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

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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
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Page 422 - The order of a differential equation is the order of the highest derivative which occurs in it.
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Page 86 - NOTK 1 — The room should be darkened, and light from all other sources, including other experiments, thoroughly cut off by screens. Discussion. — Since the intensity of illumination (Exp. 124) varies inversely as the square of the distance from the source of light, and directly as the intensity of the source, the amount of light that falls on each side of the oiled spot should be equal to the candle power of the source divided by the square of the distance of such source from the screen. Equating...
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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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