   Chapter 9.6, Problem 30E Mathematical Applications for the ...

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

Solutions

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)=10(x22x8)2.

Formula used:

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

f(x)=nxn1

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

g(x)=cf(x)

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

f(x)=u(x)+v(x)

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

f(x)=u(x)v(x)+v(x)u(x)

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

dydx=nun1dudx

Calculation:

Consider the provided function,

f(x)=10(x22x8)2

Consider (x22x8) to be u,

f(x)=10u2

Differentiate both sides with respect to x,

f(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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