Consider the function f ( x ) = 5 x 3 − 5 x on the interval [ − 2 , 2 ] . Find the average or mean slope of the function on this interval.

Question

Consider the function  on the interval . Find the average or mean slope of the function on this interval.   By the Mean Value Theorem, we know there exists at least one  in the open interval  such that  is equal to this mean slope.For this problem, there are two values of  that work.The smaller one is   and the larger one is   

Answer 1

Finding an average rate of change over a finite interval is a 7th grade slope question: (y₂ - y₁) / (x₂ - x₁).

f(-2) = - 40 + 10 = - 30 and f(2) = 40 - 10 = 30

So the average rate of change of f on [ - 2 , 2 ] = (30 - (- 30)) / (2 - (- 2)) = 15

To find at which x-values the instantaneous rate of change = 15, we need to differentiate using power rule:

f^'(x) = 15x² - 5 = 15

15x² = 20 ; x² = 4/3 and x = ± 2 / √3 or ± 2√3 / 3 in SRF