45–46. Linear approximation


a. Find the linear approximation to f at the given point a.
b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?


ƒ(x) = x²⸍³ ; a =27; ƒ(29)

Verified step by step guidance

1. Identify the function f(x) = x^(2/3) and the point a = 27 where you will find the linear approximation.
2. Calculate the derivative f'(x) using the power rule: f'(x) = (2/3)x^(-1/3).
3. Evaluate the derivative at the point a = 27 to find f'(27).
4. Use the point-slope form of the linear approximation: L(x) = f(a) + f'(a)(x - a), substituting in the values of f(27) and f'(27).
5. Substitute x = 29 into the linear approximation L(x) to estimate the value of f(29) and compare it to the actual value of f(29) to determine if it underestimates or overestimates.