The average rate of change of r with respect to θ says how fast the curve moves away from the origin as it turns.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
A positive rate
Positive means r is increasing, so the curve is spiralling outward over that interval.
A negative rate
Negative means r is decreasing, so the curve is moving toward the origin as the angle grows.
It is not the speed along the curve
It measures only the change in distance from the origin, not how fast the point travels.
Where r is zero
The curve passes through the origin wherever r = 0. On a rose, those are the points where petals meet.
Sign changes of r
When r changes sign, the curve crosses the origin and continues on the opposite side.
What it means for r to increase
If r increases as θ increases, the curve is moving away from the origin as it rotates. Describing that behaviour over an interval of θ is a distinctive AP Precalculus task with no equivalent in older courses.
Average rate of change of r
The change in r divided by the change in θ measures how quickly the distance from the origin is changing per unit of rotation. Its units are distance per radian, which is worth stating.
r can decrease through zero
When r passes through zero the curve goes through the origin and continues on the opposite side. Tracking the sign of r is what explains why some polar curves have loops.
This is not the speed along the curve
A point can move quickly along a polar curve while r barely changes, if most of the motion is rotational. The rate of change of r describes only the radial component, not the total motion.
Step 2: Try It Yourself
Tap and try it out.
- Angle45° = π/4 rad
- x-coordinate0.707
- y-coordinate0.707
- cos 45°0.707
- sin 45°0.707
The cosine is the x-coordinate and the sine is the y-coordinate. They are not merely equal — they are the same two numbers, which is why angles past 90° cause no trouble.
Step 3: Watch an Example
One step at a time.
Watch Rosa Compute an Average Rate
For r = θ, Rosa examines the interval from θ = 1 to θ = 3.
- Step 1
At θ = 1 the distance is r = 1.
Step 4: Your Turn
Practice makes it stick.
The Rate
Problem 1 of 2
r goes from 2 to 8 as θ goes from 1 to 4. What is the average rate of change?
The Direction
Problem 2 of 2
The average rate of change of r is negative. Is the curve moving toward or away from the origin? 1 toward, 2 away.
Toward or Away
1 of 8
r goes from 1 to 5 as θ goes from 0 to 2. Average rate?
2 of 8
r goes from 9 to 3 as θ goes from 0 to 3. Average rate?
3 of 8
A positive rate. Toward or away from the origin? 1 toward, 2 away.
4 of 8
For r = θ, what is r at θ = 4?
5 of 8
r = 0 at some angle. Where is the curve then? 1 at the origin, 2 far out.
6 of 8
r goes from 4 to 4 as θ increases. Average rate?
7 of 8
Sort each situation by what the curve is doing.
Tap something to move it.
- Empty
- Empty
8 of 8
r goes from 2 to 10 as θ goes from 0 to 4. Average rate?
Step 5: Quick Check
Show what you know.
Question 1 of 2
r goes from 3 to 12 as θ goes from 0 to 3. What is the average rate of change?
Question 2 of 2
What does a negative rate of change of r mean?
What You Learned
- The rate of change of r against θ says how fast the curve leaves the origin.
- Positive means moving outward and negative means moving inward.
- Where r is zero, the curve passes through the origin.