Elements of the Differential and Integral Calculus (Google eBook)

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Ginn, 1904 - Calculus - 463 pages
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Contents

Continuous variation
12
SECTION PAGE 22 Onevalued and manyvalued functions
14
Explicit functions
15
Integral rational functions
16
Explicit algebraic functions
17
Functions
18
Independent and dependent varialili
19
Notation of functions
20
Values of the independent variable for which n function is defined
21
Limiting value of a function
22
Continuity and discontinuity of functions illustrated by their graphs
23
Fundamental theorems on limits
26
Special limiting values
29
The number e
31
Introduction
37
Increments 40 Comparison of increments 41 Derivative of a function of one variable
39
Symbols for derivatives
40
Differentiable functions
41
General rule for differentiation
42
Applications of the derivative to Geometry
43
41
44
RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORMS 46 Importance of general rule
46
Differentiation of a constant
48
Differentiation of a variable with respect to itself
49
Differentiation of the product of a constant and a variable
50
Differentiation of a variable with a constant exponent
51
Differentiation of a quotient
52
Differentiation of a function of a function
57
Differentiation of inverse functions
58
Differentiation of the simple exponential function ill
60
Differentiation of the general exponential function 01
61
CO Logarithmic differentiation 02
62
Differentiation of sin v
67
Differentiation of cos v
68
Go Differentiation of sec v
69
Differentiation of esc v
70
Differentiation of arc sin v
74
Differentiation of arc cos o
75
Differentiation of arc tan v
76
Differentiation of arc cotr
77
Differentiation of arc esc v
78
Implicit functions
83
Differentiation of implicit functions
84
Chapter VII
86
Equations of tangent and normal lengths of subtangent and sub normal Rectangular coordinates
89
Parametric equations of a curve
92
Angle between the radius vector drawn to a point on a the tangent to the curve at that point
97
Lengths of polar subtangent and polar subnormal
98
Solutions of equations having multiple roots
100
Applications of the derivative in mechanics Velocity
102
Component velocities
104
Acceleration
105
Component accelerations
106
Chapter VIII
109
The nth derivative 90 Leibnitzs formula for the nth derivative of a product 91 Successive differentiation of implicit functions PAGE 109 109
110
Chapter IX
116
Tests for determining when a function is increasing and when decreasing
117
Maximum and minimum values of a function
118
First method for examining a function for maximum and mini mum values
121
Second method for examining a function for maximum and mini mum values
123
General directions for solving problems involving maxima and minima
129
Chapter X
136
Chapter XI
140
and dy considered as infinitesimals 102 Derivative of the arc in rectangular coordinates
141
Derivative of the arc in polar coordinates
143
Formulas for finding the differentials of functions
144
Successive differentials 140
146
Chapter XII
148
Comparison test for convergence
151
Chapter XIII
152
Change of the dependent variable
153
Change of the independent variable
154
Power series
155
Simultaneous change of both independent and dependent variables
156
Chapter XIV
159
Curvature at a point
160
Formulas for curvature
161
Radius of curvature
162
Chapter XV
166
The Theorem of Mean Value
167
The Extended Theorem of Mean Value
168
Maxima and minima treated analytically
169
The Generalized Theorem of Mean Value
172
Evaluation of a function taking on an indeterminate form
173
Evaluation of the indeterminate form
174
Evaluation of the indeterminate form
177
Evaluation of the indeterminate form 0 cc
179
Evaluation of the indeterminate forms 0 1 cc
180
Chapter XVI
183
Total differentials
200
141
201
Differentiation of implicit functions
202
Successive partial derivatives
205
Order of differentiation immaterial
206
Chapter XVIII
208
143
210
144
211
The evolute of a given curve considered as the envelope of its normals
213
146
214
Chapter XIX
217
Infinite series
218
Existence of a limit
220
Fundamental test for convergence
221
227
229
Chapter XX
231
Taylors Theorem and Taylors Series
232
Maclaurins Theorem and Maclaurins Series
234
Computation by series
238
Approximate formulas derived from series
240
Taylors Theorem for functions of two or more variables
243
Maxima and minima of functions of two independent variables 231 232 234 238 240 243
246
Chapter XXI
252
Method for determining asymptotes to algebraic curves
253
Asymptotes in polar coordinates
257
Singular points
259
Nodes
262
Cusps
263
Conjugate or isolated points
264
Transcendental singularities 205
265
Curve tracing 206
266
General directions for tracing a curve whose equation is given in rectangular coordinates
267
Tracing of curves given by equations in polar coordinates 209
269
Chapter XXII
271
Tangent plane to a surface
273
Normal line to a surface
275
Another form of the equations of the tangent line to a skew curve
277
Chapter XXIII
280
INTEGRAL CALCULUS Chapter XXIV
287
Constant of integration Indefinite integral
289
Rules for integrating standard elementary forms
291
Trigonometric differentials
303
Chapter XXV
309
Physical signification of the constant of integration
310
Chapter XXVI
315
Imaginary roots 191 Case I
318
Chapter XXVII
329
Differentials containing fractional powers of a + bx only 198 Differentials containing no radical except V + bx + x1 199 Differentials containing no r...
334
Conditions of integrability of binomial differentials 202 Transformation of trigonometric differentials
337
Miscellaneous substitutions
339
Chapter XXVIII
341
Reduction formulas for binomial differentials
344
Reduction formulas for trigonometric differentials
349
To find J e sin nxdx and J ex cos nxdx
353
Chapter XXIX
355
The definite integral
356
Geometrical representation of an integral
357
Mean value of px
358
Decomposition of the interval
359
Calculation of a definite integral
360
When 0 r is discontinuous
361
Change in limits
365
Chapter XXX
367
SECTION PAGE 221 Areas of plane curves Rectangular coordinates
371
5
373
Areas of plane curves Polar coordinates
376
Length of a curve
378
Lengths of plane curves Rectangular coordinates
379
Lengths of plane curves Polar coordinates
382
Volumes of solids of revolution
384
Areas of surfaces of revolution
388
Chapter XXXI
392
Definite double integral Geometric interpretation
396
Value of a definite double integral over a region
400
Plane area as a definite double integral Rectangular coordinate
402
Plane area as a definite double integral Polar coordinates
406
Moment of inertia Rectangular coordinates
408
Moment of inertia Polar coordinates
410
General method for finding the areas of surfaces
411
Volumes found by triple integration
415
Miscellaneous applications of the Integral Calculus
419
Chapter XXXII
424
Solutions of differential equations 242 Verifications of solutions 243 Differential equations of the first order and of the first degree 244 Differential e...
425
INTEGRAPH TABLE OF INTEGRALS 245 Mechanical integration
446
The integraph
448
Integrals for reference
450
INDEX
461
Copyright

Common terms and phrases

Popular passages

Page 52 - The derivative of the quotient of two functions 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 of the denominator.
Page 39 - x". It is not "dy" divided by "dx" or "d" multiplied by "y" divided by "d" multiplied by "x." In precise mathematical terms a derivative of a function is the limit of the ratio of the increment of the function to the increment of the independent variable when the latter increment varies and approaches zero as a limit.
Page 17 - The angle subtended at the center of a circle by an arc equal in length to a radius of the circle.
Page 109 - Similarly, the derivative of the second derivative is called the third derivative ; and so on to the nth derivative.
Page 130 - Assuming that the strength of a beam with rectangular cross section varies directly as the breadth and as the square of the depth, what are the dimensions of the strongest beam that can be sawed out of a round log whose diameter is d ? Solution.
Page 175 - Differentiate the numerator for a new numerator and the denominator for a new denominator.* The...
Page 19 - What is the ratio of their radii ? of their apothems ? of their perimeters ? of their areas ? 5. The diameters of two circles are d and d' respectively. What is the ratio of their radii ? of their circumferences ? of their areas ? 6. If the number of sides of a regular inscribed polygon is indefinitely increased, what is the limit of the apothem ? of each side ? of the perimeter ? of the area ? of the angle at the center ? of each angle of the polygon ? 7. How do you find the area of a regular polygon...
Page 131 - Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius 10 cm is -*- cm.
Page 12 - By the definition of a function we see that the area of a square is a function of the length of a side and the distance that a falling body travels is a function of the time it falls.
Page 50 - The summation of the product of a constant and a variable is equal to the product of the constant and the summation of that variable: A collection, class, or listing of mathematical objects is called a set.

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