# Differential and Integral Calculus: An Introductory Course for Colleges and Engineering Schools, Volume 25

Longmans, Green, 1912 - Calculus - 481 pages

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### Contents

 Limits of Functions 20 The Difference between the Limit of x when x a and the Value of x when x a 22 Further Examples of Finding Limits of Functions 23 Exercises 24 Meaning of the Symbols g 00 and 0 25 Infinite Limits of Trigonometric Functions 26 Example of a Function that has no Limit 27 A Geometrical Limit 28 CHAPTER III 32 The Derivative 33 Article Paoe 23 The Equations of Tangent and Normal 36 Exercises 37 CHAPTER IV 38 Exercises 43 Exercises 44 Proof of VI when n is a Negative Integer 45 Exercises 46 Exercises 48 Exercises 49 Exercises 50 Differentiation of Trigonometric Functions 51 Exercises 53 Exercises 55 CHAPTER V 56 Successive Differentiation of Implicit Functions 57 Exercises 58 CHAPTER VI 60 Discontinuities of fx 63 Roots of Polynomials 65 Zeros of Functions 67 Exercises 68 Convexity and Concavity 70 The Flex 72 Examples of Finding Flexes and Convex and Concave Arcs 73 Exercises 75 CHAPTER VII 76 Maxima and Minima Determined by Means of the First Derivative 77 Exercises 78 Employment of the Second Derivative in Determining Maxima and Minima 79 Exercises 80 Curve Tracing 81 Exercises 83 Exercises 84 CHAPTER VIII 88 Differentiation of Logarithmic Functions 90 Differentiation of Exponential Functions 92 Exercises 94 Differentials 97 Differentials of Higher Orders 101 CHAPTER IX 103 Instantaneous Velocity 104 Exercises 107 Angular Velocity 108 Velocity of any Change of State 109 Expansion of a Metal Rod 111 Acceleration Ill S3 Exercises 112 A General Differentiation Formula 113 CHAPTER X 115 Exercises 117 CHAPTER XI 120 Geometric Interpretation of the Parameter 123 Exercises 124 Derivation of Parametric Equations 126 CHAPTER XII 128 The Epi and HypoCycloids 129 Exercises 131 The Involute of the Circle 132 CHAPTER XIII 134 Curves in Polar Coordinates 135 Exercises 137 CHAPTER XIV 140 Cartesian Coordinates 141 Polar Coordinates 143 The Method of Infinitesimals 144 Exercises 148 CHAPTER XV 149 Resolution of Velocity along a Curve 150 Exercises 152 CHAPTER XVI 153 The Circle of Curvature 155 Evolutes and Involutes 157 Properties of the Evolute 158 Exercises 161 CHAPTER XVII 163 The Law of the Mean 164 Cauchys Theorem 165 Indeterminate Forms 166 Exercises 171 Indeterminate Forms Continued 172 Exercises 173 BOOK II 175 Two General Theorems of Integration 176 Fundamental Formulae of Integration 178 Proof and Application of Formulae I and II 180 Devices for Bringing the Integrand to a Form to which the Funda mental Formulae Apply 181 Exercises 183 Proof and Application of Formula Ill IV and V 184 Additional Devices 185 Exercises 186 Article Page 129 Proof and Application of Formulae VI and VII 187 Exercises 188 CHAPTER XIX 190 Exercises 192 Some Special Methods of Integration 193 Exercises 194 Integration of cosm x sin x 195 Exercises 196 Exercises 197 Integration by Parts 198 Exercises 200 CHAPTER XX 201 Exercises 204
 Exercises 223 THE DEFINITE INTEGRAL AREAS AND LENGTHS OF CURVES Article Page 157 The Definite Integral 224 Exercises 225 The Area under a Curve Expressed as a Definite Integral 226 Exercises 230 Areas of Curves Given by Parametric Equations 231 Exercises 233 Polar Coordinates 234 Polar Coordinates 235 Lengths of Curves 236 Exercises 238 CHAPTER XXIII 240 Area of a Sector 244 Proof that Every Function has an Integral 246 CHAPTER XXIV 253 Areas of Surfaces of Revolution 255 Exercises 260 Polar Coordinates 262 Exercises 264 BOOK III 265 Relations between Two Points 266 Article Paob 182 Some Simple Loci 267 The Plane in Space 268 Exercises 270 Quadric Surfaces 271 Exercises 276 CHAPTER XXVI 278 Direction Cosines 279 Parametric Equations of the Line 281 The Normal Equation of the Plane 282 Exercises 283 CHAPTER XXVII 286 Exercises 289 Exercises 291 BOOK IV 293 Partial Derivatives 294 Exercises 295 The Tangent Plane and the Normal Line 296 Exercises 299 Exercises 301 Exercises 306 Partial Derivatives of x y z where xyz are them selves Functions of Several Independent Arguments 307 Exercises 310 Implicit Functions 311 Article Page 208 Exercises 313 The Tangent Plane and the Normal Line 314 Differentials Partial and Total 315 Remarks 319 CHAPTER XXIX 321 A Plane Area Expressed as a Double Integral 327 The Area of a Surface Expressed as a Double Integral 328 Exercises 331 The Triple Integral 332 The Triple Integral in Polar Coordinates 335 Volumes by Triple Integration 337 CHAPTER XXX 338 Center of Mass of a Plane Area 341 Center of Mass of an Arc of a Plane Curve 342 Center of Mass of a Surface of Revolution 343 Center of Mass of Any Curved Surface 344 Exercises 346 BOOK V 349 Exercises 355 Examples of Development by Taylors Theorem 358 Exercises 360 Exercises 361 Evaluation of the Indeterminate Form by Taylors Theorem 362 Criteria for Maxima and Minima Determined by Taylors Theorem 363 Infinite Forms 366 Examples of the Application of the Infinite Forms of the Theorems of Taylor and Maclaurin 369 Exercises 374 The Ratio Test for the Convergence of Power Series 377 Exercises 378 Exercises 379 Computation of Logarithms 380 Exercises 382 Exercises 386 Exercises 388 Exercises 392 The Inverse Hyperbolic Functions 393 Geometric Properties of the Hyperbolic Functions 395 CHAPTER XXXII 398 Integration of a Rational Fraction when the Denominator is Re solved into its Linear Factors 399 Exercises 402 Exercises 403 Exercises 405 Case 4 406 Article Page 264 Exercises 408 Miscellaneous Exercises 409 CHAPTER XXXIII 410 Envelopes 411 An Important Property of the Envelope 413 Exercises 415 Caustics 416 Exercises 418 BOOK VI 421 Solution Particular Integrals The Complete Primitive 422 Derivation of the Differential Equation from the Complete Primi tive 424 Exercises 425 Solving a Differential Equation 426 The Differential Equation of the First Order and Degree 427 Exercises 428 Exercises 429 Exercises 431 Exercises 433 Exercises 434 Equations Reducible to Linear Form 435 Exercises 436 Answers 451 Index 477 Copyright

### Popular passages

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Page 422 - ... of the third order. The degree of a differential equation is the exponent of the highest...
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Page 108 - Figure 3-7 represents the position of the ladder, with the distance from the base of the wall to the foot of the ladder labeled...
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