   Chapter 2.9, Problem 42E

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.

To determine

(a)

To estimate:

g1.95 and g(2.05) by using a linear approximation.

Explanation

1) Formula:

Linear approximation of g(x) at a:

Lxga+g'(a)(x-a)

2) Given:

g2= -4,g'x=  x2+5 for all x

3) Calculation:

To find linear approximation of g(x) at 2, substitute a = 2 in Lxga+g'(a)(x-a)

Lxg2+g'(2)(x-2)

To find  g'2, substitute x = 2 in g'x=  x2+5

g'2=  22+5 =  9 =3

Now, substitute g'2=3  and g2= -4 in Lx

To determine

(b)

To explain:

Estimates in part (a) are too large or too small.

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