Math · Calculus I · Concept
The Mean Value Theorem and Rolle’s theorem
The Mean Value Theorem (MVT) says that a function’s average rate of change over an interval is matched, somewhere inside, by its instantaneous rate: some tangent line is parallel to the secant line. Rolle’s theorem is the case of equal endpoint values. The theorem explains why f′ = 0 means constant and f′ > 0 means increasing, and it bounds how much a function can change.
The statement
If f is continuous on [a, b] and differentiable on (a, b), there is at least one c in (a, b) where the derivative equals the average rate of change.
A tangent parallel to the secant
The right side is the slope of the secant line through (a, f(a)) and (b, f(b)), so some tangent between them has the same slope. For motion: if a trip averages 60 km/h, the speedometer read exactly 60 km/h at least once.
Rolle’s theorem
When f(a) = f(b), the secant is horizontal, so some c has f′(c) = 0: the graph has a horizontal tangent between two equal heights. Tilting the picture turns Rolle’s theorem into the Mean Value Theorem, which is how the theorem is proved.
Why the conditions matter
|x| on [−1, 1] has equal endpoint values but no horizontal tangent, because it is not differentiable at 0. A function with a break can fail as well. When a condition fails, the conclusion may still hold or may not; the theorem simply makes no promise.
Consequences
If f′(x) = 0 on an interval, f is constant there, and if f′(x) > 0, f is increasing. Two functions with the same derivative differ by a constant, which is why every antiderivative carries + C.
Bounding change
If |f′(x)| ≤ M on [a, b], then f cannot change by more than M times the length of the interval: a speed limit caps the distance traveled.
Common mistakes
- Skipping the conditions: f must be continuous on [a, b] and differentiable on (a, b).
- Giving a c outside (a, b): the theorem promises a point strictly between the endpoints.
- Expecting exactly one c: there may be several.
- Confusing the average rate of change, (f(b) − f(a))/(b − a), with the average of the function’s values over the interval.
Key terms
- Mean Value Theorem
- If f is continuous on [a, b] and differentiable on (a, b), then f′(c) = (f(b) − f(a))/(b − a) for at least one c in (a, b): some tangent is parallel to the secant through the endpoints.
- Rolle’s theorem
- The Mean Value Theorem when f(a) = f(b): a function continuous on [a, b] and differentiable on (a, b) with equal endpoint values has f′(c) = 0 for some c in (a, b).
- Average rate of change
- The change in output divided by the change in input between two distinct points. Geometrically it is the slope of the secant line.
- Secant line
- A line through two points on a curve; its slope is the average rate of change between them. In trigonometry, “secant” also names the function sec θ = 1/cos θ.
- Tangent line
- The line that matches a curve’s direction at a point, with slope equal to the derivative there. It can cross the curve; in trigonometry, “tangent” also names tan θ = sin θ/cos θ.
- Differentiable
- Having a derivative at a point. A function is continuous wherever it is differentiable, but not the other way round: |x| is continuous at x = 0 with no derivative there.
Work through an example
Find every c in (0, 2) that satisfies the Mean Value Theorem for f(x) = x³ − x on [0, 2].
Find the c the Mean Value Theorem promises →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
See the parallel lines in Graph Trace the slope in Derivative Tracer Check the derivative in Math Open worked example on a board Mean Value Theorem in Math ReferenceYour existing work stays on this device. Examples open as editable copies.