Mean value theorem
Guarantee a tangent slope equal to the interval’s average rate of change.
Mean value theorem is one of 1 theorems formula in the calculus section of this library, and it is used at ap · university level.
Why mean value theorem works
Tilt the picture until the chord joining the two endpoints is level. A continuous function on a closed interval has to reach a highest and a lowest value, and at an interior high or low point the slope is zero. Tilting back turns that zero slope into the chord's slope.
What each symbol means
$a,b$ are endpoints and $c$ is an interior point.
Mean value theorem: when it holds
$f$ must be continuous on $[a,b]$ and differentiable on $(a,b)$.
When it stops applying
Both hypotheses do real work. The absolute value function on the interval from −1 to 1 has average rate 0 but no interior point with zero slope, because its corner destroys differentiability, and a jump anywhere inside would break it just as surely.
Mean value theorem: a worked example
For $f(x)=x^2$ on $[1,3]$, average slope is $4$, so $c=2$.
The mistake to avoid
What people do: Assuming the guaranteed point is the midpoint of the interval.
Why it goes wrong: That is true for a parabola by coincidence only; for the cubic x³ on the interval from 0 to 3 the average slope is 9 and the point is the square root of 3, about 1.732, not 1.5.
Do this instead: Set the derivative equal to the average rate and solve, then check the solution really lies strictly inside the interval.
Mean value theorem: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a,b$ are endpoints and $c$ is an interior point.
- Check the conditions before substituting. $f$ must be continuous on $[a,b]$ and differentiable on $(a,b)$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Theorems
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where mean value theorem comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about mean value theorem
Is the guaranteed point unique?
Not at all. A wavy function can match the average rate at several interior points, and the theorem only promises that at least one exists.
How does this compare with Rolle's theorem?
Rolle is the special case where the two endpoint values are equal. The average rate is then 0 and the guaranteed tangent is horizontal.
Why is continuity needed on the closed interval but differentiability only on the open one?
The proof only needs the graph to join up at the two ends, while the point it produces always sits strictly inside, where a derivative is required.
What is it actually used to prove?
It is what guarantees that a function whose derivative is zero everywhere must be constant, and it underpins the error bounds used for Taylor polynomials.