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Cogito

AP Calculus BC · Chapter 2 · Lesson 3

Practice Worksheet

Slopes of Polar Curves

Converting to find a tangent.

12 problems · about 21 minutes · CHA-3.G

NameDate

Figure — use these to answer the problems

A polar point is (r cos θ, r sin θ). Treating θ as a parameter is what makes the slope computable.
(0.707, 0.707)
  • Angle45° = π/4 rad
  • x-coordinate0.707
  • y-coordinate0.707
  • cos 45°0.707
  • sin 45°0.707
Angle in degrees45
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. dy/dθ = 15 and dx/dθ = 5. What is dy/dx?

    Answer

  2. What does dr/dθ measure?

    • How fast the curve moves away from the origin.
    • The slope of the tangent line.

  3. For r = 3 at θ = 0, what is the x-coordinate?

    Answer

  4. For r = 3 at θ = 0, what is the y-coordinate?

    Answer

  5. dy/dθ = 6 and dx/dθ = 2. What is dy/dx?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. dy/dθ = 0 and dx/dθ = 4. Horizontal or vertical tangent? 1 horizontal, 2 vertical.

    Answer

  2. dy/dθ = 5 and dx/dθ = 0. Horizontal or vertical? 1 or 2?

    Answer

  3. Which rule is needed to differentiate r cos θ when r depends on θ? 1 product rule, 2 power rule.

    Answer

Stretch Yourself

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

4 problems

  1. Match each quantity with what it measures.

    Draw a line from each item on the left to its match on the right.

    • dr by d theta
    • dy by dx
    • One half r squared integrated
    • How fast the curve leaves the origin
    • The slope of the tangent line
    • The enclosed area

  2. dy/dθ = 12 and dx/dθ = 3. What is dy/dx?

    Answer

  3. The Conversion

    For r = 2 at θ = 0, what is the x-coordinate?

    Answer

  4. The Confusion

    Is dr/dθ the slope of the tangent line? 1 yes, 0 no.

    Answer