Final answer:
The student is attempting to find the linear approximation L(x) for the function f(x) = (1.01)^x at x = 3. By calculating f(3) and f'(3), the linear approximation L(x) can be constructed and evaluated.
Step-by-step explanation:
The question appears to ask for the linear approximation of a function at a given point. To find the linear approximation L(x) of the function f(x) = (1.01)^x at a, we first evaluate f(a) which is the value of the function at a, and f'(a) which is the derivative of the function at a. The equation for the linear approximation is L(x) = f(a) + f'(a)(x - a). In this case, to evaluate (1.01)^3, we let x = 3 and calculate f(3) and f'(3), then plug them into the linear approximation formula to find L(3).
To calculate the derivative, we would use the power rule. For the function f(x) = (1.01)^x, the derivative f'(x) if x is a constant, would be f'(x) = ln(1.01) * (1.01)^x. For a = 3, we then have f(3) = (1.01)^3 and f'(3) = ln(1.01) * (1.01)^3, which we can use to evaluate L(x).