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·|approximation - exact| / |exact|, where the exact value is given by a calculator.
f(x) = 2/(x + 1); a = 0; 2/1.2
a. l(x) = 2 - 2x
b. using the linear approximation, 2/1.2 ≈ 1.600
(round to three decimal places as needed )
c. the percent error in the approximation is □%.
(round to three decimal places as needed.)

Explanation:

Step1: Find the exact value

The function is \( f(x)=\frac{2}{x + 1} \). When \( x = 0.2 \) (since \( \frac{2}{1.2}=\frac{2}{1+0.2} \)), the exact value \( f(0.2)=\frac{2}{0.2 + 1}=\frac{2}{1.2}\approx1.6667 \) (using a calculator).

Step2: Calculate the percent - error formula

The approximation value from part b is \( L(0.2)=2-2\times0.2=1.6 \).
The percent - error formula is \( 100\times\frac{\vert\text{approximation}-\text{exact}\vert}{\vert\text{exact}\vert} \).
Substitute the values: \( 100\times\frac{\vert1.6 - 1.6667\vert}{\vert1.6667\vert} \).
First, calculate the numerator: \( \vert1.6 - 1.6667\vert=0.0667 \).
Then, \( 100\times\frac{0.0667}{1.6667} \).
\( \frac{0.0667}{1.6667}\approx0.0400 \).
Multiply by 100: \( 100\times0.0400 = 4.000\% \).

Answer:

\(4.000\)