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)=\frac{2}{x + 1} ; a = 0, \frac{2}{1.2} )

a. ( l(x)=2-2x )
b. using the linear approximation, ( \frac{2}{1.2} approx square ).
(round to three decimal places as needed.)

Explanation:

Step1: Substitute \(x = 0.2\) into the linear approximation formula

We have the linear approximation \(L(x)=2 - 2x\). Substitute \(x = 0.2\) (since \(\frac{2}{1.2}\) corresponds to \(x=0.2\) in the function \(f(x)=\frac{2}{x + 1}\)) into \(L(x)\).

$$L(0.2)=2-2\times0.2$$

Step2: Calculate the value

$$L(0.2)=2 - 0.4=1.6$$

Answer:

\(1.600\)