Consider the function f(x)=7-4x2 on the interval [-2,7].

Question

Consider the function f(x)=7-4x² on the interval [-2,7].

(A) Find the average or mean slope of the function on this interval, i.e.

(f(7)+f(-2))/(7-(-2))=

(B) By the Mean Value Theorem, we know there exists a  in the open interval (-2,7) such that f'(c) is equal to this mean slope. For this problem, there is only one c that works. Find it.

 c= 

Answer 1

At x = -2, f(x) = -9 so the point is (-2, -9)

At x = 7, f(x) = -189 so the point is (7, -189)

To find the average slope, just use the slope equation, m = (y₂ - y₁)/(x₂ - x₁) which is exactly what you have listed except you use the function notation "f(x)" instead of "y"

Average Slope = (-189 - -9)/(7 - -2) = -180/9 = -20

The derivative of f(x) = 7 - 4x² is found using the power rule. If you haven't learned the power rule you will need to use the limit of the difference quotient. Either way, you should find out that f '(x) = -8x

To find the x value, -8x = -20 or x = -20/-8 = 5/2 so f '(5/2) = -20 or in other words c = 5/2.