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Chapter 2: Q. 58 (page 185)

Use the definition of the derivative to find the derivatives described in Exercises 55-58.

Givend2fdx2=2-x2, findf(3)(4)andd4dx4(f(x))

Short Answer

Expert verified

The value is-2

Step by step solution

01

Step 1. Given information

Given functiond2fdx2=2-x2

02

Use the definition of derivative

Calculating, we get

limz鈫�xf(2)(z)-f(2)(x)z-x=limz鈫�x2-z2-2-x2z-xlimz鈫�xf(2)(z)-f(2)(x)z-x=limz鈫�x-z2-x2z-x=limz鈫�x-(z-x)(z+x)z-x=limz鈫�x-(z+x)=-(x+x)=-2xf(3)(4)=-2(4)=-8d4dx4(f(x))=limz鈫�xf(3)(z)-f(3)(x)z-xd4dx4(f(x))=limz鈫�x(-2z)-(-2x)z-x=limz鈫�x-2(z-x)z-x=limz鈫�x-2=-2

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Most popular questions from this chapter

In Exercises 69鈥�80, determine whether or not fis continuous and/or differentiable at the given value of x. If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

fx=x2,ifx鈮�12x-1,ifx>1,x=1

Each graph in Exercises 31鈥�34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

Use the definition of the derivative to find ffor each function fin Exercises 39-54

f(x)=1x2-1

Use the differentiation rules developed in this section to find

the derivatives of the functions

f(x)=4-x7

Suppose h(t) represents the average height, in feet, of a person who is t years old.

(a) In real-world terms, what does h(12) represent and what are its units? What does h' (12) represent, and what are its units?

(b) Is h(12) positive or negative, and why? Is h'(12) positive or negative, and why?

(c) At approximately what value of t would h(t) have a maximum, and why? At approximately what value of t would h' (t) have a maximum, and why?
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