/2) dx = 2/ (1 + 3^2) d2/. 16. (« ^2 + 2/^) dx = 2 xy dy. 17. f^^x = c. 18. ^^^y = x. dx dx 19. In Ex. 1 above, draw a figure to represent the direction of the integral curves at various points. Hence solve the equation geomet- rically. 356 DIFFERENTIAL EQUATIONS [X, § 183 20. A point moves so that the angle between the x-axis and the direc- tion of the motion is always double the vectorial angle. Determine the possible paths. ^^^ xy ^ ^^ ^ > 0. X2 + 2/2 21. Proceed as in Ex. 20 for a point moving so that its radius vector always makes equal angles with the direction of the motion and the x-axis. A71S. r = c sin 0. 22. The speed of a moving point varies jointly as the displacement and the sine of the time. Determine the displacement in terms of the time. Ans. s = ce-*<=08'. 23. Find the value of y if its logarithmic derivative with respect to x is x2. 184. Type III. Linear Equations. This name is applied to equations of the form (1) |+i'.= «, where P and Q do not involve y, but may contain x. Its solu- tion can be obtained by first finding a particular solution of the reduced equation, (1*) ^ + Py*==0, ax where 7j* is a new quantity introduced for convenience in what follows ; and where Q is replaced by zero. In (1*) the vari- ables can be separated (see Ex. 2, § 182), and we get jt -\PdX y* =e •' as a particular solution, the constant C of integration being given the particular value 0. If we make the substitution (2) y = V'y*j X, § 184] LINEAR EQUATIONS 357 « where t> is a function of x to be determined, the equation (1) becomes dx dx The first term vanishes by (1*) leaving y*^ = Q, or dv = ^dx==[Qj'""ldx. dx y Hence v= r-^c/x + c= fcgJ^'^^jdaj + c and (3) y=vy* = e-^'"'^^jlQe^'''yix + c This equation expresses the sohition of any linear equation. It should not be used as a formula; rather, the substitution (2) should be made in each example. Example 1. Given dy (1)' ^+3x22/ = x6, dz the reduced equation in the new letter y* = y/v is (1*)' ^ + 3 x^y* = 0, whence y* = e"**. Hence the substitution y = v • y* becomes (2)' y = ve-'\ whence ^ = e"*' ^" - 3 va;2e-,»^ dx dx and (1) takes the form Te-.-'l^ _ 3 vx^e-^'l + 3 ar^ [vg-x''] = arS. This reduces, as we foresaw in general above, to the form e-x»^ = a:5 or '^ = xfie'', dx dx 358 DIFFERENTIAL EQUATIONS [X, § 184 whence V -Ix^e^^dx + c = -I [a;=e^= - e^']+ c, or, returning by (2) to ?/ : (3) ' y = ve~^' = i [a;3 - 1] + ce-< Check : Differentiating both sides, (4) ^ = a;2-3x2ce-^*; dx eliminating c by multiplying (3)' by 3 x^ and adding to (4), ^+3x2y=a;6. dx The result (3)' may also be obtained by direct substitution in (3) from (1)'. Suificient practice in the direct solution, as in the preceding ex- ample, is strongly advised. 185. Extended Linear Equations. This name is often given to equations of the form dy/dx+Py= Qy\ Putting z = ?/'"" reduces it to a linear equation in z. Example 1. Given ^+y^xy^. Tut z = y-^. dx X Then ;^ = -2,-3^, or f^ = -(l/2),3^ dx dx . dx dx Thus -(l/2)2/3g + ^ = x2/3 and ^_25=_2x. dx X Here P = -?, ^ Pdx =-2\ogx, eJ"^''" = x-2; so that z = x^( ( - ^dx + cW- 2 x2 logx + cx2 = 2/-2, and finally x'^y- (c — 2 log x) = 1. Check this result. X, §186] MISCELLANEOUS EXERCISES 359 EXERCISES LXXIV. — LINEAR EQUATIONS Solve the following linear equations and check each answer : 1. ^Iy.— xy = e'V2, 3, ^ + 2/ cos X = sin 2 x. dx dx 2. ^ + 3 3-2^ = 3 3:5. 4. .r '!l>+y= log x. dx dx Solve the following extended linear equations, checking each answer : 5. ^ + y = ,f, 7. ^ + re ^ r^ sin e. dx X de 6. ^ + M = .rw3. 8. xy"- '^-y^ = x?. dx dx Solve the following equations, checking each answer : 9. cos2 x^^ + y = tan x. 10. /^^' = (l+r^) sin $. dx dd 11. ^ = -s + L 12. ^+2/ = .-. dt dx Ans. s = ce-' — 1 + «. Ans. ye^ = x + c. 13. dy - ydx = !iinxdx. 14. sec »d/- + (r - 1)^^ = 0. 15. (x^+l)dy=(xy + k)dx. 16. a; («y + y rfx = .r»/2 log x dx. 17. The equation of a variable electric current is L—+Bi=e, dt where L and B are constants of the circuit, i is the current, and e the electromotive force of the circuit. Calculate i in terms of t, 1°, if e is constant ; 2°, if e = eo sin tx)t. A71S. 2° i = ^" sm (wt -)+ ce-i^/% = arc tan (w L/i?) . 186. Other Methods. Nonlinear Equations. A variety of other methods are given in treatises ou Differential Equations; some of these are indicated among the exercises which follow. Noteworthy among these are the possibility of making advan- tageous snbstitntions ; and — what amounts to a special type of substitution — the possibility of writing the given equation in 360 DIFFERENTIAL EQUATIONS [X, § 186 the form of a total differential, dz = 0, where 2; is a known function of x and y, which leads to the general solution z = constant (see Exs. 4-11, below). Equations not linear in y' may often be solved. If the given equation can be solved for y', several values of y' may be found, each of which constitutes a differential equation : the general solution of the given equation means the totality of all of the solutions of all of these new equations (see Exs. 15-16, p. 362). EXERCISES LXXV.— MISCELLANEOUS EXERCISES 1. Solve the equation 2 y dy/dx + xy"^ = e". [Hint. Put y'^ = v; then dv/dx = '2,y dy /dx, and the equation becomes dv/dx + xv^e", which can be solved by previous methods.] 2. Solve the equation cos ydy + sin y sec^ x dx = tan x dx. [Hint. Put v = siny, ?/ = tana;; then dv = cos ydy, du=»ec^xdx; the equation becomes dv +vdu = u dn/(l + w^), which is linear.] 3. Solve the following equations, using the indicated substitutions: (a) y^dy + (y^ + x) dx = 0. (Put t^ = y^ ) {b) sdt ~tds = 2s{t- s)dt. (Fut s = tv.) (c) xdy — ydx = {x-- y-) dy. (Put y = vx.) (d) m2«2 (udv + v du) = (v + u2) dv. Put uv = x, V- y.) 4. Solve the equation (3 x^ + y) dx + (x + S y^) dy = 0. [Hint. If we put z = x^ + xi/ + y«, this equation reduces to dz = 0; for dz = idz/dx)dx + {dz/dy)dy. But dz=0 gives z = const., hence a:8 + j;y + j,8=c ^ is the general solution. Such an equation as that given in this example is called an exact differential equation.] 5. Solve the equation xdy — ydx = 0. Hint. This equation can be solved by previous methods; but it is easier | to divide both sides by x^ and notice that the resulting equation is d{y/x) = 0; { hence the general solution is y/x = c. A factor which renders an equation exact (l/a;2 in this example) is called an integrating factor. 6. Solve the equation (x^ + 2 xj/'^) dx + (2 x-y + y-) dy = 0. [Hint. Put z = x^f-.i + x^y^ + y^/S.] X, § 186] MISCELLANEOUS EXERCISES 361 7. Solve the equation (s + < sin s) ds + (t — cos s) dt = 0. [Hint. Arrange: sds+ [t sin sds — cos sdt] + tdt = 0; integrate this knowing that the bracketed term is — d{t cos s).] 8. Solve the equation xdy — (y — x) dx = 0. [Hint. Arrange: [xdy — ydx^+xdx = 0; divide by x^, and compare Ex. 5.] 9. Show that [/(.r) + 2 xy^] dx + [2 x^y +

C-^ constants.] These are all special forms of the general equation of the sec- ond order <^{x, y, dy/dx, d-y/dx^) = 0. [IV] We shall consider other special forms also, some of which include the above; namely, the cases that arise when one or more of the quantities x, y, dy/dx, are absent from the equation. (See § 191, p. 371.) 188. Type I. This type of equation arises in problems on motion in which the tangential acceleration d?s/df is propor- tional to the distance passed over (see § 89, p. 156) : (1) g=±*-, a form which is equivalent to [I], written in the letters s and t. If we multiply both sides of tins equation by the speed V = ds/dt and then integrate with respect to t, we obtain (2) I -dt= I ±k-s—dt: ^ ^ J dt de J dt ' but we know that J dtde J dt J 2 2\dt) 364 DIFFERENTIAL EQUATIONS [X, § 188 and r± kh ~dt = ± k^ Csds= ± Ms2 + c' ; hence (2) becomes* Case 1. If the sign before /cs is +, (3) becomes (4) v='^^ = kVs'+C„ whence f ^^ = Ckdt + Co, (5) log(s4-Vs2+Ci) = A:«+C2; or, solving for s, (6) s = Ae^* 4- Be-'^', where 2 A = e^'' and 2 B = — Cie'^' are two new arbitrary- constants. By means of the hyperbolic functions sinh ?i = (e" — e-")/2 and cosh ■« = (e" + e~")/2 this result may also be written in the form (7) s = a sinh (kt) + b cosh (kt), where b + a= 2 A and h — a — 2 B. Case 2, If the sign before k^ is — , Ci must be negative also, or else v is imaginary ; hence we set Cj = — a^ and write (42) v= i^ = ^V(?^^ kdt + O2, * This is often called the energy integral, for if we multiply through hy the mass m, the expression mv^/2 on the left is precisely the kinetic energy of the body. X, § 188] SECOND ORDER 365 whence or solving for s : (62) s = a sin ( kt + C-i) = A sin kt -\- B cos kt, where A=: a cos Cj and B = a sin C2 are two new arbitrary constants. Equation (6,) is the characteristic equation of simple har- monic motion ; the amplitude of the motion is a, the period is 2 ir/k, and the phase is — Cj/k. The differential equation (1) was first found in § 88, p. 155, "We now see that the general simple harmonic motion (60) is the only possible motion in which the tangential acceleration is a negative constant times the distance from a fixed point ; i.e. it is the only possible type of natural vibration under the assumptions of § 90, p. 157. EXERCISES LXXVI. — TYPE I 1. Solve each of the following equations : (c)g=4.. (<:=-»- 2. Find the curves for which the flexion (cPy/dx-) is proportional to the height (y). 3. Determine the motion described by the equation of Ex, 1 (a) if the speed V ( = ds/iU) and the distance traversed s are both zero when t = 0. 4. Proceed as in Ex. 3 for Ex. 1 (b), and explain your result. 5. Write the solution of Ex. 1 (a) in terms of sinh t and cosh t. De- termine the arbitrary constants by the conditions of Ex. 3, and show that the final answer agrees precisely with that of Ex. 3. 6. Determine the motion described by the equation of Ex. 1 (b) if V = 2 and s = 10 when « = : if t? = and s = 5 when t = 0. 366 DIFFERENTIAL EQUATIONS [X, § 189 189. Type II. Homogeneous Linear Equations of the Second Order with Constant Coefficients. The form of this equation is where A, B, C are constants. The type just considered is a special case of this. one. Fol- lowing the indications of the results we obtained in § 188, it is natural to ask whether there are solutions of any one of the types we found in the special case : Trial of e^^. If we substitute 2/ = e** in (1) we obtain the equation : (2) \_Ak- + Bk+(J]e^ = 0. The factor e** is never zero ; hence k must satisfy the quad- ratic equation (1*) Ak'2 + Bk + C=0, which is called the auxiliary equation to (1). If the roots of (1 *) are real and distinct, i.e. if (3) D = B'-4.AC>0, then these roots ki and k^ are possible values for k, and the gen- eral solution of (1) is (4) y = Cie^'i'*' + Cae'^^"', since a trial is sufficient to convince one that the sum of two solutions of (1) is also a solution of (1) ; and that a constant times a solution is also a solution. Trial oi y = e^^ • v. If (3) is not satisfied, the substitution (5) y=e''' -v changes (1) to the form (6) ^g + [2K.4 + ^]^ + [.lK^+5K+C]. = 0, X, § 189] SECOND ORDER 367 which becomes quite simple if we determine k so that the term in clv/dx is zero : (7) 2kA+B=0, whence k = -B/2A; then (6) takes the form ^ ^ d-r^ 4.4- ' where K = V4 AC - B'/(2 A)=^-D/2A is real if (9) 0=^^-4. AC<0, zohich is the case ive could not solve before. If D<0, the solutions of (8) are (10) v=Ci sin (K.r) + C.2 cos (Ka;), by (62), § 188, p. 365 ; hence the solutions of (1) are (11) y = e>^v~e*i^[Ci sin (Kx) + Co cos (Ka?)], where k = - B/{2 A) and K = V^D/(2 A) ; these values of k and K are most readily found by solving (1 *) for A;, since the solutions of(l*)arek = (-B± VI>)/(2 ^) = k ± K V^l. If i> = 0, K = and the solutions of (8) are (12) v=CiX+C2; hence the solutions of (1) are (13) J/ = eKx . V = cKa;[CiX + C'2], where k = —B/(2 A) is the solution of (1 *) ; since when D = 0, (1 *) has only one root k = ~ B/{2 A). It follows that the solutions of (1) are surely of one of the three forms (4), (11), (13), according as D = R- -i AC is +, ~, or ; that is, according as the roots of the auxiliary equation (1 *) are real and distinct, imaginary, or equal; in resume: 368 DIFFERENTIAL EQUATIONS [X, § 189 D=B^-iAC Character of KOOTS OF (1») Values of Roots OF (1*) Solution of (1) + Real, unequal fci, k.2 (4) - Imaginary K ± K V^ (11) Equal K (13) Such solutions as (11) have been forecasted from the work of § 92, p. 162, where an equation (in the letters s and t) of pre- cisely the type (11) was studied. Indeed, if y and x are re- placed by s and t, and if k is negative, (11) expresses precisely the most general form of damped vibration, studied in § 92. Examples 1 2 3 Equation (1) 3j/"_4 2/'+2/=0 32/"-4y+|2/=0 3 2/"-42/'-H2 2/=0 Auxiliary equa- tion (1*) 3A;2-4A:+1=0 3*2-4 A:+f=0 3i-2-4A- + 2=0 Roots of (1*) 1, 1/3 2/3, 2/3 K2±V^) Solution of (1) 2/=cie^+C2e^/3 2/ = e2x/3^Ci + C2X) ?/ = e2'/3(CiC0S EXERCISES LXXVII. — LINEAR HOMOGENEOUS. TYPE H 1. 2/" - 4 J/' + 3 2/ = 0. 2. 2/" + 32/' + 22/ = 0. 3. 5 y" - 4 2/' -}- 2/ = 0. 4. 9 ?/" + 12 2/' + 4 2/ = 0. 5. y" -2y> + y = 0. 6. y" + y' + y = o. 7. y" -2y' + 3y = 0. 8. 3 J/" + 6 2/' -h 2 2/ = 0. 9. 2/'' - 9 2/' + 14 y = 0. 10. 2 2/" — 3 y' + 2/ = 0. 11. 6 2/" - 13 2/' + 6 2/ = 0. 12. 2/"-3 2/' = 0. 13. 2/"-4 2/ = 0. 14. y" + 9y = 0. 15. 2/" + A-2/' = 0. 16. ?/" ± *y = 0. X, § 190] SECOND ORDER 369 17. If a particle is acted on by a force that varies as the distance and by a resistance proportional to its speed, the differential equation of its motion is d-r/dt- + h dx/dt + ex = 0, where c > if the force attracts, and c < if the force repels. Solve the equation in each case. 18. If in Ex. 17, 6 = c = 1, and the particle starts from rest at a dis- tance 1, determine its distance and speed at any time t. Is the motion oscillatory ? If so, what is the period ? Solve when the initial speed is Vp. 19. If in Ex. 17, 6 = 1 and c =— 1, discuss the motion as in Ex. 18. 190. Type III. Non-homogeneous Equations. This type is of the form : where A, B, C, are constants, and F(x) is a function of x only. We proceed to show that this form can be solved in a manner exactly analogous to § 184, p. 356 ; first write down the reduced equation in the new letter y* : doi? dx and solve (1*) by the method of § 189. Let y* = <^(x) be any one particular solution of (1*) (the simpler, the better, except that ?/* = is excluded). Then the substitution (2) y = {x).u transforms (1) into (3) \A"{:x) + B<^\x) + C4>{x)\u+ \2A\x)+ B{x)\^ + A^(x)p{=F{x); ax^ 2b 370 DIFFERENTIAL EQUATIONS [X, § 190 but, since {x) satisfies (1*), the first term of (3) is zero ; and if we now set du/dx = v temporarily, this equation can be written as the linear equation : .^. dv ( 2A^'(x) + B(x^ ^.^ F(x) ^ ^ dx \ A(x) j A4>(xy which is precisely of the form solved in § 184. Comparing (4) with (1), § 184, we have Having found -y by § 184, we have u= i vdx +c^, y = u{x) = (ic) j v +2y = e'. Ans. y = eV6 — Cie-^* + c^e-'. 4. y" -2y' +y= z. Ans. y = x+2 + e=^(ci + Czx). 5. y" + y = sinx. Ans. ?/ =— |a;cos x + Cisinx + C2C0SX. 6. y" — y' — 2y = sin x. Ans. y —^^{cos,x — 3sinx) + Cig-^ + o^e^'. 7. 2/" + 4 y = x2 + cos X. Ans. 2/ = ^(2 x2 — 1) + 1 cosx + Ci cos 2 x + C2 sin 2 x. 8. y" -2y' = e^' + 1. ^ms. y = ^ x(e2* _ 1) + d + C2e2». 9. J/" - 4 y' + 3 2/ = 2 e^^. .4«s. y = xe*= + Cic' + de^. 10. If a particle moves under tlie action of a periodic force through a medium resisting as the speed, the equation of motion is d-s/dfi + Ads/dt = 5 sin C t. Express s and the speed in terms of «. If ^ =: 5 = C = 1, what is the distance passed over and the speed after 5 seconds, the particle starting from rest ? 191. Type IV. One of the quantities x, y, y' absent. Type IV„ : {y") = 0. Solve for y", to obtaiu a solution, say y" = a. Then integrate twice. The general solution for each value of y" is of the form y = ^ axr + CiX + C2. In problems of motion, this type is equivalent to the statement that (x, y', y") = 0. The substitution m = y' z=dy/(lx, fbn/d.v = d-y/dxr = y", reduces the given equa- tion to an equation of the^rs^ order in ?». x, dm/dx. Solving, if possible, one gets a relation of the form /(»i, x, c) = 0. This 372 DIFFERENTIAL EQUATIONS [X, § 191 is again an equation of the first order in x and y, and may be integrated by methods given in Part I, §§ 182-186. The interpretation in motion problems is particularly vivid and beauti- ful. Thus V = ds/dt and j^ = dv/dt = d?s/dt'^ ; hence any equation in j„, V, t, with s absent, is a differential equation of the first order in v. Solving this, we get an equation in v and t ; since v — ds/dt, this new equation is of the first order in s and t. Example 1. 1 + x + x2 ^^ = 0. dx^ Setting dy/dx = m, 1 + x + x'^ ^ = 0. dx 1 4- X Separating variables, — dtn = ^ dx. x2 Integrating, — m = 1- log x + Ci. X Integrating again, y = log x — x log x + (1 — Ci)x — C2. Interpret this as a problem in motion, with s and t in place of y and x, and jr = dv/dt = d^s/dt"^. Example 2. In a certain motion the space passed over s, the speed v, and the acceleration j^, are connected with the time by the relation 1 + -y^ — jy = ; find s in terms of t. Placing j = dv/dt, the equation dt is of the first order. The variables can be separated, and the integral is tan-i V = t + cior V = tan (t + Ci), which is itself a differential equation of the first order if we replace v by ds/dt. Integrating this new equation : (ds = ftan (t + ci)dt + c-z, or s = - log cos {t + c{) + c^. In such a motion problem we usually know the values of v and s for some value oi t. If v = and .s =10 when « = 0, for example, Ci must be zero (or else a multiple of tt) and C2 must be 10 ; hence s= — log cos t-\r 10. Example^. 1 + x ^ + x2^= =1 + xw + x^^. dx dx^ dx This can be written dm/dx ■\-m/x — — l/x^ X, § 191] SECOND ORDER 373 which is linear in m and x, the solution being m = — log X + -^-i . X X The second integration gives y=-i [log X]2 + Ci log X + C2. Interpret this as a motion problem, and determine Ci and co to make y = 10 and m = 3 when x = 1. T3rpe IVc : jc missing. (y, y', y") = 0. The substitution m = y' gives , ,, fZu' dy' dy dm dx dy dx dy and the transformed equation is an equation of the first order in y and m. We solve this and then restore y' in place of m, whereupon we have left to solve another equation (in x and y) of the first order. This is precisely the way in which we solved Type I, § 188, Type I being only an important special case of Type IV^. Example 1. If the acceleration jr is given in terms of the distance passed over (compare § 188), we have This is transformed by the relation dv dv ds dv (which is itself a most valuable formula) into in which the variables can be separated ; integration gives }v'^=({s)ds + c, which is called the energy integral (see footnote, p. 364). 374 DIFFERENTIAL EQUATIONS [X, § 191 The work cannot be carried further than this without knowing an exact expression for (/>(s). When 0(s) is given, we proceed as in § 188, replacing v by ds/dt and integrating the new equation : J ^^■2^,p(s)ds = t+k. + 2c Unfortunately the indicated integrations are difficult in many cases ; often they can be performed by means of a table of integrals. One case in which the integrations are comparatively easy is that already done in § 188. EXERCISES LXXIX. — TYPE IV 1. w"2- 4x2 = 0. Ans • y = ± l/3x3 + CiX+C2. Ans .2y = cie- + e-Vci + Co. Ans ■ y = XV9 + Ci log X + C2. Ans . S2 = : «2 + Cit + C2. 6. da;2 = ± k^y. 8. dx^~ ^e^y. 10. dx^ = X + 3 sin x. 12. doi^' =['-(l)T- 2. 2/" = Vl + 2/'2. 3. xy" + 2/' = x2. 4. s&^sldf^^idsldty^^ 1. 5. ^ = 1-. dv^ Vi ' dx2-' • 9. ^ = x2cosx. dx2 11. ^ = e^-cos2x. dx* 13. Show that Ex. 12 is equivalent to the problem, to find a curve whose radius of curvature is unity. 14. The flexion {d^y/dx^) of a beam rigidly embedded at one end, and loaded at the other end, which is unsupported, is k{l — x), where A; is a constant and I is the length of the beam. Find y, and determine the constants of integration from the fact that ?/ = and dy/dx = at the embedded end, where x = 0. 15. Find the form of a uniformly loaded beam of length Z, embedded at one end only, if the flexion is proportional to ^2 _ 2 Zx + x2, where X = at the embedded end. 16. Find the form of a uniformly loaded beam of length I, freely sup- ported at both ends, if the flexion is proportional to P — 4 x2 in each half, where x is measured horizontally from the center of the beam. X, § 193] HIGHER ORDER 375 PART III. GENERALIZATIONS 192. Ordinary Equations of Higher Order. An equation whose order is greater than two is called an equation of higher order; the reason for this is the comparative rarity in applica- tions of equations above the second order. There seems to be a natural line of division between order two and higher orders, which is analogous to the natural demarkation between space of three dimensions and space of higher dimensions. We shall state briefly the generalizations to equations of higher order, however, since they do occur in a few problems, and since it is interesting to know that practically the same rules apply in certain types for higher orders as those we found for order two. 193. Linear Homogeneous Type. The work of § 189 can be generalized to any linear homogeneous equation with constant coeflBicients : Thus if we set y = e*', as in § 189, we find (1*) A;" + ajfc"-' + • • • + a„_, k + a„ = 0, again called the auxiliary equation. Corresponding to any real root k^ there is therefore a solution e''i'; if all the roots are real and distinct, the general solution of (1) is (2) 7/=(7,e*i^ + C,e*^ + ..- + C,.eV, where k^, k,, •••, k\ are the roots of (1). Curiously enough, the chief difficulty is not in any operation of the Calculus ; rather it is in solving the algebraic equation (1*). It is easy to show by extensions of the methods of § 189 376 DIFFERENTIAL EQUATIONS [X, § 193 that any pair of imaginary roots of (1*), A;=k±KV— 1 cor- responds to a solution of the form f (3) 2/ = e" [ C sin (A"^) + C ' cos (Kx)], which then takes the place of two of the terms of (2). Finally, if a root k = k of (1*) occurs more than once, i.e. if the left-hand side of (1*) has a factor (k — k)^, the correspond- ing solution obtained as above shoidd be multiplied by the polynomial (4) Bo-hB,x-hB2x'-\--'-{- B^_,xP-\ where p is the order of multiplicity of the root (i.e. the expo- nent of (k — k)"), and where the B's are arbitrary constants which replace those lost from (2) by the condensation of several terms into one. The proof is most easily effected by making tlie substitution y = e'^' • m, whereupon the transformed differential equation contains no derivative below d^u/dx^ ; hence u = the polynomial (4) is a solution of the new equation, and y = e** times the polynomial (4) is a solution of (1). This work may be carried out by the student in any example below in which (1*) has multiple roots. J: t This fact is often made plausible by the use of the equations qv.\/—i — COSM+ V— Isinw, e~"N/-i= cos u — V—1 sin u ; these equations can be derived formally by using the Taylor series for e", cos u, sin u, vrith v=uV—i, but they remain only plausible until ^fter a study of the theory of imaginary numbers. The solutions e* ± A'V— i are indicated formally by (2) ; hence it is plausible that (3) is correct. A more direct process which avoids any uncertainty concerning imaginaries is almost as easy. For the substitution ?/ = e«'« (see § 189) gives a new equation in u and x which, together with its auxiliary, has coefficients of the form {d'>^A(k)/dk") -f-n!, where A{k) represents the left-hand side of (1*). Now 5" V— 1 is a solution of the new auxiliary by development of A{k) in powers of (k — K) ; hence u = sin (Kx) and (t = cos (Kx) are solutions of the new differential equation, as a comparison of coefficients demonstrates. This process constitutes a rigorous proof of (3) . J To avoid using imaginary powers of e, if that is desired, substitute y = e'''' [cos (Kx) -»- V— 1 sin (K^)]u, when the multiple root is imaginary, X, § 194] HIGHER ORDER 377 These extensions of § 189 should be verified by the student by a direct check in each exercise. KXAMPI.K 1 2 3 (1) 2/"'-2/'=0 yiv+6j/"'+12y"+8 2/'=0 2/"' + 8r/ = (1*) ^•3 - ^• = A:*+6F+12A:2 + 8A; = A,-3 + 8 = k = 0, 1, - 1 0,-2,-2,-2 -2, 1±V3\/^1 y ci + coc^ + cse-* ci 4- e-'^Ccs + C33; + dx'^) Cie-^ + c^(c2Cos\/3x + C3 sin V3 x) 194. Non-homogeneous Tjrpe. The non-homogeneous type (1) T^ + "1 TVl + • • ' + ""-1 7 + ^^ = -^(^) dx" dx" ^ dx cannot be solved in general by an extension of § 190. But in the majority of cases which actually arise in practice,* a suffi- cient method consists in differ entiatiyg both sides of (1) re- peatedly until an elimination of the ?v(//<^hand sides becomes possible. The new equation will be of higher order still : (2) dx"" dx""'^ ^A dy + -L = 0, hd its rigid-hand side is zero. Solve this equation by § 193, and then substitute the- result in (1) for trial ; of course there 5^~will be too many arbitrary constants; the superfluous ones are determined by comparison of coefficients, as in the examples below. Example 1. y'" + y' = sinx. Differentiating both sides twice and adding the result to the given equation : yy + 2y"' + y' = 0. * For more general methods, see any work on Differential Equations ; e.g. Forsyth, Differential Equations. 378 DIFFERENTIAL EQUATIONS [X, § 194 The auxiliary equation k^ + 2k^-{-k = has the roots k=0, k = ± V'^ (twice). Hence we first write as a trial solution y, the solution of the new equation : y^ = ci + (C2 + C3X) cos x + (C4 + Cscc) sin x ; substituting this in the given equation, we find — 2 C3 cos x — 2 C5 sin x = sin x, whence C3 = and C5 = — 1/2 ; substituting these values in the trial solution y, gives the general solution of the given equation : ) 2/'" + 2/" + 2/' = 0. (e) 2/i^ - 2/" = 0. (c) 2/'" + 2/' = sin a;. (/) f" + v" = e^- 14. The following equations, though not linear, may be solved by first setting y' =p or y" = q or y'" — r. (a) y' = y" + Vl + 2/"^. (c) 1 + x + xY" = 0. (b) y" + y"'x={y"yx*. {d) xy^" + y'" =x^. 15. Solve the equation x'^j/" + xy' — y — log x. [Hint. Put x = e^ \ then dy ^ dv . (Jz^ ^l dy . d^_d n dv\ _ dz ^ 1 UVy , greater than or equal to. >, greater than. <, less than or etjual to. <, less than. n! (or [n), factorial n = n(n— 1) ••• 3 • 2 • 1. q.p., approximately. | a |, absolute or numerical value of a. 3. Signs peculiar to The Calculus and its Applications : (a) Given a plane curve y =f(x) in rectangular codrdinates (.r, y) ; Hi = slope = dy/dx — f (x) = y' = first derivative ; see p. 23. [Also occasionally D^y, f^, y, p, by some writers.] a = angle between positive a;-axis and curve = tan-^ ni. Ay, A-y, •••, A">/, first, second, •••, n"' differences (or increments) of y. dy =f(x) • Ax, d-y =/"(x) • A?, ••-, d^y =f('0(x) • Ax", first, second, •••, ?i"i differentials of y. rr = relative rate of increase, or logarithmic derivative ; see p. 146 ; = /(^) -/(^) = {(ly/dx) - y = d {log y)/dx = r^ -f- 100. Vp = percentage rate of increase = 100 • r,. b = flexion = d-y/dx^ =/"(.r) = ?/" = second derivative ; see p. 71. d'^y/d.f" =/<")(x) = y("'> = n^^ derivative. K= curvature — l-i-B; R = radius of curvature = 1 -^ A'; p. 170. 1 SIGNS AND ABBREVIATIONS [1,3 J"/(x) dx = indefinite integral of /(.r); see p. 96. \ f(x) dz= \ f(x) dx = definite integral off(x); see p. 115. s = length of arc ; s \ = arc between x = a and x — b. :=6 = area between y =0, y-f(x), x = a,x = b; see p. 116. o:-]; (6) Given a curve p =f(0) in polar coordinates (p, 0) : ^j/ — Z (radius vector and curve) = ctn-i [(dp/dd) -=- p] = ctn-i [d (log p)/de'].

= angular speed. g = acceleration due to gravity. (d) Problems in space; functions z =f(x, y, ••■) of several variables : Previous notations are generalized when possible without ambiguity, exceptions are p = dx/dx =fx; q = dz/cy = fy ; r = d'^zlcx? = /„ ; s = cHjdx dy = f,y ^fy^; t = d^ldf- = /,„. [The notation {dz/dx), used by some writers for dz/dx is ambiguous.] 4 Other letters commonly used. with special meanings : If =: ratio of circumference to diameter of circle = 3.14159---. e = base of Napierian (or hyperbolic) logarithms = 2.71828---. M= logio e = modulus of Napierian to common logarithms = 0.434- •■ ^ = " sum of such term as " ; thus : ^'"a;^ = «i" -\- a.? + ■■■ aj. (a, p, 7), — direction angles of a line in space. (?, m, n), — direction cosines ; I = cos «, etc. S. H. M. — simple harmonic motion. e or e, — eccentricity of a conic ; also phase angle of a S. H. M. II, AJ EXPONENTS AND LOGARITHMS 3 a, — amplitude of a S. H. M. (rt, b), — semiaxes of a conic ; (a, b, c), semiaxes of a conicoid. A = difference (of two values of a quantity). p — density ; also radius vector, radius of curvature, radius of gyration. 5. Trigonometric, logarithmic, hyperbolic, and other transcendental functions : See Tables, II, A ; II, F, 3 ; II, G ; II, H ; and consult Index. 6. Inverse function notations : If y =f(x), then f~^{y) = x; f-^ denotes an inverse function. [This notation is ambiguous ; confusion with {/(x)}"^ = 1 -^f{x).'] sin-i X or arc sin x, — inverse of sin x, or anti-sine of x, or arc sine x, or angle whose sine is x. [Other inverse trigonometric functions, and hyperbolic functions, follow the same notations. See Tables, II, G, 18 ; H,7.] TABLE II STANDARD FORMULAS A. Exponents and Logarithms. (The letters B, b, etc. indicate base; L, I, •■• indicate logarithm; JV, », • •• indicate member ; base arbitrary when not stated. See § 73, p. 130.) Laws of Exponents Rules or Logarithms (1) N = BL; in particular (1)' L = logeN, i.e. iV^= B'^es^; and: 1=50; B=Bi; \/B= E'K logl = 0; log2,5=]; log|,(l/5)=-l. (2) B^B' = B^+i. (2)' log (iV . n) = log .V + log n. (3) Bi-^ Bi = B^-i. (3)' log (.V-f- n) = log X- logji. (4) {B'-Y = B^i-. (4)' log (i\r») = 71 log X. (5) N =BL, B = fe*, X = 6*i. (5)' logfc X = log6 B . log^ .V. B=e, 6 = 10 gives *=0.4.34294r}= J/"=logioe ; \ogiay=Jf- loge X 5=10, 6=e gives *=2.302585=l-=-Ji/=lope10; log, 3^=(l-=-J/') log,oxV. 6=JV gives L=]A, l=log65. logjft; e.(7., log«10=l-=-log,oe. L=x gives 10*=«»-^-«'; «z=io*x. 2f=x gives logger = 3/' • log, a- ; log, ir=(l + JT) log,oa!. (6) y = ex" gives v = nu + k, Ji = logm x, v = logio y, k = logio c. (j) y = ce" gives v = mx -\-k, v = logi,,?/, m = a logio e = aM, k = \ogioc. 4 STANDARD FORMULAS [II, B B. Factors. (1) a2 - 62 = (a _ 5)(a + 5). (2) (a ± 6)2 = a^±2ab + b^. (3) a"- ft- = (a - 6)(a'»-i + a"-^ 6 + a'^-^b^ + ■■■ + 6»-i). (4) a2n+l + 52n+l = („ + 5)((i2n _ (^2n-lft + ... + 62») , See also Tables, IV, Nos. 16, 20, 21, 49, 50. (5) Polynomials : if /(a) = 0, /(x) has a factor x — a; in general : /(x) -=- (x — a) gives remainder /(a). (6) (a ± 6)" = a» ±^a''-^b + "^^'^^ a"-262+ ... + (±l)»6n. See II, B, 1, p. 7. ^ ^ ' ^ C. Solution of Equations. (1) ax2 + 6x + c = 0, roots: x — A ± ^ft^jzl^^ = _ A ± V^^, 2a 2a 2a 2a where f real D = b" — iac; roots of (1) are ] coincident >0 when D \ =0 <0 [ imaginary (2) X" + /»i.r"-i + pax"-! + •• +i)„-ix +p„ = 0. Roots : Xi, Xo, •■•, x„ ; then ^Xj = — i^i, X^«3^y =i'2i ^x,x,Xt = — ps, etc. (3) /(x) — 0(x)=:O: roots given by intersections of y =/(x), y= | 6 | : (a + 6)"=a» 4- ^a"-^& + "^'^~ ^^ a"-^&'H ••• + C„,r«"-''6''+ ••• (forever), (c) Special cases : — ! — =(1 ± j)-! = 1 T a- + ■'•= T ■T^+ a^ T ■•• ; ( I •'• I < !)• (Geometric progression.) l±a- Vr^^ =(1 ± ,r)^/2 = 1 ± l.r - _l_.r2 ± -i^.r^ _...;(! ^ |< 1). vrr^ 92 . -^ : 2. Arithmetic series : n +(a + d) + (a + 2 d)+ ■■■ -[-{a +(n— l)d); last term = 1 = a +(n — l)d ; sum = s = n(a + l)/2. 3. Geometric series : a + ar + ar- + nr^ + •••. (a) n terms : Z = ar'^-^ ; s = = a • ?• — 1 r —\ (6) infinite series, | r | < 1 : s = a/(l - r). 4. 1 + 2 + 3 + 4+ - +(H-l)+?i = h(h + 1)/2. 5. 2 + 4 + 6 + 8 + ••• + (2 ?« - 2) + 2 H = ?i(« + 1). 6. 1 + 3 + 5 + 7 + ■•• +(2;i-3) + (2?j-l) ^ n\ 7. 12 + 22 + 32+ ... +(« - l)2 + n2 = ,j(„ + l)(2 7i+ l)-3! 8. 13 + 2^+33+ ... +(n- 1)3 + k5 = [„(„+ l)/2]2. 9. 1 + 1/1!+ 1/2! + 1/3!+ .- =lim^l+-y = c = 2.71828.... 10. <" = 1 + x/1 ! + a;V2 '• + a;'/-"^ ! ••• ; (all x) ; a' = e^'-'g". 11. log,(l ±x) = ±a;-x2/2±x3/3-x^/4±rV5 ; (-li = 7r<7V4 = i r/2 ; arc C.S = r • a, (a in radians) = jrra/lSO, (a in degrees); Chord 2)5 = 2 r sin a = 2r8in (1^/2); Sector ODCB = -^^ irr^, (a In degrees) ; Triangle I)OB = r^ sin a cos o = (l/2)r»sin2o; Segment £>FBC-= r^ [ira/lSO -(sin2a)/21. /(I +cosa) ; — cos a)/2 ; cos o)/2. ii, b, seiniaxes ; r, >•', radii, e ='\/aJ - l^ ; (eccentricity); p = 62/,. = a(l - c')/*- ; a = tan-» ('^) = eccentric angle ; a* = a cos a, y = 6 .sin a ; 2,S , . 2.S a- = a cos — p , y = 6 sin — - , a) = a sin 4", y = 6 cos <<> ; ij» = w/2 - a. /■ + /• = const. = 2 a. .4 = irrtft ; .S = OA />;=—- a = —- cos -' - ; 2 2 a ' j:v5^ rt* — a!' [ Tablet.] = n J Q v^l — f * cos' a t/o : where cos o = x/ii. rc5/>=PV^^'l; F' a l^[MF N 1 / \ n / ^ 6. Parabola. e= V P. 7. Prism. 8. Prismoid p. 125). a2 &2 2_ (origin at O) (pole at F). p = LN/2 ; LiV^= latus rectum. y^ = Ipx, (origin at O) ; P = -. — ^-— s. (pole at F). B = area of base ; A = heiglit. £ = lower base (area) ; M— middle section ; 7*= upper base ; /t= height. r-' — r = const. = 2 a ; 5=Sector OVP = ^ log (^ + ^) =¥"••'-©-¥•'■■'-■(1)^ a!=a.cosh?^, y = &8inh— ; ab (lb or if tan <^ = sinh ^^ , ab X = asee(j>, y = b tan <^. Area ONPM= § ^2 a;3'2pi/j ; Arc OP = /"^ Vl +{:y/p)idy. (See Tables, p. 3S, No. 45 (a). (See also Taft/es, IV, G, p. A6.) [The volume of each of the solids mentioned below, except (16), follows this formula, though not all are prismoids.] 9. Pyramid (any sort). 10. Bight Circular Cylinder. A = area of base ; h = height. r = radius of base ; h = height ; ^=base (area). r=A./>/ii. A (curved) = 2 nj-h ; A (total) ,= 2 nrh + 2 nr^ ; II, F] MENSURATION Dimensions or EiirATioNS 11 11. Right Circular Cone. Si'c Kig. 2 1 , p. s7. Urn a.=r/li ; cos a=/i/s ; .sin a = t/x. 12. Frustum of Cone. JJ=lower base (area); 7'=iiiii)t'r base. 13. Sphere. (a) Entire Sphere. {b) Spherical Seg- ment. Other notations as above. (c) Spherical Zone. r = radius of base ; A = height ; B = base ; « = slant height ; a = half vertex angle. r = radius lower base ; li = radius upper base ; /( = height; «=slant height. + (5 - ?„)» = ,J ; r = radius ; d = diameter ; C = great circle (area). a = radius of base of seg- ment ; h = height of segment. h = height of zone ; «, 7) = radii of base A (curved)= TrrV/J + As ^ (total) =7rr(«-f /■) ; V=-nrVi/Z = Bh/i. A. (curved) =it»(R + r)\ V=7rh(^Ifl + llr + r*)/S. ^ = 4 nri = 7r(/2 = 4 C; >^ = 47rH/.} = 7r(/3/6 = ^ . r/S = 4 0/3. (j2 = /( (2 r - A) ; A = 2 ttM = 77 («2 + /*2) ; F= ffA. (3 rt2 + A2)/6 = nh^ (3 r - A)/3. ^ = 2 ttM ; V = nh (S 1 -fc sin-lar lst<2 •1th Q t/2 -7r/2 ,r/6 7r/4 n/S 0.G2 -sin-U+X-) cos-la- UtQ 2dQ n/2 TT 7r/3 ir/4 n/6 0.96 7r-COS-l(+X-) tan-l.r \»tQ 4th (? «o ttA -7r/4 0.4fi 0.62 0.71 ,r/6 >Tr/4 -tan-l(+^-) H. Hyperbolic Functions. 1. Definitions. (See figures III, E, Jo, pp. 20, 28 ; and V, C, Binh X = (e* — e-'^)/2 ; cosh x = (e* + c-')/2 ; tanh X = sinh x/cosh x = (e* — e~*)/(e* + e**) ; ctnh X = 1/tanh x ; sech x = 1/cosh x ; csch x = 1/sinh x. = Gudermannian of x = grdx = tan-i (sinh x) ; tan

= gd X, sinh x = tan (^, cosh x = ctn 0, tanh x = sin 0. I. Analytic Gecmetry [(x, y) or (a, h) denote a point ; {a-^, y{) and (3-2, ^2) two points ; etc.] 1. Distance I = P1P2 = \/(x2 - xi)2 + (^2 - yiY = Vax^ + a/. 2. Projection of P1P2 on Ox = Ax = X2 — xi = i cos a, where «=Z(Or, P1P2). 3. Projection of P1P2 on Oy = Ay = yo — yi = I sin a. 4. Slope of P1P2 = tan a= {y^ — 2/i)/(x2 — Xi) = A?// Ax. 5. Division point of P1P2 in ratio r : (xi + r Ax, yi + r Ay). 6. Equation Ax + By + C = 0: straight line, (rf) y = mx + b : slope, in ; y-intercept, b. (6) y — yo = m (x — Xo): slope, }?i ; passes through (xq, j/o)- (c) (y-2/i)/(2/2-2/i)=(x-Xi)/(x2-Xi): passes through (xi,?/i), (x2,?/2). (c?) X COS a + y cos ^ =p: distance to origin, p ; a — Z(Ox, n) ; /3 = '^{Oy^ n); n = normal thi-ough origin. [General equation Ax + By + C = reduces to this on division by \^A^ + &.] 7. Anglebetweenlinesof slopesTOi, TO2=tan-i[(TOi — m2)/(l + mim2)]. [Parallel, if TOi = m2 ; perpendicular, if l+TOim2=0, i.e. if mi = — l/mo.] 8. Transformation x = x' + h, y = y' + k. [Translation to (/;, k).'] 9. Transformation x = ex', y = ky'. [Increase of scale in ratio c on X-axis ; in ratio k on y-axis. ] 10. Transformation, x = x' cos ^ — y' sin tf, y = x' sin d + y' cos 6. [Rotation of axes through angle 6.'] 11. Transfoi-mation to polar coordinates (p, d): x = p cos 0., y — p sin $. Reverse transformation : p = Vx-^ + y^, 6 = tan-i (y/x). II, J] ANALYTIC GEOMETRY 15 12. Circle : (x — a)'^ + (2/ — b)'^ = r- ; center, (a, 6) ; radius, r ; or {x — a) = r cos e, {y — b) = r sin ^. (d variable.) 13. Parabola : y'^ = 2px: vertex at origin ; latus rectum 2 p. 14. Ellipse : x-/a^ + 2/-/^"^ = 1 = center at origin ; seniia.xes, a, b. (See II, F, 4, p. 9.) 15. Hyperbola : x^/a^ — y'^/b'^ = 1 ; center at origin ; semiaxes, a, b ; asymptotes, x/a ± y/b = 0. See II, F, 5, p. 10. (a) If a = ft, X- — y- = a" ; retangular hyperbola. (6) xy = k; rectangular hyperbola ; asymptotes : the axes, (c) 2/ = (a.»: + 6)/(ca;+rf), rectangular hyperbola; asymptotes: x= ~d/c, y = a/c. 16. Parabolic Curves: y = ao + aix + a^x^ + ••• + anX". [Graph of polynomial ; see also Figs. A, B, pp. 17, IS.} 17. Lagrange Interpolation Formula. Given y — f(or), the poly- nomial approximation of degree » — 1 [parabolic curve through n points, (a:i, yi), (x-2, yo), •-, (a-„, «/„)] is y = P(X) = UiPiix) + ViP>>{3C) + - + ynPni.^), where the polynomials pi(x), p^{x), •••, p«(x) are Pi(x) = (a^ - a:i)(a; - xo) — (x - x.-i)(x - Xj+j) ...(x-Xn) (Xf - Xi) (X,- - Xo) ... (X,- - X,_i) (Xi - Xf+i) •• • (Xj - x„) [Numerator skips (x — x,) ; denominator skips (Xj — x,). Proof by direct check.] [For a variety of other curves, see Tables, III, pp. 17-32.} Formulas of Solid Geometry, §§ 102-3 ; pp. 31.5-10 ; see also Figs. "^III, N1-X5, p. 31. When possible the preceding formulas of plane geom- etry are so phrased that an additional term of the kind indicated gives the analogous formula of solid geometry. In particular, see G d, p. I4. J. Differential Formulas. [See (a) List nf Dijfi'rential Formulas of Elementary Functions, pp. 40, 173. (6) List of Standard Integrals, p. 174, and Tables, IV, p. 33. Reverse these to obtain Differential formulas. (c) List of Standard Applications of Integration, Tables. IV, H, p. 46. (d) Infinite Series, Taylor's Formula, etc., see Tables, II, E, pp. 7-8. 16 STANDARD FORMULAS [II, J 1. 2/ = /(a?) : dy =f'(x) dx, f'(x) = dy -^ dx = dy dx. 2. F{x, y)=0: F^dx + Fydy = 0, or dy=- IF^ ^ Fy] dx ; F^=dF/dx, F,= cF/dy. 3. 35 =/(«), y = " f -f"') - (/')']A^< -/'• 4. Transformation ac = /(<) : y = ^(x) becomes y = (t, u) : y = F(x) becomes u = ^{t). (a) dy/dx becomes ^^^U 1^ or \^± + ^-±. ^I -^ r^+ §^ . ^]. ^ ^ di d< La« du dtA \_dt du dt\ (b) d^y/d£'^ becomes d [dy/dx]/dt -i- dx/dt ; [compute as in 5 (a)]. 6. Polar Transformation x = pcose, y = p sin . dx = cos dp — p sm 0d0 ; dy — sin dp + p cos dO, d^x = cos 0d^p-2 sin dp dO -pcosO d0^, d'^y = sin d'^p + 2 cos ^ dp dd - p sin dff-. 1. z = F(x, y): dz = F^ dx + F, dy = p dx + qdy; [see I, 3 (d), 'p. ^]. 8. Transformation x =f(u, u), y — \ du dv/ [Similar expressions for c-z/dy'^ and higher derivatives.] TABLE III STANDARD CURVES A. Curves y = xn, all pass through (1, 1) ; positive powers also through (0, 0); negative powers asymptotic to the j/-axis. Special cases : ji = 0, 1 Chart of ?/ = x" for positive, negative, fractional values of n 1 1 ' 1 \\ w ' 1 /' / 1 1/ ^z 1 1 1 11 J '\[\ ., X 2 /! /I I / T-. 1 1 ' ! V, T , 2 -'/_t/j ■-'/I .x7 ' /' 1 \ ' 1 \ ^1 1 ii i" n iimH yi './ 1 1 1 1 1 1 1 ' \> -\ 1 4i-: ~ "/^/ ' / 1 i 1 \ 1 J \ 11 u /_/ 7 7 "^ 7 ' ' / 1 \'* i \\~ V t t 'I I/I 7 i '1/ \\ \- ' - \ A V T J 1 ' :'/ L Ij.i ll. 111 \ 1 1 \ TUP / / - ^-^fl \ V ' K,r ■ > il \ ' 1 ' n\ \ I " ^T^ ' \\ j y -i ■sW / ' / ! . . zW'\iv\\\\ / /i 1/ X- 'U^^ \ p '~ ~7i \V^ AW /■' \ y^ "li 1/ _ AvA ,'\\\\\ / 1 p^ \ t^X li / 1 / '^1 JV-.U-1 \ \ 1 / ~i \\^ ^-^ \>^ \^ A / Evt 1 Fo»er9 U\ -^N -rn ^/ M , > . \\ 1 / r"- I'T'"' "', v.// ,^ ^^^^ ^^/ ^^=4l3i: ^v' ■"^! 1 1 {{111 1 1 1 i 1 i ^ OdJ I'owen all.i Iluots -^ j / yi^b^ 1 ",01 "^ U- «X-U^ 1 r...ma,.i„| 1 ^ ^/ \ 1"' ^•> 4— :i= b^^fe^ \ ]3rd C u Jnihl /, ■• './ ' v< t'^ *tii^- \[ i [A y . \sx li V^l "; 1^'" ' k % M ^ "^" \*lS ^ Vi- n = 1.11, - Adi.baU^ EkpA,.lon for Air -^^^ -f- ^ -^y NtT ^z- j*-K' 1 1 M ''^^'A^ 's£snn ^i\' 1 liX^^t^ n '\. fv^ \ p/\ '\Jy "^ fT= — ir^:^ \~7^^^^IA^^^ \ 1 //; 3t_d ^"^^^^VT^ — >-f--' yf'C^ cSi T^ sT ' -1-— 4>'4/vf^ ^. '^^ ^^ 7 \ "^ ^~-^'"', ) "' ii itifi=§v \ " '^ V 7 >\* ^^^\^ 7 \\^\ \ ~ " ^ i 1% 1 I 1 ^^-<3 ■/ i 4 \\ '^ A , \M ^^ J V i i 1 '"i^ it " AH- ^w ;? ^ve 1 ltou(>. i 1 4it +- t ^^ t if V.' r laiipear lu 1 1 n: " iH '%■' •g_5 Ih QuDdriin 1 1 \\\ 1 1 \\ \\ '\ / -t-r 1 1 l'\ 1 _L Fia. A 17 18 STANDARD CURVES [III, A are straight lines ; n = 2, 1/2 are ordinary parabolas ; re = — 1 is an ordinary hyperbola ; n = 3/2, 2/3 are semi-cubical parabolas. The curves pv^ = c occur in the theory of g-as expansion, where /^ = pressure; •» = volume ; c and m constants. In isothermal expansion (p. 29'2) jn = 1, whence pv = c orp = c»— 1; (?i = — 1 in Fig. A). Choose scales so that y = ^Vc and « = a-. Inadiabatia ':■ done compressing J" "2 j}dc = worA- B. Logarithmic Paper ; Curves y = .r", y = Jcx**^. Logarithmic paper is used chiefly in experimental determination of the constants k and n ; and for graphical tables. In Fig. B, k = 1 except where given. y q^B iMimwt \ \ 1 i^ M/!!l/ 1/ V ■ /' 8 ^ >kl,l 1 i\\v. ^ o M r f 1 / / .° '^^> :]TissiA\\v V W fi / / / ^^MJiApvY^ V vj i\ / 'v / ( / m^B^ \ %J. Ifl mill ' i^)/^ /> -<" '~ -/ ___.m^V^^^ ^ \ jfj-^- A ^ *'/ ^/ 1/ / ' / ' ■^\ ' \N p^>.\ 5/1 ■'^ 1 n 1/ /i /^ / / , '' \i\ ^X\ \^ \ ^3!"^ /!/ 11/ / / a, y. ^i >v ' .15 \['i .-lb :. :iiAii:^, — >.\z ^^J<^ife. iJ^ .. --- -45"^" 1 1 U'^ ^ — ' I' <^ / / 1 |\ \ '-^K J KN (, Vfi>ri <&' /, i // f 1 ^ ^ •^ ^fef- nl Pjfe^*" ^1/ / 1 j\ i \ \ \ ^«^^i\ ^ ■^ ^'^C^' /^ A / / IjUVrV^ ^'^'^ \ f^ il ^v ^j^'I/tT ^ J L n :\- l \J \-? Ni \%x aralle ■'1 M^)^ _v 7 '^tK"\ 1 v.. J i| % 5i: .15 i-^T^ ^ WW ^=- --i-p ^ = S ^ % — : -^ .'-'7 c;y < ttlt-? i~i ,\ V ''^,^>;^ a?±E y / 171 1 I\^llu 1 1 ■! i:::HZ ^ ^ >, 1 i-<^i^ •la .2 .26 :i .1.5 2t 2.5 3 3.5 4 4.5 5 6 7 8 9 1» Fig. B {See §122, p. 229, above; also Williams-Hazeu, ffydraulic Tahlex ; Trautwine, En- gineers^ Handbook ; D'Ocagne, Nomographie.\ The line y = a*-— 1 gives the reciprocals of numbers by direct readings. Ill, D] ELEMENTARY FUNCTIONS 19 C. Trigonometric Functions. The inverse trigonometric func- tions are given by reading y first. :::+q: 1 ! tF EEEir m N'^ ^ w S' - #1= * |±W^ 1 jt ^|: Ff* Iv^f/ T i V ^ f" 1 ^ ;■ ^■4- <- ^^ ^ ''^ y ^ 7 "i' ■" • 7^'" - ^- \'>'^ 1 y- .\ A / 'if -:Hch. Trofthe ' L trin Fnnrlir^na r^ ^ (>; and Their Inverses 8 y =-sm .r gives x= arc _ r\ y^ vX~^ 'A -- .. I Z /-/ \ ^-^- -f^^+7-f m !/ L \ 1 M 1 ! 1 ;^ ' K - ' ' 1 /< \ : , I Fio. C D. Logarithms and Exponentials : y — logiox and y = log, a;. Note logeX = logioX logg 10 = 2.303 logio .r.. The vahies of the expo- nential functions x = 10*' and x = e" are given by reading y first. See E. 20 STANDARD CURVES [III, E E. Exponential and Hyperbolic Functions. The catenary (hyperbolic cosine) [?/ = cosh x = (e^ + e-^)/2J and the hyperbolic sine [y = sinh x = (e^— e-^)/2] are shown in their relation to the ex- ponential curves y =e'', y = e"*. Notice that both hyperbolic curves are asymptotic to »/ = e^/2. Fig. E The curve y = e~' is the standard damping curve ; see Fig. F2, and § 92, p. 160. The general catenary is y ={a/'i,){e'/'^ + e~*/")= a cosh (ar/a) ; it is the curve in which a flexible inelastic cord will hang. (Change the scale from 1 to a on both axes.) Ill, F] HARMONIC CURVES 21 F. Harmonic Curves. The general type of simple harmonic curve is 2/ = a sin (A-.r + e): Curve [ a sin (&x + «) sinx cosx 8in2jp (1/2) sin (6* - - 1.2; aiiiplitiide a 1 1 1 1/2 wave-leniarth ! 2 ,r/^- •2 IT 2. jr ./3 phase - f/'!- f/'i 0.2 A compound harmonic curve is formed by superposing simple har- monics : in Fig. Fi, j/ = sin 2 x + (1/2) sin (6 x — 1.2) is drawn. -Jl\- y-z = Bini2x = 4- 8iu(0r- 1.2) i r 'V = 8in2z+l. 8 1 ^ 1 n (Cx- i / \ A". \^ ^h\ /v. h \\. ^'A ■\/ V K w /l/i 1 ^ V ! 1 ! Such curves occur In theories of vibrations, sound, electricity. See §§ 90-92, pp. 157 22 STANDARD CURVES [III, F The simplest type of damped vibrations is y = e-" sin kx : Fig. F2 shows y — e-^''^ sin 3 x. The general form is ?/ = aer""^ sin (/tx + e) . Such damped simple vibrations may be superposed on other damped or un- damped vibrations. See §§ 92, 189, pp. 160, 368. - 4kr „l„ ' v=JJJ-, n " - ■ IPL ^ i-iauo 1 ^ ii V ?/ = L « linSa > \ • < ' <•>; , - v , ' 'H -•- e^ - I* ,' «>^ . / s \T . ' V , . , ' _ _ y _ I _ ^ _ _ _ ^ ^_ , _ _ _ _ ^ pi L ^ ^ _ _ L _ _^ _ _ - - - - / - - ^ - - - 1^- - - - - - - - [^ - ^ ^ '^ -^ - 4^ ^H - !- - - ^ V - s s. , ^t^ ^ " -■ - _ _ A _ 0| _ _ _ _ _ _ ^ 1^ _ _ _ 3 - _ _ -i _ 2j .BVn4i^- _ _ - ^ ^ ^ - - - "1 - A - - ^- Kt - Y/ - - - - : r :: ^ ^ r - ^ - n - '- - - \ ' ' Si t ' A ,^ - r - •'' 1 ^1 / i ^ 1 ', - ,' « / ■' - - - ' .-/ \ ? ■ - . ^ »^> •^ - ^ \A 1 Fig. F2 G. The Roulettes. A roulette is the path of any point rigidly connected with a moving curve which rolls without slipping on another (fixed) curve. J 1 ^ Ihe Cycloid i^'^ ) / M N Fig. Gi Figure Gi shows the ordinary cycloid, a roulette formed by a point P on the rim of a wheel of radius a, which rolls on a straight line OX See also Fig. 69, p. .307. The equations are [ X = ON— MN =ae- asmi \ y = NC — EC = a — a cose, where e = Z NOP. Ill, G] ROULETTES 23 Figure Go shows the curves traced by a point on a spoke of tlie wheel of Fig. II, or the spoke produced. These are called trochoids ; their equations are Ix = (10 — b sin^, \y = a— b cos 0, The Trochoids where b is the distance PC- If ft > a, the curve is called an epitrochoid; if 6 < ff, a hypotrochoid. Figure G3 shows the epicycloid ; 3;= (a + 6)costf-/)Cosr^^-±-^(?1, y]= (a + ft) sin ^ - 6 sin f^-i-^ 0~\, Epicycloid Fig. G, formed by a point on the circumference of a circle of radius ft rolling on the exterior of a circle of radius a. 24 STANDARD CURVES [III, G Figure G4 shows the special epicycloid, a = b, ix = 2a cos — a cos 2 ff, \y = 2asind — a sin 2 d, which is called the cardioid ; its equation in polar coordinates (p, / = is also a critical point, but the surface cuts throu^'h its tangent plane there, along the curve A" = — 2 ; yi = a^ _ 3 ar + 2. These curves are drawn by means of the auxiliary curve q=ai'—&x, itself a type of cubic curve ; then y =^/q — A; is readily computed. Coiltpnr JLines of the'~-^urface z-x' -3x + z=k '\ 4) if 1 ^\ < 1 \ icii Jff (( m = 2 (jN ^ n- . ■^t— k^ zS-^ \ y^' < i \K- = ^^\\ m Curv s: X -3x- k m " 1 4-3-1 t: 6,-4,-2, 0, 2, 4 2: The Btrophoid 1 l.^ ixillary Curves -3xaj,d2/l4-x 1 1 Fig. 1. Figure !•> shows the contour lines of the surface 2 = :r8— 8 r + .v'- (r— 4) tor z ~ k = — a, — 4, - 2, 0, 2, 4 ; that is, the cubic curves -.(.rs. ■/0/(4 -.-•). The surface has a maximum at (- 1, 0). At (1. 0) the horizontal tangent plane ? = -2 cuts the surface in the strophoid y* = (a^ - Sai + 2)/ (4 — a") whose equation with the r ew origin 0* is jf* — a-'^ (3 + x') / (3 - a;'). The line ar = 4 Is an asymptote for each of the curves. 26 STANDARD CURVES [HI, 1 Figure I3 shows another cubic : the cissoid, famous for its use in the ancient problem of the " duplication of the cube." Its equation is or p = 2 a tan e sec 6. 2 a — X It can be drawn by using an auxiliary curve as above ; or by means of its geometric definition : 0P= QB, when Oy and AB are vertical tangents to the circle OQA. Figure I4 shows the conchoid of Nicomedes, used by the ancients in the problem of trisection of an angle. Its equation is \x-a} asec0 ± b. Conchoid Folium of Descartes Fig. I4 Fig. Is Figure Ij shows the cubic x^ + y^ — S axy = 0, called the Folium of Descartes J see Exs. 1, p. 46; 11-12, p. 63. Fig. I. Figure le shows the witch of Agnesi : y = 8 a^/(x^ + 4 a^) ; see Exs. 3, p. 163 ; 5 (ft), 5 (d), P- 166 ; 5, p. 180 ; and see III, J, below. Ill, I] QUARTICS — CONTOUR LINES 27 Figure I7 shows the Cassinian ovals, defined geometrically by the equation PF • PF^ = k- ; or by the quartic equation, lix - ay- + 2/2] [(X + ay- + y^] = ^4, where a = OF ( = 1 in Fig. I7) . The special oval k- = a^ is called the lemmiscate, (x- + y-)'^ = 2a^ (x- — y-) or p- = 2 a- cos 2 0. \ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 M 1 1 1 1 1 1 1 1 Contour Lin es on Surface 2< = -1 (Cx+a)-+^2] 1 ^ '" 1 1 1 1 1 1 1 1 1 1 1 M 1 1 1 1 1 1 k- •i -. 1 ^^ --A A- ==p== -"t=" = ---'v-^i^'^^" X-'' "^^ "^ ^y \_. 1— -_^ -^ \^ / 1^}^^ riri -^■ 4J^ ^^^ '^^sN V- J~l t/L^^^ ^""r^" L j/ /^^ -J — -v " . [V V :; u^^^-(L5^ .;>>[ r '/}■!■' \ \ 'rt -. AXX-^X- /n 1 '^ 1 1 r^ \ \\ 04 FTHiX \j 1 T l?^ i hV D I/I -i -^ ^c^^trzi 7 -aJZ iV?"-'' V •' \l ^ 1 \/ 1 11 ^ 5 S^>4!'J^ is.^ ' >_ ^^~>— f-^ ^ /a I ^^^^=4-ru z>-p^ "T^H 1^ ' ^"^"^^^ ■^ ^< 1 y V^ ^^sXiii I-/ ^■'~>j~ ---v^' 1 ^-r^i / y -S ^^ -^=- " I 1 s ^-,ir _ A;" = 1 5 \^^^ y . 15. X— _,.„ 1 ^-' 1 , r — —* ^ 1 ^ I'-^l 1 1 1 1 l-H-n MM Cagslnian Ova la li:x.-a)=- (Figure dra -1/- 1 [(X4a)- + V" ) = A* (oc PF- PF'- it-l 1 M 1 1 1 1 «„fora=n 1 1 1 1 1 1 1 1 Fig. I, The ovals are also the contour lines of the surface 2*=[(x-a)2 + y2][(x + a)2 + 2/2], which has minima at (x = ± a, y = 0), and a critical point with no extreme at origin. 28 STANDARD CURVES [III, J V^ y ^v JT /, ^^ -jTf w^ ^ ^1 ^, X J. Error or Probability Curves. Figure Ji is the so-called curve of error, or probability curve h ,,2,2 where h is the measure of precision. See Tables, IV, H, 148, p. 45; and V, G, p. 54. Figure Jo shows the very- similar curve Fig. Ji ■ " y = sech x = 2/(e^ + e-^). In some instances this curve, or the witch (Fig. le) , may be used in place of Fig. Ji. Any of these curves, on a proper scale, give good approximations to the probable dis- tribution of any ac- cidental data which tend to group them- selves about a mean. Fiu. J2 K. Polynomial Approximations. Figure Ki shows the first Taylor polynomial approximations to the function y = sin x. (See § 134, p. 2-58.) M-] 1 4+/.0- U 71 h-L : «-, V 1 ,N\irh T -- -i -- ^^--- - - "fl^, u',r " t Vx^>. ■" A _ : : 'tf- - ' - - - - ^/^ ' }^~ '^ ' :^ ^ li^'l^^' " ^ ^*A '' "J^r"" '^^~^^•^ r +-f - -'A - ' " V ■'T-^L. ■_p5'<-i2r: tc— ^-^iii- L._L-- _j.:::;::±ji±Ei:±± Ill, L] APPROXIMATION CURVES 29 Figure Ko shows the Simpson-Lagrange approximations : (1) bj a broken line ; (2) by an ordinary parabola ; (3) by a cubic, which however degenerates into a parabola in this example. (Lagrange Interpolation For- mula, Taft^es, p. 15.) The fourth approxi- mation is so close that it cannot be drawn in the figure. In practice, the division points are taken closer together than is E feasible in a figure. 1 1 1 1 M 1 1 1 1 1 M M 1 M i 1 1 1 I [ 1 I ! i • M rp n Simpson - Lagrange Polynomial Approximations \-[y[-\ 4- -U- -pioi ij- s.lixp HT --i ±._:^ f-r-- --4 + ----: r^d"- ntU -No.2;r . .^.^.-io-i^ ■'fc"*! II ^^ t:.^n.i :^[ ^ -i-l<^ .^C f4i T^ < > ^ -y'^Y - ^-^^ A t -J: i 1 1 P >C \t No .3 S^ J J . ::z^_ :il! ... ::^ :: -S,^ ^ Tt ^ d- Jtu- 1^^ -7^ - - -jf --3 --2 - ^ 3 .^41 : ± iL :± Fig. Kj No. 1. y = -,r, 0<.)-<-; y = ix - A.r2 = 1.2T3a-- AObx^, Q'-""/" 1 a \x \^^^\^,.''& __X- \\\ i'/V M 1" / Ii! / Fig. Na Fig. Ns Elliptic Paraboloid Fig. IS!* Fig. Ns TABLE lY STANDARD INTEGRALS Index: A. Fundamental General Formulas, p. 33. B. Integrand — Rational Algebraic, p. 34. C. Integrand Irrational, p. 37. (a) Linear radical r = Vax + h, p. 37. (b) Quadratic radical V± r- ± a^, p. 37. D. Binomial Differentials — Keduction Formulas, p. 39. E. Integrand Transcendental, p. 30. (a) Ti-igonometric, p. 39. (b) Trigonometric — Algebraic, p. 4S. (c) Inverse Trigonometric, p. 4.h (d) Exponential and Logarithmic, p. 43. F. Important Definite Integrals, p. 44. G. Approximation Formulas, p. 45. H. Standard Applications, p. 46. A. Fundamental General Formulas. 1. If = — -, then W = V + constant. [Fundamental Theorem.) 2. If I M dx - 1, then — = U. [General Check] •^ dx 3. \cu dx = c\u dx. 4. ( [u + v] dx = \u d.r + \v dx. 5. ( II dv = UV — \ V du. [Parts] 6. {(f(u)du] = r/[d)(.r)] ^'^^-^^ rfX. [Substitution.] LJ J u=,uz) J dx 34 STANDARD INTEGRALS [IV, B B . Integrand — Rational Algebraic . 7. fic»»dic = ^^, w^-1, sees. J n+1 Notes, (a) / (Any Polynomial) dx, — use 3, 4, T. (6) /(Product of Two Polynomials) «?«,— expand, then use 3, 4, 7. (c) Jc rfaj = ca;, by 3, 7. 8. C^ = loge iC = (logio x)(log, 10) = (2.302585) logio «. J oc Notes, (a) J (l/a-"") cfx, — use 1 with w = — m if ??! :^ 1 ; use 8 if m = 1. (P) / [(Any Polynomial)/a''"] dx, — use short division, then 7 and 8. 9. r_^=larctan^ = ltan-i^ = lctn-i J a'^+3c'^ a a a a a ; = — — Ctn-l— [+ const.]. a a 10. r _^ = _L log ^^:^=-i- log ^^:^[+ const.]. Note, All rational functions are integrated by reductions to 7, 8, 9. The reduc- tions are performed by 3, 4, 6. No. 10 and all that follow are results of this process. 11. C(ax + ?))»(;x = - '^^^ + ^^"^\ n^-l. (See No. 12.) [From 7.] J a n + 1 12. r — ^ = 1 log (ax + 6). [From 8.] J (ax + b) a Notes, (o) \ ^^ '^ dx, — use long division, then 7 and 12. i ax+ b ^^^ r Any Polynomial ^^^ _ ^^^ ^^^^ division, then 7 and 12. •• ax + b 13. r ^?£ = 1 Til („j^l), [From 11.] J (ax + 6)"' a (ni-l)(ax + &)"'-! 14. f_E^?^ — = lr_^+log(«x+ &)]• [Fromll, 12.] Notes. (<() f "^''' + -^ tfa-, — combine A times No. 14 and B times No. 13, m = 2. J (<«• + 6)2 (Any Polynomial) J («» + rfflj, — use long division, then 7 and 14 (a); or use 15 IV, B] RATIONAL ALGEBRAIC 35 15. [ ( F(x,ax + b)dx~\ = I C f (^^-^ , u\ du. [From 6.] Notes, (a) Restatement : put u for ai + b, ^^^-^ — for a:,— for rfce. a a (ft) >/(««• + m d n ; use 1 (h) if m < n ; see also 51-54. 16. 1 = 1 [— « ^— 1. (aa? + ft) (cic + d) ad— be Lax + b cx + dj Notes, (a) [ — , — use 16, then 12. Special cases, — see 10 andl6 (6V (ft) r iLr = ^_rri (L — Idx. (Special case of 16 (a).) (c) r Ax + B rfa5, — use 16, then long division, 12. ^ ' i(ax + b)(cx+d) (^^ , (An.vPolvnomi.il) ^ ^^^^ jg ^^^^ ^ division, 7, 12. J Uix + b)icx +ii) (e) If ad - be = 0, 18 can be used. Notes. («) Restatement : PutMfor221+i; -^fora- a; ?< — (i (6) r — — — = — ^— r ^"""^" •* .7-"(rtJ- + ft)"* ftm+n-lJ u"> I ,\ r dx u- a log ;<■ . r 18. f ^^ = -J— tan-1 X Jg , if a > 0, 6 > 0. [See 9.] J ax'^ + b ^/Wh ^ b -J^[u- -a' M - -«; (u-a)a ; *" for a r + 6 (^ bdu -«)2 for daj. n+n-2 _, du • then U8< 8 6. dx «J - if' ?{ + 2 a« lopr « x^U'x + ft) 2ft3 log ^ax-v-b ^ if „ > 0, 6 < 0. [See 10.] 2 V- a6 Vax+V-b dx Notes, (a) f ° . - use 18 (2nd part) ; ft = - c. (ft) f — ^ - - I -^ . J ««' - c I ' ' ^ 'J c - nx' J (7x2 - c 36 STANDARD INTEGRALS [IV, B J ax/ + b 2 a Notes, (a) [ \ . , — use long division, then 18. (6) r±liL±L^ ax, — use 18, 19. (0 J^ aa-2 + 6 OQ 1 1 r m'^ _ wix — ?i ~[ (WIX + 7l)(( Notes. («) f- — — -tt > > 21. ax2 + 6x + c = arx + ^]'-^!^il^. L 2 a J 4 a Notes, (a) f f , '^f , 1 ^ = f ^^^^^ — ; — , then 18. ^ J «a'2 + b;e 4- c I , , , ^ J „ b' - 4, [If the cubic has a double factor, set ti = that factor, then use 17 c] 22 f ^^^ ^ 1 L J (ax2 + 6)2 2 a ax-' H + 6 23. f ^ f ^ = ^ + J- (• '^■^- ; then 18. J (rtx2 + by 2 &(«x2 + fe) 2 6 J «x2 + & ' 24. C r^\, =ro^f-1 ; then 7 or 8. J{ax^ + b)^ L2 a J 11^' ju = axni f <^a: _ J X 2 }>i, - 3 C dx J (ax2+ft)"' 2ft(m-l) (fflx2 4-&)"-i 2(m - 1)6 J (ax2 + M".-i 25. 2(m-l)6J (ax2 + 6)' Notes, (a) Use 25 repeatedly to reach 23 and thence 18. (ft) Final forms in partial fraction reduction are of types 12, 24, 25 (by use of 21). IV, C] IRRATIONAL ALGEBRAIC 37 C. (a) Integrand Irrational : involving /• = \ aj- + b. 26. I \ F {X, Vajc + 6) dj- \ ^ = ( r( '"' " ^ , r]^ dr, 2r \ Vrtx + h d.r = \ r— dr = — r', r = y/ax + b. J J a 3 a (.>■ Vax + 6 d.r = -( (?•* - bf^) dr = — [--^'\. J d^ J d^ \_b 3 J f ^' ='Udr = '^r. -^ Vax + b a J a 30. r dx ^ r^^r ; use 9 or 10. J .■ ->/„.. A.h J r- — b Vox Note. Vi/a- + // == ((/.i- + ?<) V(Z7+7, ; (%/,/,»• + /,)3 = (^ax + b) v')/(ar + d) : use 50, tlien 49, then 32-48. (b) Substitution of « =\/ {aoe + b)/{cx + d) is successful ^vithout 50. D. Integrals of Binomial Differentials — Reduction Formulas. Symbols : « = ax" + 6 ; a, ft, p, m, n, any numbers for which no de* nominator in the formula vanishes. 52 . (x"\ax'' + b)Pdx^ ? [x'^+^up + 7ipb (x"*uP-^dxV J in + np + 1 J \x^(ax^ + b)P(lx = — — ! [- x'"+hip+^ + (m + n + np+l) (x"'uP+^ dx]. bn{p + 1) J 53. (x^'^iax^^ +b)Pdx = — -[.v^+ij/p+i — a(m + n + np + 1) (x^^^uPdxl. (?n-|-l)ft J 64. (xr*^{ax'^ + h)Pdx = — ^- ^[.r'"-"+i?t/'+i -(7)1 - 71 + l)ft rj7»»-"MPrfx]. a{rn + np + 1) J Notes, (a) These reduction formulas useful when p, m, or n are ft-actional; hence applications to Irrational Integrands. (fi) Repeated application may reduce to one of 32-48. (c) Do not apply if j?, m, ?), are all integral, unless ?! ^2 and p large. Note 11, 15, 17-25w En. Integrand Transcendental : Trigonometric Fiinctlons. 55. I sinu^rfa? =— cos J7. 56. Tsin'^ xdx=— ^ cos x sin x -f- ^ x = — J sin 2 x + \x. Note. J" sin* kx dx, — set kx = u, and use 56. Likewise in 55-78. 57. rsin»xdx=-?inri£^2^ + !Lziirsin»-2xdx. J n n J Note. If n is odd, put sin' a; = 1 — cos' a; and use 62. 'xdx. 40 STANDARD INTEGRALS [IV, E 58. (cosxdx = 8mx. 59. i cos-xdx = ^sinxcosx + ^ a; = ^ sin2 a; + i*. 60. (cOSnxdX = ""^"-^ ^ '^'" ^ + 'J—^ (cOSn-^xdX. J n n J Note. If n is odd, put cos^a; = 1 - sin^sc and use 63. 61. j sinxcosxfZx = — ^cos2x = ^sin2x[+ const.]. 62. f sin x cos^xdx = - ^"^""^^ ^ , n 9^ - 1. •/ n + 1 63. rsin'>xcosxdx = ?i5!^ „^_ 1. J 71+1 64. fsiu" X COS" xdx^ ^'""^'^-""^"-^ ^ + ^^1^1-1 fsinnxcos-^^dx — sin«— 'xcos™+ix , w— 1 T . m + n m+ nJ Note. If n is an odd Integer, set sin' a- = 1 - cos' x and use 62. If ;» is odd, use ( 65. fsin (mx) cos (nx) dx = - ^"" ^^^^ + "^ '"J - ^""^^"^ " ^^^^^ , J 2(m+?0 2(m-?i) 66. fsin Onx) sin (n,r) dx = ^^^I0ri-n)x] _ s in[(»i + >t)x] »^ 2(m - ?i) 2(ra + 71) 67. rcos(77ix)cos(7U-)(Zx^^^"'^("^- "^■'"-1 +"'"l^("^ + ")-'^J, m=^±n. •' 2(»?l — ?l) 2(771 + 71) 68. j tan xdx = — log cos x. 69. ftan"- x dx = tan x — x. 70. ftan^x^x = ^^""~^^- _ C tsin''~2 ^.4^.,^ J 71 - 1 J 71. j ctii X f?x = log sin x. 72. rctn2 ,/• dr = — ctn x — x. 73. (ctW'xdx =- ^^""~'^ - rctn"-2.rd>:. 74. Jsec X dx = log tan ( f + 7 j = log (s^c x + tan x) [ 4- const. ], m ^ ± n. IV, E] TRIGONOMETRIC 4] 75. ( CSC X dx = log tan ? = — log (esc x + ctn x) [ + const. ]. 76. fsec- xfix = tan x. 77. (cac'^xdx = — ctn z. TseC" X CSC" a; dr = \ ~ (See also 04.) J J sin" X COS"" X m-l J 78. ?ft — 1 1 sec'"""'^ csc"~' .r sec'"--xcsc"xdx ■ .sec"-i X csc"-ix + "' + " - ^ fs n — 1 J ' X csc"-2 X dx. n- 1 Notes, (a) In 64 and TS and many others, m and 7^ may have negative values. (6) To reduce J[sin"ir/cos»" a-] t/« take 7n negative in 64. (c) To reduce J"(cos»"a'/siu"a']rfa; take n negative in 64. 79. SubatitutioBB : 1 u = dti sinx C08X tanx X dx <„ sincT cos a; cfa; u v/1 - «J « sin-»!t (/!/ V. - »J Vi -«t cos a- — sin X dx Vl - h2 M Vl - «2 cos-i u rf« Vi _ u^ (3) tan X ffctxdx « 1 « tan-' « du ! + «» Vl + «2 Vi + «2 W sec 3! tan x dx V>/'-l 1 Vh2 _ 1 see-' K du «V«i- I ! '"" ;; l.e.^-dx 2u 1 + «2 1 - »/J 1 + «J ,-«a 1 + «2 Replace ctn x, sec x, esc x by 1/tan x, 1/cos x, 1/sin x, respectively Notes, (a) J /"(sin x) coaxdx, — use 79, (1). (b) / F(cos 35) sin x dx, — use "9, (2). (c) J'/f(tana-)sec»a!da-, — u8e-9, (3). (d) Inspection of this table shows de»irahle ttubsiUutionii from trigonoiiietrio to algebraic, and conversely. Thus, if only tan z, sin'z, cos' a; appear, use 79, (8). 42 STANDARD INTEGRALS [IV, E 80. r ^ = ^ sin-i^ + ^"'"^. ^ a + 6 sin X Va'^ — 6^ a + h&mx 1 i^gL=j4!^,^i±^tan(2Z21,i{«2<52. Vft^ _ a2 5 + V62 - a'-^ + a tan (a;/2) 81. r ^ = 2 ^^^_, r J^36 ^^^ q , «2 > 62 . Ja + bcosx V(j2 _ 52 L^a + & 2 J 1 ipg V^ + g + V^ - g tan (x/2) ^ ^2 ^^3 V62 _ a^ V6 + a — -v/fc — a tan (x/2) . f ^ = 1 logtan^±^. « = sin-i ^ ■ J a sin a; + 6 cos X Va^ + h'^ 2 Va"^ + 6^ •> a + b &mx ^ a + h s\ax b ^ b •' u+Z» sin as then use 82 a, 80. (c) Many others similar to (a) and (6); e.y. J[sin«/(a + & cosa;)] dte, — use V9, (2). id) f , „ ^"',, ;- and like forms, — use 79, (3) ; see 79, note d. ^ ^ J o2 sin2 X + h^ cos2 x («) As last resort, use 79, (5), for any rational trigonometric integral. Eft. Integrand Transcendental : Trigonometric-Algebraic. 83. \ x™ sin xdx=— x"' cos x + to j x™-'^ cos x dx. 84. \ x™ cos X dx = x™ sin x — to i x"'-i sin x dx. Notes, (a) / aj sin tc c?a) = — « cos a? + J cos a; dx, — use 58. (6) /«"' sin vidx, — repeat 83 to reach 58. (c) J (Any Polynomial) sin » rfa;, — split up and use 88. (d) For cos SB repeat («), {h), (c). 35 rsinxdx^ -sinx ^ _J_ fcosx^^^ ^ _^ ^^ J X™ (to — 1 ) X™-! TO — 1 J X™-! 86. r52?^^ = ^os^ L_ r!i5^dx, TO^l. J x" (m — 1) x™-! TO — 1 J x™-! IV, E] TRANSCENDENTAL 43 87. ^•^rfx=j'[l-^ + ^-...]d.c;seeII,E,13,p.^. 88. f^2if,,.,^rrl_^ + £!_...l,;^.seeII,E,14,p.5. J X J Lx 2 ! 4 ! J Note. Other tiigouoinetric-algebraic combinations, use 5 ; or "9 followed by 89-94. Ec. Integrand Transcendental : Inverse Trigonometric. 89. I siii-i xdx — x siu-i x + Vl — x^. [From 5.] 90. I cos-i xdx = x cos-i x — Vl — x^. 91. I tan-i xdx = x tan-^ .r — | log (1 + x^). 92. r.,"sin-ixdx = 5*^^«"^li-^---L^ fl^-:!!^^, then53or54,32,36. 93. fx" cos-i X dx = ^""^^ '^"^"' ^ + -— r -'"' ^' ^^^ , then 53 or 54, 32, 3d. J « + 1 M + 1 J Vl-x'^ 94. r.-"tan-ixdx^y'^'^^""'^---i- f?!^lii?^ then 19 (c). J K + 1 ?J + 1 J 1+ X2 ' ^ ^ Notes, (a) Replace ctn-'a; by - - tan-' j- ; or by tan-i (l/.c) and substitute \/x = u. {b) Replace sec~>a; by cos"' (I/a"), csc"'j by sin-i(l/^) and substitute \/x = u. (c) J(.^ny Polynomial) sin-^xdx, split up and use 92. (Similarly for cos-'sr, etc.) ('') J/W sin~»«rfa;, - use (.5) with u = sin-»a5. (Similarly for cos"iiB and tan-»(w ) (#) Other Inverse Trigonometric Integrands, use 79 or 5. Erf. Integrand Transcendental : Exponential and Logarithmic 95. Ca'dx = -*^^ = ~^- logiQ e = , "— 0.4343. J logea logio« logio« 96. \e''ilx = e^. Notes, (a) J e^^dx = <*'-=-*. (i) Notice a* = eClog*")* = e*', A = log.a. 97. rx"e*^df = -x^e^ - " (x^-^e^Ulx. Notes, (a) J Lr«*^ (c) / (Any Polynomial) e^'dx, split up and use 97. 44 STANDARD INTEGRALS [IV, E 98 C^ dx= —- \ ^ T-l^ dx (repeat to reach 99). J a;"* (m — 1) x"*"! m — 1 J a:"*"^ 99. C— (Zx = ( [I + 1 + -^ + — + --Idu, ti = kx ; see TaftZes, V, H J X J Lu 2 ! ;J ! J , /^/^ r -^ • 7 fc, A sin nx — n cos «x 100. \ e*^ sin nx dx = e*'' — — — . J K^ + rfi ,M C 1,^ 7 kx ^ COS mx + m sin mx 101. I e''^ cos m.r (Zx = e** r-— ! ; . 102. I log X dx = X log X — X. 103. i:(\ogxY'^ = ^^^^&^y^,n^-l. J X n + 1 104. C^^ = r^^' , tt = log X ; see 99 and TaUes, V, H. J log X J u 105. f .X" log X dx = x"+i rl^g^ ^-1 . 106. fe** log X fZx = - e*» log x - J f — cZx, see 99. J k k J X F. Some Important Definite Integrals. 107. i -^^ = — - — , if »W > 1 (otherwise non-existent). J\ xm m—l 108. r ^at ^ _JL . Jo a2 + fc2a;2 2 a6 109. I icwe-* rta? = r (rt + 1) = n ! if n is integral. See V, F, p. 54. (a) In general, r (w + ] ) = m • T (n)- as for »i !, if n > 0. (?.) r (2) = r (1) = 1, r(iA') = Vn. r (« + 1) = n («). 110. rxni-.r)"dx = I^(»^ + ^>^(» + ^>. > r (m + n + 2) 111. i sinnoc ■ smnixdx = \ cosna'cos'inxdx-—0, \{ tn^^n, ^ Jo Jo IT-, if m and ?i are integral. 112. I sin2 nx dx = \ cos'- nx dx = 7r/2 ; n integral, see .5G, 59. Jo Jo IV, G] DEFINITE INTEGRALS 45 113. f"e-*^dr = l/*- 114. P[(sinnj)/j]dx = 7r/2. 115. pe-*' sin nx dx = n/^k'^ + n-), if k > 0. 116. P e-*' cos mx dx = A•/(^- + m-) , if A- > 0. 117. Ce-^^x" dx = ^^ ^^^ "^ ^^ = ^^ , if n is integral. See 109. 118. Pf^-*'^^^.r = V^/(2^•)• 119. J^ e-^^^= cos mx dx = -^-r , if ^• > 0. 120. r-ii?^-= P— ^^- = — • 121. r(loga-)«d:<; = (-l)»n! Jo e*r4.e-Ai Jo cosli ix 2 k Jo 122. p^'log sin a: dx = p 'log cos x dx = - | log 2. 123. r'"''sin-^"+i xdx = P''"cos2»+' xdx = „ !'t'^"'^", («> positive Jo Jo^ 3.5.7...(2n + l)^ ^^^^^^^ ^ 124. ("'"'■sin^" xdx = (""'"cos^" xdx = ' ' ' /"/""" "^^ ^ ("' positive ^' ^^ 2.4.b...2« 2 jj^^^g^^^ G. Approximation Formulas. 125. Cf(x)dx=f{c)(b-a), a' (y)\'dy. (f) Parameter equations : ds =\/ dx^ + dy^. (d) Polar equation : ds =\^J^~+~^d9^. 134. Volumes of Solids : (dF. (a) Frustum (area of cross section A): dV= Adh\ V = iAdh where h is the variable height perpendicular to the cross section A . (6) Solid of revolution about se-axis : dV = ■ny''- dx. (c) Solid of revolution about y-axis : d F = -n^dy. (d) Rectangular coordinate divisions : dV=dxdydz\ (e) Polar coordinate divisions . dV=- p^ sin 9 dp d<^ dO. IV, H] APPLICATIONS 47 135. Area of a Surface : J jsec «|» dx dy, where \{/ is the angle between the element ds of the surface and its pro- jection dxdy. ((/) Surface of Kevolution about te-axis : .4 = J 2 iry ds. {b) Surface of Revolution about y-axis: A=l'2irxds. 136. Length of tioisted arcs : \ ds, (n) Kectangular Coiirdiuatcs : d-t = N/rfa-J + dy^ + dzi. (6) Explicit Equations i/ =/(x), s = dO». 137. itfass 0/ a body : 3I=(dM=(pdV, where p is the density (mass per unit volume). (a) Ifp is constant: Jr=pidr; see 13-1. (b) On any curve : dV=ds,ifp = mass per unit length. (c) On any surface (or plane) : dV= dA, if p = mass per unit area. 138. Average value of a variable quantity q : A. V. of q. : (a) throughout a solid : q =/(.x, y, z) \ A. V. of q. = p2 dV-r- J rf T. (6) on an area A: A. V. of q. = ^qdA ^ ^dA. (c) on an arc « : A.V. of q. = jqds -i- Jds. 139. Center of Mass, (x,y,~z): x=\xdM-^(dM, with similar formulas for y and z. See dM., 137. (a) for a volume : dJf= pd F. (b) for an area : rfJ/"= p dA. (c) for an arc : dM = pds. 139.* Theorems of Pappus or Guldin : (a) Surface generated by an arc of a plane curve revolved about an axis in its plane = length of arc x length of path of center of mass of arc, (h) Volume generated by revolving a closed plane contour about an axis in its plane = area of contour x length of path of its center of mass. 1^0. Moment of Inertia: I=(r'idM. (See 137, 139.) (a) For plane figures. Tz+ Ty= T„, where Tx, I\i. h "re t.iken about the tr-axis, the j/-axis, the origin, respectively. (ft) For space figures, /i + Iy+ Im= /o- (o) /, = /- + («- S)'Jf, where 1^ is taken about a lino || to the xaxis. 48 STANDARD INTEGRALS [IV, H 141 . Badius of Gyration : k'i = I -i- M = TrS dM -4- (dM. [In 140 and 141, r may be the distance from some fixed point, or line, or plane.] 142. Liquid pressure : ^^ = i p^ dA^ where p is the total pressure, dA is the elementary strip parallel to the surface ; h is the depth below the surface ; and p is the weight per unit volume of the liquid. 143. Center of liquid pressure : h= \ Ji^dA -^ i hdA, • 144. Work of a variable force : W — \f cos + ds, where /is the numerical magnitude of the force, ds is the element of the arc of the path, and \p is the angle between / and ds. 145. Attraction exerted by a solid: F= fef ^^^^^ , where k is the attraction between two unit masses at unit distance, m is the attracted particle, dM is an element of the attracting body ; r is the distance from m to dM. Components F^, Fy. F^ of F along 0«, Oy, Oz are : Fx = kmj — , Fy = kmj ■, Fz = k>n^ -^ , . ■where a, 3, y are the direction angles of a line joining )n to dJ/. 146. Work in an expanding gas : W = \p dv. 147. Distance s, speed v, tangential acceleration jr: JT= ijvdt^ i I CsdtXdt. [Similar forms for angular speed and acceleration.] 148. Errors of observation : y d.r, where y is the J, „ ^. „ „. „-j,"" — - -• ~" (b) The usual formula y = (/t/Vi^) e"*'^" gives: P = (/i/Vn) j e '''-''■ dx, where h is the so-called measure of precision. (c) Probability of an error between x = — a and x =+ a: P (a) =1 y dx. (d) Probable error = (0.411) /h = value of n for which P(.a) = 1/2. (e) Jfean error = C xydx-r- C ydx = \/(liV7) V. NUMERICAL TABLES A. TRIGONOMETRIC FUNCTIONS [CharacU'ristics of Logaiithuis omitted — deteriuine by the usual rule from the vahu' i:a..ia,.s De Sine Tangent I'OTANGEXT NK 1 -rees Value loBxo \alue lo^'io Value log.o Value 'uV'io ! 0000 0° .0000 -X .0000 -00 00 00 1.0000 0000 !X)° 1.5708 .0175 1° .0175 2419 .0175 2419 57.2SX» 7581 .994 19.081 2806 .998(5 9994 87° 1.5184 .0{;98 4"^ .0(j98 8436 .(Xi'.H) 8446 14.301 1554 .9976 998!' 86° 1.5010 .0873 5° .0872 9403 .0875 9420 11.4:?0 0580 .i>9()2 9983 85° I.IS.T. .1047 li" .1045 0192 .1051 0216 9.5144 9784 .9i>45 imc, 84° i.4(i(;i .1222 7° .1219 0859 .1228 0891 8.1443 9109 .9925 9i)(58 83° 1.44.S(i .1396 8° .1392 1436 .1405 1478 7.1154 8522 .9903 {)958 82° 1.4312 .1571 9^ .15(>4 1943 .1584 1997 6.3138 8003 .9877 9946 81° 1.4137 .1745 10° .1736 2397 .1763 24<)3 5.(5713 7537 .9848 9934 80° 1.3963 .1920 11° .1908 2806 .1944 2887 5.1446 7113 .9816 9919 79° 1.378S .2094 12° .2079 3179 .2126 3275 4.7046 6725 .9781 95t04 78° 1.3614 .2269 13° .2250 3521 .2309 36;U 4.3315 6;«56 .9744 9887 77° 1..34:'.!i .2443 14° .2419 3837 2493 3968 4.0108 6032 .9703 9869 76° 1.. 32(55 .2618 15° .2588 4130 .2679 4281 3.7.321 5719 .9659 9849 75° 1..3090 .2793 16° .2756 4403 .28(57 4575 3.4874 5425 .9(513 9828 74° 1.2915 .2t)67 17° .2924 4659 .3057 4853 3.2709 5147 .9563 9806 73° 1.2741 .3142 18° .3090 4900 .3249 5118 3.0777 4882 .9511 9782 72° 1.2566 .3316 19° .3256 5126 .3443 5370 2.9042 4630 .9455 9757 71° 1.2392 .3491 20° .3420 5311 .3640 5611 2.7475 4389 .9397 9730 70= 1.2217 M65 21° .3584 5543 .3839 5842 2.6051 4158 .933<5 9702 69° 1.2043 .3840 22° .3746 5736 .4040 60()4 2.4751 3936 .^)272 9672 68° 1.1868 .4014 23° .3907 5919 .4245 6279 2..35.59 3721 .9205 9(540 (57° 1.1(594 .4189 24° .4067 6093 .4452 6486 2.2460 3514 .9135 9607 6<)° 1.1519 .4363 25° .4226 6259 .46(i3 6687 2.1445 3313 .9063 9573 (55° 1 .1.345 .45:38 26° .4384 6418 .4877 6882 2.0503 3118 .8988 9537 64° 1.1170 .4712 27° .4540 6570 .5095 7072 1.9626 2928 .8910 9499 63° 1.099(5 .4887 28° .4695 6716 ..5317 7257 1.8807 2743 .8829 9459 62° 1.0821 .5061 29° .4848 685(i .5543 74;?8 1.8040 25(52 .874(5 9418 61° 1.0(547 .5236 30° ..5000 6«)0 .5774 7614 1.7321 2.386 .8660 9.375 (50° 1 .0472 ..5411 31° .5150 7118 .(5009 7788 1.6643 2212 .8572 9331 59° 1.0297 .5585 32° .5299 7242 .()249 7958 1.(5003 2042 .8480 9284 58° 1.0123 .5760 33° .5446 7361 Ami 8125 1.5399 1875 .8387 92m 57° .{•948 .5934 34° .5592 7476 .6745 8290 1.4826 1710 .8290 9186 5(5° .9774 .(;109 35° .5736 7586 .7002 8452 1.4281 1548 .8192 9134 55° .9.-99 .()283 3<)° .5878 7692 .7265 8613 1.37(54 1387 .80iX) 9080 54° .9425 .6458 37° .6018 7795 .7536 8771 1.3270 1229 .7986 9023 53° .9250 .6632 38° .6157 7893 .7813 8928 1.2799 1072 .7880 8(H5.5 52° .907(5 .6807 39° .6293 7989 .8098 9084 1.2319 0916 .7771 8i)05 51° .8901 .r,981 40° .6428 8081 .8391 92.38 1.1918 0762 .7660 8843 50° .8727 .7156 41° .6561 8169 .8693 9392 1.1501 0608 .7547 8778 49° .8552 .73:«) 42^ .(i691 8255 .9004 9544 1.1106 04.5(5 .7431 8711 48° .8378 .7505 43° .6820 8338 .9325 96tt7 1.0724 0303 .7314 8641 47° .8203 .7679 44° .6917 &418 .9657 9848 1.0355 0152 .7193 8.569 4(5° .8029 .7854 45° .7071 8495 1.0000 0000 1.0000 0000 .7071 8495 45° .7854 Value lo4 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 60 7782 7789 7721 8(527 6 73 8<)33 8639 8645 8(>51 8657 8663 8669 8675 8681 8686 6 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 6 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 6 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 6 77 88(>5 8871 8876 8882 8887 8893 884 8910 8915 6 78 8921 8927 8932 8938 8943 8»49 8954 8i>60 89(>5 8971 6 79 8976 8982 8987 8993 8'.)98 9004 9009 9015 9020 iX)25 5 80 W31 9036 9042 9047 9053 9058 9063 9069 9074 9079 5 81 ;X)85 9090 9096 9101 9io<; 9112 9117 9122 9128 9133 5 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 5 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 i»238 5 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 5 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 5 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 93i)0 5 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 5 88 9445 9450 9455 9460 9465 94489 5 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 5 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 958(5 5 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 J1633 5 92 9638 9643 9<)47 9(>52 9657 9661 9666 9671 9(575 9680 5 93 9685 9689 9(594 9699 9703 9708 9713 9717 9722 9727 5 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 5 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 5 96 9823 9827 9832 98:« 9841 9845 9850 9854 9a59 9863 5 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 <)908 4 98 9912 9917 9921 9926 t)9:«) i)934 9939 9943 iW8 9<)52 4 99 9956 9961 9965 9969 9974 9978 9983 •)987 9991 it<)96 4 52 NUMERICAL TABLES [V,C C. EXPONENTIAL AND HYPERBOLIC FUNCTIONS e^ e -x siiih * cosl X a, logea- Value logio Value logio Value logio Value logio 0.0 — 00 1.000 0.000 1.000 0.000 0.000 — 00 1.000 0.1 -2.303 1.105 0.043 0.W5 9.957 0.100 9.001 1.005 0.002 0.2 —1.610 1.221 0.087 0.819 9.913 0.201 9.304 1.020 0.009 0.3 -1.204 1.350 0.130 0.741 9.870 0.305 9.484 1.045 0.019 0.4 -0.916 1.492 0.174 0.670 9.826 0.411 9.614 1.081 0.034 0.5 -0.693 1.649 0.217 0.607 9.783 0.521 9.717 1.128 0.052 0.6 -0.511 1.822 0.261 0.549 9.739 0.637 9.804 1.185 0.074 0.7 -0.357 2.014 304 0.497 9.696 0.759 9.880 1.255 0.099 0.8 -0.223 2.226 0.;547 0.449 9.653 0.888 9.948 1.337 0.126 0.9 -0.105 2.460 0.391 0.407 9.609 1.027 0.011 1.433 0.156 1.0 0.000 2.718 0.434 0.368 9.566 1.175 0.070 1.543 0.188 1.1 0.095 3.004 0.478 0.333 9.522 1.336 0.126 1.669 222 1.2 0.182 3.320 0.521 0.301 9.479 1.509 0.179 1.811 0.258 1.3 0.262 3.669 0.565 0.273 9.435 1.698 0.230 1.971 0.295 1.4 0.336 4.055 0.608 0.247 9.392 1.904 0.280 2.151 0.333 1.5 0.405 4.482 0.651 0.223 9.349 2.129 0.328 2.352 0.372 1.6 0.470 4.953 0.695 0.202 9.305 2.376 0.376 2.577 0.411 1.7 0.531 5.474 0.738 0.183 9.262 2.646 0.423 2.828 0.452 1.8 0.588 6.050 0.782 0.165 9.218 2.942 0.469 3.107 0.492 1.9 0.642 6.686 0.825 0.150 9.175 3.268 0.514 3.418 0.534 2.0 0.693 7.389 0.869 0.135 9.131 3.627 0.560 3.762 0.575 2.1 0.742 8.166 0.912 0.122 9.088 4.022 0.604 4.144 0.617 2.2 0.788 9.025 0.955 0.111 9.045 4.457 0.649 4.568 0.660 2.3 0,833 9.974 0.999 0.100 9.001 4.937 0.690 5.037 0.702 2.4 0.875 11.02 1.023 0.091 8.958 5.466 0.738 5.557 0.745 2.5 0.916 12.18 1.086 0.082 8.914 6.050 0.782 6.132 0.788 2.6 0.956 13.46 .1.129 0.074 8.871 6.695 0.826 6.769 0.831 2.7 0.993 14.88 1.173 0.067 8.827 7.406 0.870 7.473 0.874 2.8 1.030 16.44 1.216 0.061 8.784 8.192 0.913 8.253 0.917 2.9 1.065 18.17 1.259 0.055 8.741 9.060 0.957 9,115 0.960 3.0 1.099 20.09 1.303 0.050 8.697 10.018 1.001 10.068 1.003 3.5 1.253 33.12 1.520 0.030 8.480 16.543 1.219 16.573 1.219 4.0 1.386 54.60 1.737 0.018 8.263 27.290 1.436 27.308 1.436 4.5 1.504 90.02 1.954 0.011 8.046 45.003 1.653 45.014 1.653 5.0 1.609 148.4 2.171 0.007 7.829 74.203 1.870 74.210 1.870 6.0 1.792 403.4 2.606 0.002 7.394 201.7 2.305 201.7 2.305 7.0 1.946 1096.6 3.040 0.001 6.960 548.3 2.739 548.3 2.739 8.0 2.079 2981.0 3.474 0.000 6.526 1490.5 3.173 1490.5 3.173 9.0 2.197 8103.1 3.909 0.000 6.091 4051.5 3.608 4051.5 3.608 10.0 2.303 22026. 4.343 0.000 5.657 11013. 4.041 11013. 4.041 log, X = (logio x)^M ; M= .4342944819. logio 6^+" = logm e^ + logjo ev. Sinhx and coshx approach e^/2 as x increases (see Fig. E, p. 20). The formula logio (e V2) = M • x — logio 2 represents login sinh x and logio cosh x to three decimal places when x >3.5 ; four places when x > 5 ; to five places when X > 6 ; to eight places when x > 10. V, E] ELLIPTIC INTEGRALS 53 D. VALUES OF doc -'o Vl-A:2sin-'e -'O v^ sine siii<}>, [Elliptic Integral of the First Kind.] 0.0 .i. = o- 1.049 1.312 1.575 0.2 0.087 0.175 262 0.325 0.78!) 1.054 1.321 1.588 0.3 0.087 0.175 0.262 0.526 0.792 1.062 1.336 1.610 0.4 0.087 0.175 0.262 0.527 0.798 1.074 1.358 1.643 0.5 0.087 0.175 0.26.3 0.529 0.804 1.090 1.385 1.686 0.6 0.087 0.175 0.263 0.5.32 0.814 1.112 1.42() 1.752 0.7 0.087 0.175 0.263 0.5;«) 0.82(5 1.142 1.488 1.8.54 0.8 0.087 0.175 0.264 539 0.839 1.178 1.566 1.9! 13 O.i) 0.087 0.175 0.264 0.544 0.8,58 1.233 1.703 2.275 10 0.087 0.175 0.265 0.549 0.881 1.317 2.028 00 i;(/c, )= pvi-fessiiiaerferr r VALUES OF dx. VI - ic2 [Elliptic Integral of the Second Kind.] [ x = sin e ' [t« = sin = 10'' .^=15'' *=30° *=45° «=60° ^0.3 1>=0.4 i>=0.6 i»=0.6 jp=0.7 p=0.8 P= r(p+i)= 1.000 0.951 0.918 0.897 0.887 0.886=v^/2 0.894 0.909 0.931 0.9 r(A; + 1) = A;r(^), if A;>0; lience r(A; + l) can be calculated at intervals of 0.1 Minimum value of r(/> + 1) is .885(30 alp = .46163. G. VALUES OF THE PROBABILITY INTEGRAL: ^^d. X .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 0. 1. 2. .0000 .8127 .9953 .1125 .8802 .9970 .2227 .9103 .9981 .3286 .9340 .9989 .4284 .9523 .9993 .5205 .9661 .9996 .6039 .9763 .9998 .6778 .9838 .9999 .7421 .9891 .9999 .796 .992 l.000( H. VALUES OF THE INTEGRAL f [Note break at x = 0.] ^ &^dx 00 X n = l 11=2 n=^3 »=4 n=5 /t=6 »i = 7 n = 8 n = x=—n/10 - .2194 — 1.823 - .0489 -1.223 - .0130 - .9057 - .0038 - .7024 - .0012 — .5598 - .0004 - .4544 - .0001 — .3738 -.0000 -.0( - .3106 - .2* .t=+n/10 x=+n -1.623 1.895 - .8218 4.954 -.3027 9.934 + .1048 19.63 .4542 40.18 .7699 85.99 1.065 191.5 1.347 l.< 440.4 10 — - =— 00. Values on each side of x = can be used safely. (* ' -^ and r — dx reduce to the integral here tabulated ; see IV, 99, 104, p.' Jo logx J x" V,I] RECIPROCALS SQUARES CUBES 55 RECIPROCALS OF NUMBERS FROM 1 TO 9.9 1 .0 .1 .2 3 .4 .5 .6 .7 .8 .9 l.O(X) 0.f)09 0.833 0.7G!) 0.714 0.fiG7 0.625 0.588 0.556 0.526 2 0.500 0.47tJ 0.455 0.4.35 0.417 0.400 0.385 0.370 0.357 0.345 3 0.833 0.323 0.313 0.303 0.2M 0.28(! 0.278 0.270 0.263 0256 0.2r>0 0.244 0.238 0.2.33 0.227 0.222 0.217 0.213 0.208 0.204 0-,'00 0.196 0.192 0.189 0.185 0.183 0.179 0.175 0.172 0.169 0.1G7 O.KA O.KJl 0.151t 0.15t) OArA 0.152 0.14it 0.147 0.145 0.143 0.141 0.139 0.137 o.i;» 0.133 0.132 O.VM 0.128 0.127 0.125 0.123 0.122 0.120 0.119 0.118 0.116 0.115 0.114 0.112 9 0.111 0.110 0.109 0.108 0.106 0.105 0.104 0.103 0.102 0.101 I2. SQUARES OF NUMBERS FROM 10 TO 99 1 2 3 4 5 6 7 8 9 1 100 121 144 169 196 225 256 289 324 3(il 2 400 441 484 529 576 625 676 729 784 841 3 iKX) i)61 1024 1089 1156 1225 129() 1369 1444 1521 4 1600 1681 1764 1849 1936 2025 2116 2209 2;»4 2401 5 3500 3601 2704 3809 2916 3025 3136 3249 3364 3481 6 .3(i(X) 3721 3844 39(i9 4096 4225 4^56 4489 4624 4761 7 4ft00 5041 5184 5329 5476 5625 5776 5929 fW84 6241 8 WOO (i561 6724 6889 7056 7225 7396 7569 7744 7921 9 8100 8281 84M 8649 88;i6 9025 9216 9409 9604 9801 I3. CUBES OF NUMBERS FROM 1 TO .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 1.00 1.33 1.73 2.20 2.74 3.37 4.10 4.91 5.83 6.86 2 8.00 9.26 10.65 12.17 13.82 15.62 17.58 19.68 21.95 24.39 3 27.00 29.79 32.77 35.94 39.:J0 42.87 46.66 50.65 54.87 56.32 4 64.0 68.9 74.1 79.5 85.2 91.1 i)7.3 103.8 110.6 117.6 5 1350 132.7 140.6 148.9 157.5 166.4 176.6 1852 1951 205.4 6 216.0 227.0 238.3 250.0 262.1 274.6 287.5 .•100.8 314.4 328.5 7 ;i43.0 357.9 373.2 389.0 405.2 421.9 4.39.0 456.5 474.6 493.0 8 512.0 5.31.4 551.4 571.8 .592.7 614.1 6;i«).l 658.5 (W1.5 705.0 9 729.0 753.6 778.7 804.4 s:«).(> 8.-)7.4 884.7 912.7 !m.2 970.3 56 NUMERICAL TABLES [V,J Ji. SQUARE ROOTS OF NUMBERS FROM 1 TO 9.9 .0 .1 .2 .3 .4 .6 .6 .7 .8 .9 0.000 0.316 0.447 0.548 0.632 0.707 0.775 0.837 0.894 0.949 1 1.000 1.049 1.095 1.140 1.183 1.225 1.265 1.304 1.342 1.378 2 1.414 1.449 1.483 1.517 1.549 1.581 1.612 1.643 1.673 1.703 3 1.732 1.761 1.789 1.817 1.844 1.871 1.897 1.924 1.949 1.975 4 2.000 2.025 2.049 2.074 2.098 2.121 2.145 • 2.168 2.191 2.214 5 2.336 2.?58 3.380 3.303 3.334 3.345 3.366 3.387 3.408 2429 6 2.449 2.470 2.490 2.510 2.530 2.550 2.569 2.588 2.608 2.627 7 2.046 2.665 2.683 2.702 2.720 2.739 2.757 2.775 2.793 2.811 8 2.828 2.846 2.864 2.881 2.898 2.915 2.933 2.950 2.966 2.983 9 3.000 3.017 3.033 3.050 3.066 3.082 3.098 3.114 3.130 3.146 Jo. SQUARE ROOTS OF NUMBERS FROM 10 TO 1 2 3 4 6 6 7 8 9 1 3.162 3.317 3.464 3.606 3.742 3.873 4.000 4.123 4.243 4.359 2 4.472 4.583 4.690 4.796 4.899 5.000 5.0il9 5.196 5.292 5.385 3 5.477 5.!i68 5.657 5.745 5.831 5.916 6.000 6.083 6.164 6.245 4 6.325 (>.403 6.481 6.557 6.633 6.708 6.782 6.856 6.928 7.000 6 7.071 7.141 7.311 7.380 7.348 7.416 7.483 7.550 7.616 7.681 a 7.746 7.810 7.874 7.937 8.000 8.062 8.124 8.185 8.246 8.307 7 8.367 8.426 8.485 8.544 8.602 8.()60 8.718 8.775 8.832 8.888 8 8.944 9.000 9.055 9.110 9.165 9.220 9.274 9.327 9.381 9.434 9 9.487 9.539 9.592 9.644 9.695 9.747 9.798 9.849 9.899 9.950 K. RADIANS TO DEGREES Radians Tenths HUNDRKDTIIS Thousandths Ten-thousandths 1 57°17'44".8 5°43'46".5 0°34'22".6 0° 3'26".3 0° 0'20".6 2 114°35'29".6 11°27'33".0 1° 8'45".3 0° 6'52".5 0° 0'41".3 3 171°53'14".4 17°11'19".4 1°43'07".9 0°10'18".8 0° 1'01".9 4 229°10'59".2 22°55'05".9 2°17'30".6 0°13'45".l 0° 1'22".5 5 28(i''28'44".0 28°38'52".4 2°51'53".2 0°17'11".3 0° l'43".l 6 343°46'28".8 34°22'38".9 3°26'15".9 0°20'37".6 0° 2'03".8 7 401° 4' 13" .6 40° 6'25".4 4° 0'.38".5 0°24'03".9 0° 2'24".4 8 458°21'.58".4 45°.50'11".8 4°35'01".2 0°27'30".l 0° 2'45".0 9 515°39'43".3 51°33'58".3 5° 9'23".8 0°30'56".4 0° 3'05".6 V, M] CONSTANTS IMPORTANT CONSTANTS AND THEIR COMMON LOGARITHMS .V=N,-M1!KK Vaiie of .V Lo.:,„ .V IT 3.14109265 0.49714987 1-f- JT 0.31830989 9.5028.501.3 ir2 9.86i)60440 0.9942997.-. VF 1.77345385 0.24S.57494 e = Napierian Base 2.71828183 0.434J944S M = logiQ e 0.43429448 9.(;;i77.S4.'.l 1-^1/= log, 10 2.30258509 0.3(i221.-)(i'.) LSO -r- TT = degrees in 1 radian 57.2957795 1.75812262 IT -^ 180 = radians in 1° 0.01745329 8.24187738 7r H- 10800 = radians in 1' 0.0002fl08882 6.4637261 TT -- (U8000 = radians in 1" 0.0000O4S4813(!811095 4.68557487 sin 1" 0.0000048481.Tj81107r) 4.68557487 tan 1" 0.00000484813tJ811152 4.(58557487 centimeters in 1 ft. 30.480 1.48401.58 feet in 1 cm. 0.032808 8.5159.'v42 inches in 1 m. 39.37 1.5951654 pounds in 1 kg. 2.204G2 0.3433;i40 kilograms in 1 lb. 0.453593 9.656(>660 g 32.16 ft./sec./sec. 1.5073 = 981 cm. /sec. /sec. 2.9916690 weight of 1 cu. ft. of water 62.425 lb. (max. density) 1.7953+ weight of 1 cu. ft. of air 0.0807 1b. (at32°F.) 8.907 cu. in. in 1 (U. S.) gallon 231 2 3636120 ft. lb. per sec. in 1 H. P. 550. 2.7403627 kg. m. per sec. in 1 H. P. 76.0404 1.8810445 watts in 1 H. P. 74.-..957 2.8727135 DEGREES TO RADIANS r .01745 10° .17453 100° 1.74533 6' .00175 6" ..H.,0. .o:M91 20" ..34907 110° 1 .91986 7' .00204 7" .00003 .3° .052:36 .-30° .52360 120° 2.09440 8' .00233 8" .00004 4^ .06981 40° .69813 V.W 2.26893 9' .00262 9" .ax)04 5° .08727 .50° .87266 140° 2.44.'H(J 10' .00291 10" .00005 6'^ .10472 («" 1.04720 1,50° 2.61799 20' .00582 20" .(MM)1() 7'^ .12217 7()0 1.22173 1(»° 2.79253 m' .00873 .30" .(KM)15 S'^ .13n, 271, 54. >n of, 43, 57, 90, 137, Gamma'T Gases, ex' 142, 292r^^^ Geometry, spkce, 315 Graphs, 2. Gudermannian, 166, 279, 13. Guldin and Pappus, Theorem, 47. Gyration, radius of. See Radius. Harmonic functions, 21. See also Trigonometric. Helicoid, 332. Helix, 332. Hooke's Law, 157. Hyperbola, 10, 15. Hyperbolic functions, 139, 140, 13, 20, 52; inverse, see Inverse. Hyperbolic logarithm. See Napier- ian. Hyperboloid, 318, 31. Hypocycloid, 24. Hypotrochoid, 23. Implicit functions, 45, 294. Improper integrals, 201. .See also Integrals. Increments, 6, 252 ; method of, 233 ; second, 234, see also Finite differ- ences. Indeterminate forms, 263, 268. Inertia, moment of. See Moment. Infinite series. See Series. Infinitesimal, 17 ; principal part, 266. Infinitesimals, higher order, 265. Infinity, 19. Inflexion, point of, 75. ntegral, as limit of sum, 116 ; funda- mental theorem, 99 ; indefinite, 96 ; notation for, 96. Integral curves, 241, 350, 380. Integrals, definite, 100, 44; double, 210; elliptic, 195, 247, 280, 9, 53; Fresnel's, 279; improper, 201, 314; multiple, 217; table of, 33; triple, 217. Integral surfaces, of a differential equation, 380. Integrand, 96. Integraph, 244. Integrating factor, 361. Integration, 96; approximate, 239, see also Approximation ; by parts, 181, 33; by substitution, 176, 33, 41 ; formulas for, 97, 174, 33; of a sum, 97 ; of binomial differen- tials, 195, 39; of irrational func- tions, 164, 37; of linear radicals, 188, 37 ; of polynomials, 97, 176 ; of quadratic radicals, 189, 37; of rational functions, 184, 34; of trigonometric functions, 178, 188, 194, 200, 39; reduction formu- las, 196, 39, 40; repeated, 206; successive, 206. Interpolation, Lagrange's formula, 15. See also Lagrange. Inverse functions, 46, 3. Inverse hyperbolic functions, 166, 247. 14, 52. Inverse problems, 347. ,See also Rates, reversed. Inverse trigonometric functions, 163. Involute, 301. Irrational functions, 29 ; differen- tiation of, 38; integration of, 164 189. Isolated point, 310. Isothermal expansion, 137, 292. Lagrange interpolation formula, 228, 15, 45. Laplace's equation, 284, 344. Law of the Mean, 251, 45; double, 267; extended, 257, see also Taylor's theorem. Least squares, 69, 229, 262, 323, 342, 6. Length, 18, 106, 46; of a space curve, 338. Lemniscate, 27. Limit { l+M". 271. (-1)" Limits, 16 ; arc to chord, 18 ; proper- ties of, 17 ; sin a to a, 19. Liquid pressure, 48. 62 INDEX Loci, 318. See also Curve tracing. Logarithmic derivative, 146, 167. See also Rates, relative. Logarithmic plotting, 229, 18. Logarithms, computation of, 7 ; graph of, 19; hyperbolic, see Logarithms, Napierian ; modulus M, 134; Napierian, 135, 53; natural, see Napierian; rules of operation, 130, 3; table of, 50. Maclaurin's Theorem, 258. See also Taylor's Theorem. Mass, 47. Maximum, 8. See also Extremes. Mean square ordinate, 248. Mensuration, 9. Minimum, 8. »See also Extremes. Modulus, of logarithms, 134. Moment, first, 224. Moment of inertia, 213, 47; polar coordinates, 214. Momentum, 82. Motion, 107, 363, 48. See also Speed, Acceleration, etc. Napierian base, e, 135. See also Logarithms. Natural logarithms. See Logarithms. Normal, 11, 59; length of, 60; to a surface, 327, 330. Notation, 1. Numbers, e, M. See Logarithms. Organic growth, law of, 143. Orthogonal trajectories, 361. Pappus' Theorem, 47. Parabola, 10. See also Curves, para- bolic. Paraboloid, 11, 32. Parameter forms, 47, 59, 107, 332. Partial derivative, 281, see also Derivative ; order of, 283. Partial derivatives, transformation, 339, 16. Partial differential. See Differential. Partial fractions, 184. Pendulum, 254, 280. Percentage rate of increase, 144. ^ P^g^^ $1.25 net. Designed particularly with the view to give the student such a working knowledge of algebra as will ordinarily be required in practical work. Graphical methods are emphasized more than is general in books of this grade. This will be found an excellent text for all students desiring technical knowledge. The book is furnished with or without answers as desired. The edition without answers will be sent if no choice is indicated. 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