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Calculus - Derivatives and Numerical Values

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Need help with calculus homework problems.

In Exercises 41-48, find dy/dx

44. 5x 4/5 + 10y 6/5 = 15
____________________________________________________________________

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

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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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Maximum Area of a Quadrilateral

In the quadrilateral PQRS shown below, the side PQ has length 5 metres, the side QR has length 6 metres, and the side RS has length 7 metres. The angle at P is a right angle, and no angle of the quadrilateral exceeds 180◦. The side PS has length x metres, where the value of x is between 0 and 12. (The quadrilateral described cannot exist for other values of x.)

See attached image.

For parts (a) and (b) (and for part (c), if you use Mathcad there) you should provide a printout annotated with enough explanation to make it clear what you have done.
If you define x to be a range variable in part (a) and wish to use x in a symbolic calculation in part (b), then you will need to insert the definition x := x between the two parts in your worksheet. (For more details, see the bottom of page 49 in A Guide to Mathcad.)
(a) Use Mathcad to obtain the graph of the function A(x).
(b) This part of the question requires the use of Mathcad in each sub-part.
(i) By using the differentiation facility, and if you wish the symbolic keyword 'simplify', find an expression for the derivative A′(x).
(ii) By applying a numerical solve block (rather than attempting to solve symbolically), find a value of x for which A′(x) = 0.
(iii) Verify, by the Second Derivative Test, that this value of x corresponds to a local maximum of A(x). (It should be apparent
from the graph obtained in part (a) that this is also an overall maximum within the domain of A(x).)
24
(c) Using Mathcad, or otherwise, calculate the maximum possible area of the quadrilateral.

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