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: \(100 \cdot \frac{0.2 - 0.17}{|0.17|} \approx 17.65\%\)