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}-\text{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 - 2x )
b. using the linear approximation, ( f(1.1) approx 2.8 )
(type an integer or a decimal)
c. the percent error in the approximation is ( square % )
(round to three decimal places as needed)

Explanation:

Step1: Calculate the exact value of \( f(1.1) \)

Given \( f(x)=4 - x^{2} \), substitute \( x = 1.1 \) into the function:
\( f(1.1)=4-(1.1)^{2}=4 - 1.21=2.79 \)

Step2: Calculate the percent - error formula

The percent - error formula is \( \text{Percent Error}=100\cdot\frac{\vert\text{approximation}-\text{exact}\vert}{\vert\text{exact}\vert} \)
We know that the approximation is \( L(1.1) = 2.8 \) (from part b) and the exact value is \( f(1.1)=2.79 \)
Substitute these values into the formula:

$$
LATEXBLOCK0
$$

Answer:

\( 0.358\% \)