Cogito
AP Calculus BC · Chapter 1 · Lesson 2
Calculus of Parametric Curves
Slope and speed without eliminating t.
12 problems · about 22 minutes · CHA-3.G, CHA-6.A
What this lesson teaches
The student computes dy/dx and arc length for parametric curves.
- dy/dx is (dy/dt) ÷ (dx/dt).
- The second derivative divides by dx/dt a second time.
- Speed and arc length both use √((dx/dt)² + (dy/dt)²).
Warm Up
Straightforward practice. Get the method working first.
5 problemsdy/dt = 18 and dx/dt = 6. What is dy/dx?
Answer 3
Why 3.
What is the common error when finding the second derivative?
Answer Forgetting to divide by dx/dt a second time.
Why The second division.
dy/dt = 10, dx/dt = 5. What is dy/dx?
Answer 2
Why 10 ÷ 5.
dx/dt = 6, dy/dt = 8. What is the speed?
Answer 10
Why √100.
x = t², so dx/dt = 2t. What is dx/dt at t = 5?
Answer 10
Why 2 × 5.
Build It Up
The same ideas with more to keep track of.
3 problemsy = t³, so dy/dt = 3t². What is dy/dt at t = 2?
Answer 12
Why 3 × 4.
dx/dt = 0 and dy/dt = 5. Is the tangent vertical or horizontal? 1 vertical, 2 horizontal.
Answer 1
Why No horizontal motion.
dx/dt = 5 and dy/dt = 0. Vertical or horizontal? 1 or 2?
Answer 2
Why No vertical motion.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the process for the second derivative in order.
Answer 1. Compute dy/dx as the ratio of the two rates. 2. Differentiate that expression with respect to t. 3. Divide the result by dx/dt again. 4. Simplify to get the second derivative.
Why The second division is the step people forget.
dx/dt = 5 and dy/dt = 12. What is the speed?
Answer 13
Why √169.
The Slope: dy/dt = 12 and dx/dt = 4. What is dy/dx?
Answer 3
Why 3.
The Speed: dx/dt = 3 and dy/dt = 4. What is the speed?
Answer 5
Why 5.