Rates of Change in Polar Functions

What it means for r to increase.

12 problems · about 21 minutes · AP Precalculus 3.15

NameDate

Figure — use these to answer the problems

As the angle sweeps, imagine r growing or shrinking. That change is what this lesson measures.

Angle in degrees45

Figure 1

Straightforward practice. Get the method working first.

5 problems

r goes from 3 to 12 as θ goes from 0 to 3. What is the average rate of change?
Answer
2
What does a negative rate of change of r mean?
- The curve is moving toward the origin as the angle increases.
- The angle is decreasing.
r goes from 1 to 5 as θ goes from 0 to 2. Average rate?
Answer
r goes from 9 to 3 as θ goes from 0 to 3. Average rate?
Answer
A positive rate. Toward or away from the origin? 1 toward, 2 away.
Answer

The same ideas with more to keep track of.

3 problems

For r = θ, what is r at θ = 4?
Answer
r = 0 at some angle. Where is the curve then? 1 at the origin, 2 far out.
Answer
r goes from 4 to 4 as θ increases. Average rate?
Answer

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

4 problems

Sort each situation by what the curve is doing.
Write each item under the heading it belongs to: The rate of change of r is positive · r is increasing with theta · The rate of change of r is negative · r is decreasing with theta
Moving away from the origin
Moving toward the origin
r goes from 2 to 10 as θ goes from 0 to 4. Average rate?
Answer
The Rate
r goes from 2 to 8 as θ goes from 1 to 4. What is the average rate of change?
Answer
The Direction
The average rate of change of r is negative. Is the curve moving toward or away from the origin? 1 toward, 2 away.
Answer