Question

a. write the equation of the line that represents the linear approximation to the following function at the given point a
b. use the linear approximation to estimate the given quantity
c. compute the percent error in the approximation, ( 100 cdot \frac { | \text { approximation - exact } | } { | \text { exact } | } ), where the exact value is given by a calculator
( f ( x ) = 4 - x ^ { 2 } ) at ( a = 1 ; f ( 1.1 ) )
a. ( l ( x ) = 5 - 2 x )
b. using the linear approximation, ( f ( 1.1 ) approx )
(type an integer or a decimal.)

Explanation:

Step1: Substitute \(x = 1.1\) into the linear approximation formula

We have the linear approximation \(L(x)=5 - 2x\).
Substitute \(x = 1.1\) into \(L(x)\):
\(L(1.1)=5-2\times1.1\)

Step2: Calculate the value

First, calculate \(2\times1.1 = 2.2\).
Then, \(5-2.2=2.8\).

Answer:

\(2.8\)