The Mean Value Theorem

Somewhere the instantaneous rate matches the average.

12 problems · about 21 minutes · FUN-1.B

What this lesson teaches

The student states and applies the Mean Value Theorem and verifies its hypotheses.

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.

Straightforward practice. Get the method working first.

5 problems

f(x) = x² on [0, 10]. What is the average rate of change?
Answer 10
Why 10.
What does the Mean Value Theorem guarantee?
Answer Some point exists where the instantaneous rate equals the average rate.
Why Existence of such a point.
f(x) = x² on [0, 6]. Average rate of change?
Answer 6
Why 36 ÷ 6.
Same interval with f′(x) = 2x. What is c?
Answer 3
Why 2c = 6.
f(x) = x² on [1, 3]. Average rate of change?
Answer 4
Why (9 − 1) ÷ 2.

The same ideas with more to keep track of.

3 problems

f(a) = f(b). What is the average rate of change?
Answer 0
Why The numerator is zero.
Does the theorem say where c is? 1 yes, 0 no.
Answer 0
Why It only promises existence.
A function has a corner inside the interval. Does the theorem apply? 1 yes, 0 no.
Answer 0
Why Differentiability fails.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Put the application of the theorem in order.
Answer 1. Check continuity on the closed interval. 2. Check differentiability on the open interval. 3. Compute the average rate of change. 4. Solve f prime of c equal to that average.
Why The hypotheses are verified before anything is computed.
f(x) = x² on [2, 8]. Average rate of change?
Answer 10
Why (64 − 4) ÷ 6.
The Average Rate: f(x) = x² on [0, 4]. What is the average rate of change?
Answer 4
Why 4.
The Guaranteed c: Same function and interval, with f′(x) = 2x. What is c?
Answer 2
Why 2.