Final answer:
To estimate the value of f(7.1) using the linear approximation, we can use the formula f(x) = f(a) + f'(a)(x - a), where a = 7. Plugging in the given values, we find f(7.1) = 6.84.
Step-by-step explanation:
To estimate the value of f(7.1) using the linear approximation, we can start by finding the slope of the tangent line to the graph of f at x = 7. Since f is differentiable at x = 7, the slope of the tangent line is equal to the derivative of f at that point. We can use the formula for the linear approximation, which is f(x) = f(a) + f'(a)(x - a), where a = 7 in this case. Plugging in the given values, f(7) = 6.5 and f'(7) = 3.4, we can calculate f(7.1) as follows:
f(7.1) = f(7) + f'(7)(7.1 - 7)
f(7.1) = 6.5 + 3.4(0.1)
f(7.1) = 6.5 + 0.34 = 6.84