If f is continuous on [a, b] and differentiable on (a, b), then some c inside satisfies f′(c) = (f(b) − f(a)) ÷ (b − a).
Step 1: Let's Learn
Read it, or press Listen and follow the words.
What it says
At some instant the rate of change equals the average rate over the interval.
A concrete reading
Average 70 mph over an hour and at some moment your speedometer read exactly 70. This is how average-speed cameras justify a ticket.
Both hypotheses matter
Continuity on the closed interval and differentiability on the open one. A corner inside breaks the conclusion.
Existence only
The theorem promises that such a c exists. It does not say where, nor how many there are.
Rolle's Theorem
When f(a) = f(b), the average rate is zero, so some c has f′(c) = 0. That special case is Rolle.
Somewhere the instantaneous rate matches the average
If f is continuous on [a, b] and differentiable on (a, b), there is a point where f′ equals the average rate of change over the interval. Drive at an average of 60 mph and at some instant your speedometer read 60.
Both hypotheses must be stated
Continuity on the closed interval and differentiability on the open one. An exam answer invoking the theorem without verifying both loses credit, and the hypotheses genuinely matter — |x| on [−1, 1] is a counterexample.
Rolle's theorem is the special case
If additionally f(a) = f(b), the average rate is zero, so there is a point with a horizontal tangent. Rolle is the mean value theorem with the endpoints level.
What it proves
That a function with zero derivative throughout an interval is constant, and that a positive derivative means increasing. Those facts feel obvious and are proved by the mean value theorem, which is why it matters.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 1)
- Slope of the tangent2
Step 3: Watch an Example
One step at a time.
Watch Diego Locate the Guaranteed Point
Diego applies the theorem to f(x) = x² on [0, 4].
- Step 1
The function is a polynomial, so it is continuous and differentiable everywhere.
Step 4: Your Turn
Practice makes it stick.
The Average Rate
Problem 1 of 2
f(x) = x² on [0, 4]. What is the average rate of change?
The Guaranteed c
Problem 2 of 2
Same function and interval, with f′(x) = 2x. What is c?
Apply the Theorem
1 of 8
f(x) = x² on [0, 6]. Average rate of change?
2 of 8
Same interval with f′(x) = 2x. What is c?
3 of 8
f(x) = x² on [1, 3]. Average rate of change?
4 of 8
f(a) = f(b). What is the average rate of change?
5 of 8
Does the theorem say where c is? 1 yes, 0 no.
6 of 8
A function has a corner inside the interval. Does the theorem apply? 1 yes, 0 no.
7 of 8
Put the application of the theorem in order.
- 1Check differentiability on the open interval.
- 2Compute the average rate of change.
- 3Solve f prime of c equal to that average.
- 4Check continuity on the closed interval.
8 of 8
f(x) = x² on [2, 8]. Average rate of change?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f(x) = x² on [0, 10]. What is the average rate of change?
Question 2 of 2
What does the Mean Value Theorem guarantee?
What You Learned
- Continuity on the closed interval and differentiability on the open one give the theorem.
- Some interior c has f′(c) equal to the average rate of change.
- Rolle's Theorem is the case where the endpoints have equal values.