Chapter 9.6, Problem 30E

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Chapter
Section

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

# In Problems 29 and 30, complete the following for each function.(a) Find f'(x).(b) Check your result in part (a) by graphing both it and the numerical derivative of the function.(c) Find x-values for which the slope of the tangent is 0.(d) Find points (x, y) where the slope of the tangent is 0.(e) Use a graphing utility to graph the function and locate the points found in part (d). f ( x ) = 10 − ( x 2 − 2 x − 8 ) 2

(a)

To determine

To calculate: The derivative f(x) of the function f(x)=10(x22x8)2.

Explanation

Given Information:

The function is f(x)=10âˆ’(x2âˆ’2xâˆ’8)2.

Formula used:

According to the power rule, if f(x)=xn, then,

fâ€²(x)=nxnâˆ’1

According to the property of differentiation, if a function is of the form, g(x)=cf(x), then,

gâ€²(x)=cfâ€²(x)

According to the property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

fâ€²(x)=uâ€²(x)+vâ€²(x)

According to the product rule, if f(x)=u(x)â‹…v(x), then

fâ€²(x)=uâ€²(x)â‹…v(x)+vâ€²(x)â‹…u(x)

According to the property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nunâˆ’1dudx

Calculation:

Consider the provided function,

f(x)=10âˆ’(x2âˆ’2xâˆ’8)2

Consider (x2âˆ’2xâˆ’8) to be u,

f(x)=10âˆ’u2

Differentiate both sides with respect to x,

fâ€²(x)=ddx(10

(b)

To determine

To graph: The derivative f(x) of the function f(x)=10(x22x8)2 and the numerical derivative of the function to check the results in part (a).

(c)

To determine

To calculate: The x-values for which the slope of the tangent of the function f(x)=10(x22x8)2 is 0.

(d)

To determine

To calculate: The points (x,y) where the slope of the tangent of the function f(x)=10(x22x8)2 is 0.

(e)

To determine

To graph: The function f(x)=10(x22x8)2 and locate the points for which the slope of the tangent of the function is 0 using graphing utility.

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