Final answer:
The linearization of the function f(x) at x = 3 is given by f(x) ≈ 44.3 + 42.7(x-3).
Step-by-step explanation:
The linearization of the function f(x) at x = 3 can be found by using the formula for linear approximation, which is given by f(x) ≈ f(a) + f'(a)(x-a), where f'(a) is the derivative of f(x) evaluated at x = a. In this case, the function is f(x) = 4.90x² + 14.3x - 20.0, and we want to find the linearization at x = 3.
First, find the derivative of f(x) using the power rule: f'(x) = 2(4.90x) + 14.3 = 9.8x + 14.3.
Next, evaluate f(3) and f'(3): f(3) = 4.90(3)² + 14.3(3) - 20.0 = 44.3, and f'(3) = 9.8(3) + 14.3 = 42.7.
Finally, substitute these values into the linear approximation formula to get the linearization: f(x) ≈ 44.3 + 42.7(x-3).