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Cogito

AP Calculus BC · Chapter 1 · Lesson 2

Practice Worksheet

Calculus of Parametric Curves

Slope and speed without eliminating t.

12 problems · about 22 minutes · CHA-3.G, CHA-6.A

NameDate

Figure — use these to answer the problems

A parametric curve has a tangent at each point, found by dividing the two rates rather than eliminating t.
-8-8-6-6-4-4-2-222446688
y = 1x² + 0x + 0
  • Point(2, 4)
  • Slope of the tangent4
a1
Point of tangency2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. dy/dt = 18 and dx/dt = 6. What is dy/dx?

    Answer

  2. What is the common error when finding the second derivative?

    • Forgetting to divide by dx/dt a second time.
    • Getting the sign wrong.

  3. dy/dt = 10, dx/dt = 5. What is dy/dx?

    Answer

  4. dx/dt = 6, dy/dt = 8. What is the speed?

    Answer

  5. x = t², so dx/dt = 2t. What is dx/dt at t = 5?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. y = t³, so dy/dt = 3t². What is dy/dt at t = 2?

    Answer

  2. dx/dt = 0 and dy/dt = 5. Is the tangent vertical or horizontal? 1 vertical, 2 horizontal.

    Answer

  3. dx/dt = 5 and dy/dt = 0. Vertical or horizontal? 1 or 2?

    Answer

Stretch Yourself

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

4 problems

  1. Put the process for the second derivative in order.

    Write 1 to 4 in the boxes to put these in order.

    • Compute dy/dx as the ratio of the two rates.
    • Differentiate that expression with respect to t.
    • Divide the result by dx/dt again.
    • Simplify to get the second derivative.

  2. dx/dt = 5 and dy/dt = 12. What is the speed?

    Answer

  3. The Slope

    dy/dt = 12 and dx/dt = 4. What is dy/dx?

    Answer

  4. The Speed

    dx/dt = 3 and dy/dt = 4. What is the speed?

    Answer