Elements of the Differential Calculus: With Examples and Applications

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Ginn, 1891 - 258 sider
 

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Innhold

The inclination of a parabola to the axis of A
9
Fundamental object of the Differential Calculus
10
CHAPTER II
11
Classification of functions
12
Derivative of a function of a function of the variable
18
Derivative may sometimes be found by solving an equation
25
Maxima and Minima of a Continuous Function
31
Article Tage 40 Derivative zero at a maximum or a minimum
33
Sign of derivative bear 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 3f i 48 Use of auxiliary variables Examples
38
41 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 l+j by the Binomial Theorem
50
This series is taken as the base of the natural system of logs 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 x completed
54
Differentiation of tea Examples
55
Circular measure of an angle Reduction from degree to cir
57
CHAPTER V
65
Integration by parts Examples
69
Length of an arc of a parabola Example
75
To find actual curvature conveniently an indirect method of differentiation must be used
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

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Side 91 - The angle formed by a tangent and a chord is measured by half the intercepted arc.
Side 171 - F'P'+ P'F, by the definition of an ellipse. Take away from the first sum F'P + BF, and we have left PB ; take away from the second sum the equal amount F'A + P'F, and we have left P'A ; .-. PB=P'A; and the right triangles PAP...
Side 168 - Art. 162; T'PR= T'PF, and the tangent at any point of a parabola bisects the angle between the focal radius and the diameter through the given point. 164. To find the area of the sector of a parabola included between two focal radii. Take points of the parabola between the extremities of the bounding radii, and join them with the focus, thus dividing the area in question into smaller sectors, of which the sector FPP' in the figure of the last article may be taken as a type.
Side 172 - EXAMPLE. Prove that a tangent to an hyperbola bisects the angle between the focal radii drawn to the point of contact. 168. To find the area of a segment of a parabola cut off by a line perpendicular to the axis. Compare the required area with the area of the circumscribing rectangle. We can regard the...
Side 3 - Ja; representing a single quantity. It is to be noted that as an increment is a difference, it may be either positive or negative. 5. If a variable which changes its value according to some law can be made to approach some fixed, constant value as nearly as we please, but can never become equal to it, the constant is called the limit of the variable under the circumstances in question.
Side 37 - Show that u^a? - 3a?+6x+7 has neither a maximum or a minimum value ; and that is neither a maximum nor a minimum when x = 0. (5) A person in a boat, three miles from the nearest point of the beach, wishes to reach, in the shortest possible time, a place 5 -a five miles from that point, along the shore.

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