Mathfolis

Mean Value Theorem

Unit 5 · Analytical Applications of Differentiation

What is the Mean Value Theorem?

The Mean Value Theorem (MVT) states: if f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c ∈ (a, b) where the instantaneous rate of change equals the average rate of change over the interval.

Mean Value Theorem

There exists c ∈ (a, b) such that
f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}

Rolle's Theorem (Special Case)

If f(a) = f(b), then there exists c ∈ (a, b) such that
f′(c)=0f'(c) = 0
AP Tip: MVT guarantees existence but does not give a unique c. Always verify both conditions (continuity on [a, b] and differentiability on (a, b)) before applying MVT.
Type 1

Verify & Apply MVT

Check that f is continuous on [a, b] and differentiable on (a, b), then find all c in (a, b) where the instantaneous rate equals the average rate of change.

Example 1
Verify that the Mean Value Theorem applies, then find all values of c ∈ (−1, 2) that satisfy the conclusion of the MVT.
f(x)=x3−2x,[−1, 2]f(x) = x^3 - 2x, \quad [-1,\, 2]
Example 2
Verify that the Mean Value Theorem applies, then find all values of c in (1, 9) that satisfy the conclusion of the MVT.
f(x)=x,[1, 9]f(x) = \sqrt{x}, \quad [1,\, 9]

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Type 2

Rolle's Theorem

When f(a) = f(b), MVT guarantees at least one c in (a, b) where f′(c) = 0 — a horizontal tangent is guaranteed somewhere between the equal endpoints.

Example 3
Verify that Rolle's Theorem applies, then find all values of c ∈ (1, 3) where f′(c) = 0.
f(x)=x2−4x+3,[1, 3]f(x) = x^2 - 4x + 3, \quad [1,\, 3]

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Type 3

MVT in Context

In word problems involving position, velocity, or other rates, MVT guarantees there is a moment when the instantaneous rate equals the average rate over the interval.

Example 4
A car travels 120 miles in 2 hours. The car's velocity function v(t) is continuous and differentiable on [0, 2]. What does the Mean Value Theorem guarantee about the car's instantaneous speed at some moment during the trip?

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Mean Value Theorem | AP Calculus AB — Mathfolis