A First Course in Infinitesimal Calculus

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Longmans, Green, and Company, 1903 - Calculus - 439 pages
 

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Contents

Constants
16
Classification of functions
17
Geometrical representation of functions of one variable
19
Limits
20
Notation
24
CHAPTER III
30
Orders of magnitude Orders of infinitesimals Orders of infinites
31
Theorems on limits and infinitesimals
35
Fundamental theorems of the calculus
36
Notation
41
Differentials
49
The derivative of the product of two or more functions
57
B Logarithmic and Exponential Functions
66
Inverse Trigonometric Functions
76
Partial derivatives Notation
79
rectangular co÷rdinates
84
Order of partial differentiations commutative
85
polar co÷rdinates
88
polar co÷rdinates
89
Applications involving rates
91
Rolles theorem
93
Theorem of mean value
94
Small errors and corrections relative error
96
Applications to algebra
98
Geometric derivatives and differentials 99106
99
CHAPTER VI
107
The nth derivative of some particular functions
111
Successive differentials
112
Leibnitzs theorem
113
Application of differentiation to elimination
114
CHAPTER VII
116
Maximum and minimum values of a function Critical points on the graph and critical values of the variable
117
Inspection of the critical values of the variable for maximum or minimum values of the function
120
Practical problems in maxima and minima
123
rectangular co÷rdinates
127
two variables
139
CHAPTER X
148
Integration as summation The definite integral
154
CHAPTER XIV
157
Integration as the inverse of differentiation The indefinite integral
160
Geometric or graphical representation of indefinite integrals
166
CHAPTER XI
171
ART PAGE 103 Elementary integrals
172
General theorems in integration
173
Integration aided by substitution
175
Integration by parts
177
Further elementary integrals
180
Integration of f2dx when fa is a rational fraction
184
Integration of a total differential
188
CHAPTER XII
192
Volumes of solids of revolution
199
Derivation of the equations of curves
203
CHAPTER XIII
206
Differential expressions involving Va + bx
207
A Expressions of form Fx Vx2 + ax + bdx B Expressions of form Fx V 202 + ax + bdx
208
To find am
211
CHAPTER XV
230
several variables
232
rectangular co÷rdinates
234
rectangular co÷rdinates
235
polar co÷rdinates
238
CHAPTER XVI
240
polar co÷rdinates
242
rectangular co÷rdinates
245
polar co÷rdinates
248
Areas of surfaces of revolution
249
Areas of surfaces z fx y
253
Mean values
255
CHAPTER XVII
260
Order of contact
261
Osculating circle
264
The notion of curvature
265
Total curvature Average curvature Curvature at a point
266
ART PAGE 147 The curvature of a circle
267
The circle of curvature at any point of a curve
268
polar co÷rdinates
271
Evolute of a curve
272
Properties of the evolute
273
Involutes of a curve
276
CHAPTER XVIII
277
Locus of ultimate intersections of the curves of a family
278
Theorem
280
To find the envelope of a family of curves having one parameter
281
Envelope of a family of curves having two parameters
284
Rectilinear asymptotes
286
Asymptotes parallel to the axes
289
Oblique asymptotes
290
polar co÷rdinates
292
Singular points
293
To find multiple points cusps and isolated points
296
Curve tracing
298
CHAPTER XIX
300
Questions concerning infinite series
301
Study of infinite series
303
Definitions Algebraic properties of infinite series
304
Tests for convergence
307
Integration of infinite series
310
Differentiation of infinite series
312
Applications of the integration and differentiation of series
313
CHAPTER XX
318
Another form of Taylors theorem
323
Maclaurins theorem and series
324
Application of Taylors theorem to the determination of condi
331
Exact differential equations Integrating factors
337
Orthogonal trajectories
343
Note A Hyberbolic functions
363
COLLECTION OF EXAMPLES
381
INTEGRALS FOR Review EXERCISES AND FOR REFERENCE
401
FIGURES
409
ANSWERS
415
INDEX
433
Copyright

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Page 50 - PROP. II. Product of Constant and Function. The differential coefficient of a product of a constant and a function of x is equal to the product of the constant and the differential coefficient of the function, or, stated...
Page 393 - The lower corner of a leaf, whose width is a, is folded over so as just to reach the inner edge of the page. Find the width of the part folded over, when the length of the crease is a minimum.
Page 320 - From eight times the chord of half the arc, subtract the chord of the whole arc, and divide the remainder by 3, and the quotient will be the length of the arc, nearly.
Page 56 - 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 280 - A straight line moves so that the sum of the squares of the perpendiculars on it from two fixed points (▒ c, 0) is constant (= 2fc'2) : show that y2 y'2 its envelope is the conic — - ( + ^ = 1.
Page 227 - Differentiating the first of these equations with respect to x, the second with respect to y, and the third with respect to z...
Page 143 - Nova methodus pro maximis et minimis, itemque tangentibus, quae nee fractas nee irrationales quantitates moratur, et singulare pro illis calculi genus, die in den Actis Erudit.
Page 37 - 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 120 - ... the following problems the cones and cylinders are supposed to be right cones and cylinders on circular bases.] 48. Determine the greatest cylinder that can be inscribed in a given cone.
Page 121 - From a given circular sheet of metal it is required to cut out a sector so that the remainder can be formed into a conical vessel of maximum capacity ; prove that the angle of the sector removed must be about 66░.

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