Question

Find the average rate of change of f(x) over the interval [5, x]. For how many values of x in the interval does the instantaneous rate of change of f(x) equal the average rate of change of f(x) over that interval?

Answer 1

To find the average rate of change of f(x) over the interval [5, x], evaluate the function at the endpoints and calculate the slope of the secant line. To find the values of x where the instantaneous rate of change equals the average rate of change, find the points where the derivative of f(x) is equal to the average rate of change.

Step-by-step explanation:

To find the average rate of change of f(x) over the interval [5, x], we need to evaluate the function f(x) at the endpoints and then calculate the slope of the secant line connecting the two points. The formula for average rate of change is:

Average Rate of Change = (f(x) - f(5))/(x - 5)

To find the values of x for which the instantaneous rate of change of f(x) equals the average rate of change over the interval, we need to find the points where the derivative of f(x) is equal to the average rate of change we calculated.