Part 1 of 3
HW Score: $10 \%, 2$ of 20 points
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.
Points: 0 of 1
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.
\[
f(x)=10-3 x^{2} \text { at } a=2 ; f(1.9)
\]
a. $L(x)=\square$
Answer
Final Answer: The linear approximation of the function at \(a=2\) is \(L(x) = \boxed{22 - 12x}\).
Step 1 ：The function given is \(f(x) = 10 - 3x^2\).
Step 2 ：The derivative of a constant is 0 and the derivative of \(x^2\) is \(2x\), so the derivative of the function is \(f'(x) = -6x\).
Step 3 ：Substitute \(a=2\) into the function and its derivative: \(f(2) = 10 - 3(2)^2 = 10 - 12 = -2\) and \(f'(2) = -6(2) = -12\).
Step 4 ：The equation of the tangent line is: \(L(x) = -2 - 12(x - 2)\).
Step 5 ：Simplify this equation to get: \(L(x) = -2 - 12x + 24 = 22 - 12x\).
Step 6 ：Final Answer: The linear approximation of the function at \(a=2\) is \(L(x) = \boxed{22 - 12x}\).
