The Mean Value Theorem

You’ll be able to

State the hypotheses and conclusion of the Mean Value Theorem
Verify that a function qualifies for the MVT on an interval
Find the guaranteed point where the instantaneous rate equals the average rate

Average rate meets instantaneous rate

The Mean Value Theorem (MVT) guarantees that a smooth function must, at some point, have an instantaneous rate of change equal to its average rate. Formally: if f is continuous on [a, b] and differentiable on (a, b), then there is at least one c in (a, b) with f′(c) = [f(b) − f(a)]/(b − a). Geometrically, the tangent line at c is parallel to the secant line joining the endpoints.

The hypotheses are not optional

The MVT needs both conditions: continuous on the closed interval and differentiable on the open interior. A corner or a break inside the interval voids the guarantee. Rolle’s Theorem is the special case where f(a) = f(b): then the average rate is 0, so some c has f′(c) = 0 — a horizontal tangent. Always check the hypotheses before invoking either theorem.

Mean Value Theorem

f′(c) = [f(b) − f(a)] / (b − a) for some c in (a, b)

Requires f continuous on [a, b] and differentiable on (a, b). The right side is the average rate of change (secant slope).

Worked example

For f(x) = x² on [1, 3], find the value c guaranteed by the MVT.

1.f is a polynomial, so it is continuous on [1, 3] and differentiable on (1, 3) — the hypotheses hold.
2.Average rate of change: [f(3) − f(1)]/(3 − 1) = (9 − 1)/2 = 4.
3.The derivative is f′(x) = 2x; set it equal to the average rate: 2c = 4.
4.Solve: c = 2, which lies in (1, 3) as promised.

Answer: c = 2. The tangent line at x = 2 has slope 4, exactly parallel to the secant from (1, 1) to (3, 9).

Watch out

The MVT gives a c in the open interval (a, b), never at the endpoints. And it fails without differentiability throughout the interior — a function with a corner inside [a, b] is not guaranteed such a c.

Checkpoint

A car travels 120 miles in 2 hours. What does the Mean Value Theorem guarantee about its speed?

Checkpoint

Which condition must hold for the Mean Value Theorem to apply to f on [a, b]?

On the exam

To use the MVT on the exam, first state that f is continuous on [a, b] and differentiable on (a, b), then compute the average rate and set f′(c) equal to it. Solving for c without confirming the hypotheses loses justification credit.

Answer the 2 checkpoints as you read.