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Calculus - Derivatives and Numerical Values

Need help with calculus homework problems.

In Exercises 41-48, find dy/dx

44. 5x 4/5 + 10y 6/5 = 15
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54. a. By differentiating x 2 - y2 =1 implicitly, show that dy/dx = x/y

b. Then show that d2y/dx2 = - 1/y3.

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Numerical Values of Derivatives
56. Suppose that the function f(x) and its first derivative have the following values at
x = 0 and x = 1

x f(x) f'(x)
0 9 -2
1 -3 1/5

Find the first derivatives of the following combinations at the given value of x
a. f(x), x = 1
b. , x = 0
c. f , x =1
d. f(1 - 5 tan x), x = 0

e. f(x) , x = 0
2+cosx

f. 10sin f2(x), x = 1

62. If x1/3 + y1/3 = 4, find d2y/dx2 at the point (8,8)

68. For what value or values of the constant m, if any is

f(x) =

a. continuous at x = 0?

b. differentiable at x = 0?

Give reasons for your answers

Attachments

Solution Preview

The solution file is attached.

In Exercises 41-48, find dy/dx
44. 5x 4/5 + 10y 6/5 = 15
5(4/5) x^(-1/5) + 10(6/5) y^(1/5) (dy/dx) = 0
4 x^(-1/5) + 12 y^(1/5) (dy/dx) = 0
dy/dx = -4 x^(-1/5) / 12 y^(1/5) = -1/[3 (xy)^(1/5)]

54. a. By differentiating x 2 - y2 =1 implicitly, show that dy/dx = x/y
b. Then show that d2y/dx2 = - 1/y3.
(a) 2x - 2y(dy/dx) = 0
dy/dx = 2x/2y = x/y
(b) d^2y/dx^2 = [y - x(dy/dx)]/y^2
= [y - x(x/y)]/y^2
= [y^2 - x^2]/y^3
= -1/(y^3)

Numerical Values of Derivatives
56. Suppose that the function f(x) and its first derivative have the following values at
x = 0 and x = ...

Solution Summary

The derivatives and numerical values in calculus are determined. Complete, Neat and Step-by-step Solutions are provided in the attached file.

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