The Mean Value Theorem

Somewhere the instantaneous rate matches the average.

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

NameDate

Figure — use these to answer the problems

Move the tangent point until its slope matches the line joining two points on the curve. The theorem says you always can.

y = 1x² + 0x + 0

Point of tangency1

Figure 1

Straightforward practice. Get the method working first.

5 problems

f(x) = x² on [0, 10]. What is the average rate of change?
Answer
2
What does the Mean Value Theorem guarantee?
- Some point exists where the instantaneous rate equals the average rate.
- Exactly where that point is.
f(x) = x² on [0, 6]. Average rate of change?
Answer
Same interval with f′(x) = 2x. What is c?
Answer
f(x) = x² on [1, 3]. Average rate of change?
Answer

The same ideas with more to keep track of.

3 problems

f(a) = f(b). What is the average rate of change?
Answer
Does the theorem say where c is? 1 yes, 0 no.
Answer
A function has a corner inside the interval. Does the theorem apply? 1 yes, 0 no.
Answer

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

4 problems

9
Put the application of the theorem in order.
Write 1 to 4 in the boxes to put these in order.
- Check continuity on the closed interval.
- Check differentiability on the open interval.
- Compute the average rate of change.
- Solve f prime of c equal to that average.
f(x) = x² on [2, 8]. Average rate of change?
Answer
The Average Rate
f(x) = x² on [0, 4]. What is the average rate of change?
Answer
The Guaranteed c
Same function and interval, with f′(x) = 2x. What is c?
Answer