Question

Conjecture a formula for f'(a) that depends only on the value a. That is, in the same way that we have a formula for f(x (recall f(x)=8s-x2), see if you can use your work f'(a) = 0 above to guess a formula forf,(a) in terms of a.

Answer 1

To conjecture a formula for f'(a), we differentiate the given function f(x) = 8s - x^2 to get f'(x) = -2x. Then, we substitute 'a' into f'(x) to get f'(a) = -2a. This formula gives us the derivative at any given point 'a'.

Step-by-step explanation:

The student is attempting to conjecture a formula for f'(a), the derivative of a function f(x), at a specific point a. Given that the formula for f(x) seems to be f(x) = 8s - x^2, the derivative of the function, f'(x), would be calculated using basic differentiation rules. To find f'(a), we substitute a into the derivative f'(x).

First, we find the derivative of f(x) with respect to x:

f'(x) = d/dx (8s - x^2)

The derivative of a constant times a variable (8s) is zero since s does not depend on x. The derivative of -x^2 with respect to x is -2x. Therefore:

f'(x) = 0 - 2x

To find f'(a), we substitute a for x:

f'(a) = -2a

This formula for f'(a) depends only on the value a, which is what was requested in the question.