5.1 The Mean Value Theorem

The theorem

If f is continuous on [a, b] and differentiable on (a, b), then there is some c in (a, b) with f'(c) = (f(b) − f(a))/(b − a) — the instantaneous rate equals the average rate. Geometrically, some tangent line is parallel to the secant line.

Rolle's Theorem

Rolle's Theorem is the special case where f(a) = f(b): then some c has f'(c) = 0 (a horizontal tangent).

Key terms

Mean Value Theorem

Some c has f'(c) = (f(b) − f(a))/(b − a).

Hypotheses

Continuous on [a, b], differentiable on (a, b).

Geometric meaning

A tangent parallel to the secant line.

Rolle's Theorem

If f(a) = f(b), some c has f'(c) = 0.

Use

Guarantees a point where instantaneous = average rate.

Try an AP-style question

For f(x) = x² on [1, 3], what value c satisfies the Mean Value Theorem?

1. √5
2. 4
3. 1.5
4. 2