   Chapter 3.10, Problem 44E

Chapter
Section
Textbook Problem

Suppose that we don’t have a formula for g(x) but we know that g(2) = –4 and g ′ ( x ) = x 2 + 5 for all x.(a) Use a linear approximation to estimate g(1.95) and g(2.05).(b) Are your estimates in part (a) too large or too small? Explain.

(a)

To determine

To estimate: The value g(1.95) and  g(2.05).

Explanation

Given:

The value g(2)=4 and g(x)=x2+5.

Result used:

The linear approximation of the function at x=a is,g(x)g(a)+g(a)(xa).

Calculation:

The linearization of the function g(x) at x=2 is computed as follows,

Substitute the value a=2 in g(x)g(a)+g(a)(xa),

g(x)g(2)+g(2)(x2)

Given g(x)=x2+5 is,

Substitute x=2,

g(2)=22+5=4+5=9=3

Substitute g(2)=4 and g(2)=3 in

(b)

To determine

To explain: Whether the estimate value in part (a) is too large or too small.

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