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,
$100 \cdot \frac{\text { approximation - exact }}{\mid \text { exact } \mid}$, where the exact value is given by a calculator.
\[
f(x)=11-3 x^{2} \text { at } a=2 ; f(1.9)
\]
Answer
Compute the percent error in the approximation: \(100 \cdot \frac{0.2 - 0.17}{|0.17|} \approx 17.65\%\)
Step 1 ：Find the derivative of \(f(x) = 11 - 3x^2\) with respect to x: \(f'(x) = -6x\)
Step 2 ：Evaluate the derivative at \(a = 2\): \(f'(2) = -12\)
Step 3 ：Write the equation of the tangent line using the point-slope form: \(y = 23 - 12x\)
Step 4 ：Use the tangent line equation to estimate \(f(1.9)\): \(y = 23 - 12(1.9) = 0.2\)
Step 5 ：Calculate the exact value of \(f(1.9)\) using the original function: \(f(1.9) = 11 - 3(1.9)^2 = 0.17\)
Step 6 ：Compute the percent error in the approximation: \(100 \cdot \frac{0.2 - 0.17}{|0.17|} \approx 17.65\%\)
