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Answer :
Final answer:
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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