Evaluate \( f(8) \) and \( f(8.1) \), and use the results to approximate \( f'(8) \). Round your answer to one decimal place. The function is given by \( f(x) = x(9 - x) \).

A) \( f(8) = 40 \), \( f(8.1) = 36.9 \), \( f'(8) \approx 1.1 \)

B) \( f(8) = 36 \), \( f(8.1) = 36.9 \), \( f'(8) \approx 0.9 \)

C) \( f(8) = 32 \), \( f(8.1) = 36.9 \), \( f'(8) \approx 1.5 \)

D) \( f(8) = 36 \), \( f(8.1) = 33.9 \), \( f'(8) \approx 0.7 \)

To approximate f'(8), we found the function values at x = 8 and x = 8.1, then used these to approximate the derivative. The derivative, to one decimal place, is approximately 1.9.

Explanation:

This question is asking to evaluate the function f(x) = x(9 - x) at two points (x = 8 and x = 8.1) and then use these results to approximate the derivative of the function at x = 8.

First, we find f(8) = 8 * (9-8) = 8, and also find f(8.1) = 8.1 * (9-8.1) = 8.19. The approximate derivative, f'(8), can be found using the limit definition of a derivative, f'(a) = lim(h->0) [(f(a+h)-f(a))/h]. Essentially, f'(8) is approximately equal to (f(8.1) - f(8)) / (8.1 - 8), which turns out to be (8.19 - 8) / (0.1) = 1.9, rounding to one decimal place.

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