Cogito
AP Calculus BC · Chapter 2 · Lesson 3
Slopes of Polar Curves
Converting to find a tangent.
12 problems · about 21 minutes · CHA-3.G
What this lesson teaches
The student computes dy/dx for a polar curve using the parametric relationship.
- A polar curve becomes parametric through x = r cos θ and y = r sin θ.
- dy/dx is (dy/dθ) ÷ (dx/dθ), and the Product Rule appears because r varies.
- dr/dθ is not the slope of the tangent.
Warm Up
Straightforward practice. Get the method working first.
5 problemsdy/dθ = 15 and dx/dθ = 5. What is dy/dx?
Answer 3
Why 3.
What does dr/dθ measure?
Answer How fast the curve moves away from the origin.
Why The rate of change of distance from the origin.
For r = 3 at θ = 0, what is the x-coordinate?
Answer 3
Why 3 cos 0.
For r = 3 at θ = 0, what is the y-coordinate?
Answer 0
Why 3 sin 0.
dy/dθ = 6 and dx/dθ = 2. What is dy/dx?
Answer 3
Why 6 ÷ 2.
Build It Up
The same ideas with more to keep track of.
3 problemsdy/dθ = 0 and dx/dθ = 4. Horizontal or vertical tangent? 1 horizontal, 2 vertical.
Answer 1
Why No vertical change.
dy/dθ = 5 and dx/dθ = 0. Horizontal or vertical? 1 or 2?
Answer 2
Why No horizontal change.
Which rule is needed to differentiate r cos θ when r depends on θ? 1 product rule, 2 power rule.
Answer 1
Why Two factors, both varying.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each quantity with what it measures.
Answer dr by d theta → How fast the curve leaves the origin; dy by dx → The slope of the tangent line; One half r squared integrated → The enclosed area
Why Only one of these is a slope.
dy/dθ = 12 and dx/dθ = 3. What is dy/dx?
Answer 4
Why 12 ÷ 3.
The Conversion: For r = 2 at θ = 0, what is the x-coordinate?
Answer 2
Why 2.
The Confusion: Is dr/dθ the slope of the tangent line? 1 yes, 0 no.
Answer 0
Why No.