1. Section 10.3 – Parametric Equations and Calculus

2. Derivatives of Parametric Equations We must find a way to analyze the curves without having to convert them.

3. Derivatives of Parametric Equations By the Chain Rule. So... OR...

4. Derivatives of Parametric Equations Nothing is new. All results about derivatives from earlier chapters still apply.

5. Example 1 Find dy/dx: Find the critical points. Test the critical points and the endpoints to find the maximum y. txy 0-50 2 0

6. Example 1 (continued) Find d 2 y/dx 2 : Find the critical points of the first derivative. Check to see if there is a sign change in the second derivative. 2.798 0 Find the x and y value:

7. White Board Challenge

8. Example 2 (a) The coordinate(s) where the tangent line is vertical. (b) The coordinate(s) where the tangent line is horizontal. Find dy/dx: This occurs when: Although t=2 makes the denominator 0, t=0 is the only value that satisfies both conditions. This occurs when: Although t=2 makes the numerator 0, t=-2 is the only value that satisfies both conditions.

9. Example 2 The one-sided derivatives are equal and non-infinite. Prove that it is Continuous Since the limits equal the values of the coordinate, the relation is continuous at t =2. The limit exists The point (x,y) for t=2 exists Prove the Right Hand Derivative is the same as the Left Hand Derivative (and non-infinite) Thus the derivative exists, at t =2.

10. Arc Length of Parametric Curves Regular Arc Length Formula.

11. Arc Length of Parametric Curves

12. Example 1 We must find the limits for the integral. For most arc length problems, the calculator needs to evaluate the definite integral.

13. Example 2 Use arc length. Use the Distance Formula Coordinate at t=0: Coordinate at t=4: