Short on time? This page is the recap. For explanations, graphs and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- The Mean Value Theorem (MVT) is an existence theorem. It says a value exists; it does not say where.
- Conditions: f continuous on the closed interval [a, b]; f differentiable on the open interval (a, b).
- Conclusion: at least one c in (a, b) has f′(c) = (f(b) − f(a))/(b − a).
- Picture: a tangent line inside the interval is parallel to the secant line through the endpoints.
- Context: at some moment the instantaneous rate (for example, velocity) equals the average rate over the interval.
- If f(a) = f(b), the conclusion is f′(c) = 0 (Rolle’s theorem, a special case).
Key relationships
| Item | What to write |
|---|---|
| Average rate | (f(b) − f(a))/(b − a), the secant gradient |
| Instantaneous rate | f′(c), the tangent gradient |
| Differentiable on [a, b] | Also continuous on [a, b] (Topic 2.4), so both conditions hold |
| Finding c | Solve f′(x) = average rate; keep only solutions strictly inside (a, b) |
| Table question | Function must be stated differentiable; any sub-interval [p, q] of the table may be used |
Assumptions behind the method
- Continuity is needed at every point of [a, b], including the endpoints.
- Differentiability is needed only inside (a, b). A vertical tangent or a corner at an endpoint is allowed.
- Corners, cusps, vertical tangents, jumps, holes and asymptotes inside the interval break a condition.
The three-part justification
- “f is continuous on [a, b] and differentiable on (a, b) because …”
- “(f(b) − f(a))/(b − a) = …” with the numbers shown.
- “By the Mean Value Theorem, there is a c in (a, b) with f′(c) = … .”
Mistakes to avoid
- Taking c to be the midpoint, or the average rate to be the average of f′(a) and f′(b).
- Keeping a solution at an endpoint or outside the interval.
- Dropping the conditions or the theorem’s name.
- Saying “no such c exists” just because a condition fails. Failure only removes the guarantee.
- Mixing up the MVT and the IVT. The IVT gives values of f; the MVT gives values of f′.
- Forgetting the chain rule when finding f′.
Quick self-check
- Find the value of c guaranteed by the MVT for f(x) = x² + 2x on [0, 2]. (Average rate (8 − 0)/2 = 4; f′(c) = 2c + 2 = 4, so c = 1.)
- For f(x) = sin x on [0, π], what does the MVT give? (Average rate 0, so f′(c) = cos c = 0 with c = π/2.)
- A differentiable function has g(2) = 7 and g(6) = −1. What must be true? (g′(c) = −8/4 = −2 for some c in (2, 6).)
Next: practice questions.