Elements of the Differential Calculus: With Examples and Applications, Volum 25

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Ginn, 1895 - 258 sider
 

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Innhold

The inclination of a parabola to the axis of X
10
CHAPTER II
11
Classification of functions
12
Derivative of a function of a function of the variable
18
CHAPTER III
22
The expressions and OXec called indeterminate forms
28
Derivative zero at a maximum or a minimum
33
Sign of derivative near a zero value shown by the value of its own derivative
34
Investigation of a minimum
35
General rule for discovering maxima and minima Examples
38
Examples
39
Integration
40
Statement of the problem of finding the distance traversed by a falling body given the velocity
41
Statement of the problem of finding the length of an arc of a given curve
42
Integration Integral
44
Solution of problem stated in Article 50
46
CHAPTER IV
49
Expansion of 1+n by the Binomial Theorem
50
This series is taken as the base of the natural system of loga rithms Computation of its numerical value
52
Extension of the proof given above to the cases where m is not a positive integer
53
Differentiation of log completed
54
Differentiation of ox Examples
55
Circular measure of an angle Reduction from degree to cir
57
CHAPTER V
65
Simplification by an algebraic transformation Examples
71
CHAPTER VI
77
The derivative of z with respect to y is the quotient of the derivative of z with respect to x by the derivative of y with respect to x
78
Osculating circle Radius of curvature Centre of curvature
81
Definition of evolute Formulas for evolute
82
Evolute of a parabola
83
Reduced formulas for evolute Example
85
Every normal to a curve is tangent to the evolute
87
Length of an arc of evolute
88
CHAPTER VII
90
Equations of the cycloid referred to vertex as origin Exam ples
92
Statement of properties of cycloid to be investigated
93
Equations of tangent and normal Example
94
Motion down a smooth curve Examples
112
Definition of series Convergent series Divergent series
118
Second form for remainder Taylors Theorem
125
Maclaurins Theorem
132
Leibnitzs Theorem for Derivatives of a Product
136
Reduction of the forms oo l 0 to forms already discussed
142
CHAPTER X
149
Tangent at any point of the pedal of a given curve
156
The substitution of one infinitesimal for another
160
Theorem concerning the limit of the sum of infinitesimals
161
If two infinitesimals differ from each other by an infinitesimal of higher order the limit of their ratio is unity
162
Direction of a tangent to a parabola
163
Area of a sector of a parabola
164
The limit of the ratio of an infinitesimal arc to its chord is unity
166
Tangent to an ellipse Examples
167
The area of a segment of a parabola Examples
168
New way of regarding the cycloid
169
Area of the cycloid
171
Length of an arc of the cycloid
172
Radius of curvature of the cycloid
174
Evolute of the cycloid
176
Examples
177
CHAPTER XI
183
Definition of differential
185
Formulas for differentials of functions Examples
186
The differential notation especially convenient in dealing with problems in integration Numerical example
187
Integral regarded as the limit of a sum of differentials Defi nite integral
189
Definition of centre of gravity The centre of gravity of a parabola
190
Article Fge
192
CHAPTER XII
199
Use of partial derivatives in finding Successive complete deriv
205
Example of the change of both dependent and independent
206
Change of variable when partial derivatives are employed
215
Equations of tangent line to a curve in space Equation
221
CHAPTER XV
227
CHAPTER XVI
234

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