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The Mean Value
Theorem

Discover how instantaneous rate of change must equal average rate of change at some point: the bridge between a function's local and global behavior.

AP Calculus BC

What Is Average Rate of Change?

For a function \( f \) on an interval \([a, b]\), the average rate of change is the slope of the secant line connecting the endpoints:

Think of driving from city A to city B: your average speed is total distance divided by total time. The MVT says at some moment you were going exactly that average speed.

Visual Intuition

x y a b c Secant line Tangent at c

Why Does the MVT Matter?

Proves Theorems
Underpins results like the Fundamental Theorem of Calculus and Taylor's Theorem.
Speed Traps
If your average speed exceeds the limit, you must have been speeding at some instant.
Monotonicity
Proves that \( f'(x) > 0 \) on an interval implies \( f \) is increasing there.

Rolle's Theorem: The Special Case

Rolle's Theorem is the MVT when \( f(a) = f(b) \). If the endpoints are at the same height, there must be a point \( c \) where \( f'(c) = 0 \): a horizontal tangent.

a b c
\(f(a) = f(b)\)
\(f'(c) = 0\)

The Mean Value Theorem: Formal Statement

1
Continuous on \([a, b]\): No breaks, jumps, or holes on the closed interval.
2
Differentiable on \((a, b)\): Smooth (no corners or vertical tangents) on the open interval.
3
Conclusion: At some interior point, the instantaneous slope equals the average slope.

Checking the Hypotheses

Continuous
on \([a, b]\)
 Differentiable
on \((a, b)\)
There exists \(c\)
\(f'(c) = \text{avg. slope}\)
 Fails if not continuous Fails if not differentiable corner Both hold: MVT applies

MVT vs Rolle's Theorem

Mean Value Theorem Rolle's Theorem
Extra Condition None beyond continuity + differentiability \( f(a) = f(b) \)
Conclusion \( f'(c) = \dfrac{f(b)-f(a)}{b-a} \) \( f'(c) = 0 \)
Secant Slope Any value Zero (horizontal)
Relationship General case Special case of MVT

How to Apply the MVT

1
Verify hypotheses: Confirm \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\).
2
Compute the average slope: Calculate \( \dfrac{f(b) - f(a)}{b - a} \).
3
Find the derivative: Compute \( f'(x) \).
4
Solve for \( c \): Set \( f'(c) = \dfrac{f(b) - f(a)}{b - a} \) and solve. Keep only solutions in \((a, b)\).

Example 1: \( f(x) = x^2 \) on \([1, 3]\)

1 2 3 (1, 1) (3, 9)

Example 1: Visual Confirmation

1 2 3 (1, 1) (3, 9) c = 2

Example 2: \( f(x) = \sin x \) on \([0, \pi]\)

Example 3: \( f(x) = x^3 - 3x \) on \([-1, 2]\)

When the MVT Does Not Apply

Not Continuous
\( f(x) = \dfrac{1}{x} \) on \([-1, 1]\) has a discontinuity at \( x = 0 \). MVT does not apply.
Not Differentiable
\( f(x) = |x| \) on \([-1, 1]\) has a corner at \( x = 0 \). MVT does not apply.
\(f(x) = \frac{1}{x}\)
\(f(x) = |x|\)
 corner

Common Mistakes

✕ Forgetting Hypotheses
Applying MVT without first checking that \( f \) is continuous on \([a,b]\) and differentiable on \((a,b)\).
✕ Including Endpoints
The value \( c \) must lie in the open interval \((a,b)\). Endpoints do not count as valid solutions.
✕ Finding the Wrong Value
MVT guarantees existence of \( c \). On the AP exam, you may need to find it, but don't forget to verify \( c \in (a,b) \).

Summary: Key Formulas

Concept Formula / Statement
Average Rate of Change \( \dfrac{f(b) - f(a)}{b - a} \)
MVT Conclusion \( f'(c) = \dfrac{f(b) - f(a)}{b - a} \) for some \( c \in (a,b) \)
Rolle's Theorem MVT with \( f(a) = f(b) \implies f'(c) = 0 \)
Hypotheses Continuous on \([a,b]\), differentiable on \((a,b)\)
What MVT proves Instantaneous rate = average rate at some interior point

Question 1 of 5

Let \( f(x) = x^2 + 2x \) on the interval \([1, 4]\). What value of \( c \) satisfies the Mean Value Theorem?

Question 2 of 5

Which of the following is a necessary hypothesis of the Mean Value Theorem?

Question 3 of 5

The MVT cannot be applied to \( f(x) = |x - 1| \) on \([0, 3]\). Why?

Question 4 of 5

If \( f \) is continuous on \([2, 6]\), differentiable on \((2, 6)\), and \( f(2) = 3 \), \( f(6) = 11 \), the MVT guarantees the existence of \( c \in (2, 6) \) such that \( f'(c) \) equals:

Question 5 of 5

Rolle's Theorem is a special case of the MVT. What additional condition does Rolle's Theorem require beyond the standard MVT hypotheses?