Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. (Enter your answers as a comma-separated list.)

Question

f(x) = x², [0, 2]

C=?

Answer 1

f(x) continuous on [0,2] and differentiable on (0,2) so the MVT applies here.

We calculate the average rate of change on the closed interval by using (y₂ - y₁) / (x₂ - x₁) as follows:

f(0) = 0 , f(2) = 4 ; Avg rate of change of f on [0,2] = (4-0) / (2-0) = 2.

To find the c-value guaranteed by the MVT, such that the instantaneous rate of change when x = c is 2, we differentiate, set f^'(x) = 2, and solve:

f^'(x) = 2x = 2 and x = 1.

c = 1 is the only c-value in the interval at which the instantaneous rate of change = the average rate of change.