Suppose that f (x) is a function such that the relationship given below is true.

f (3 + h) - f (3) = 9h^2 + 8h

(a) What is f '(3)?

(b) What is the slope of the secant line through (3, f (3)) and (7, f (7))?

Please do not use shortcuts(from future chapters) and show steps so i can see whats going on

if you recall the definition of a derivative is

lim h->0 [f(x+h)-f(x)]/h

(a)
you're given:
f (3 + h) - f (3) = 9h^2 + 8h
just divide h so that you get f'(3)
and take the lim h->0

(b) just take the slope:
[f(7)-f(3)]/(7-3)

i did try that before posting the question, it gave a wrong answer.

To find the answers to both parts (a) and (b), we can follow the same approach: differentiate the function and then substitute the required values.

(a) To find f'(3), which represents the derivative of f with respect to x evaluated at x = 3, we can differentiate the given equation with respect to x.

Differentiating both sides of the equation gives:

f'(3 + h) - f'(3) = 18h + 8

As h approaches 0, we can rewrite this equation as:

f'(3) - f'(3) = 0 + 8

Simplifying this equation gives:

0 = 8

Since this is not a valid equation (0 is not equal to 8), there is no definite value for f'(3) that can be determined from the given information.

Therefore, the answer to part (a) is that we cannot determine the value of f'(3) from the given equation.

(b) The slope of the secant line through (3, f(3)) and (7, f(7)) can be found using the formula for slope:

slope = (change in y) / (change in x)

Let's calculate the values needed:

(change in y) = f(7) - f(3)
(change in x) = 7 - 3

Substituting these values into the slope formula:

slope = (f(7) - f(3)) / (7 - 3)

We need the values of f(7) and f(3) to calculate the slope of the secant line. Unfortunately, the given equation does not provide specific values for these function evaluations. Therefore, it is not possible to determine the slope of the secant line through (3, f(3)) and (7, f(7)) accurately.

Hence, the answer to part (b) is that we cannot determine the slope of the secant line through (3, f(3)) and (7, f(7)) from the given equation.