Rates of Change in Polar Functions

What it means for r to increase.

12 problems · about 21 minutes · AP Precalculus 3.15

What this lesson teaches

The student analyses average rates of change of r with respect to θ and interprets them.

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 3
Why 3.
What does a negative rate of change of r mean?
Answer The curve is moving toward the origin as the angle increases.
Why The distance is shrinking.
r goes from 1 to 5 as θ goes from 0 to 2. Average rate?
Answer 2
Why 4 ÷ 2.
r goes from 9 to 3 as θ goes from 0 to 3. Average rate?
Answer -2
Why −6 ÷ 3.
A positive rate. Toward or away from the origin? 1 toward, 2 away.
Answer 2
Why r is growing.

The same ideas with more to keep track of.

3 problems

For r = θ, what is r at θ = 4?
Answer 4
Why They are equal.
r = 0 at some angle. Where is the curve then? 1 at the origin, 2 far out.
Answer 1
Why Zero distance.
r goes from 4 to 4 as θ increases. Average rate?
Answer 0
Why No change.

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

4 problems

Sort each situation by what the curve is doing.
Answer Moving away from the origin: The rate of change of r is positive, r is increasing with theta · Moving toward the origin: The rate of change of r is negative, r is decreasing with theta
Why The sign of the rate says everything.
r goes from 2 to 10 as θ goes from 0 to 4. Average rate?
Answer 2
Why 8 ÷ 4.
The Rate: r goes from 2 to 8 as θ goes from 1 to 4. What is the average rate of change?
Answer 2
Why 2.
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 1
Why Toward.