1. Chapter 10 – Parametric Equations & Polar Coordinates 10.2 Calculus with Parametric Curves 1Erickson

2. Calculus with Parametric Curves Erickson10.2 Calculus with Parametric Curves2  Now that we seen how to represent equations with parametric curves we can find tangents, areas, arc lengths and surface areas by applying the methods of calculus.

3. Tangents 10.2 Calculus with Parametric Curves3  We can find the tangents without having to eliminate the parameter using formula 1: Erickson

4. Horizontal Tangents 10.2 Calculus with Parametric Curves4  The curve has a horizontal tangent when  This is provided that Erickson

5. Vertical Tangents 10.2 Calculus with Parametric Curves5  The curve has a vertical tangent when  This is provided that Erickson

6. Tangent NOTES Erickson10.2 Calculus with Parametric Curves6  If we think of a parametric curve as being traced out by moving particles, then dy/dt and dx/dt are the vertical and horizontal velocities of the particle.  Formula 1 says that the slope of the tangent is the ratio of these two velocities.

7. Second derivative 10.2 Calculus with Parametric Curves7  It is also useful to consider the second derivative.  Note that Erickson

8. Example 1 10.2 Calculus with Parametric Curves8  Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. Erickson

9. Example 2 10.2 Calculus with Parametric Curves9  Find dy/dx and d 2 y/dx 2. For which values of t is the curve concave upward? Erickson

10. Example 3 10.2 Calculus with Parametric Curves10  Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. Erickson

11. Example 4 – pg. 651 #30 10.2 Calculus with Parametric Curves11  Find equations of the tangents to the curve that pass through the point (4,3). Erickson

12. Theorem – Arc Length 10.2 Calculus with Parametric Curves12  If a curve C is described by the parametric equations x = f(t), y = g(t),   t  , where f ′ and g′ are continuous on [ ,  ] and C is traversed exactly once as t increases from  to , then the length of C is Erickson

13. Surface Area 10.2 Calculus with Parametric Curves13  We can adapt the surface area formula from section 8.2 to use with parametric equations.  Formula 8.2.5 states that in the case where f is positive and has a continuous derivative, we define the surface area of the surface obtained by rotating the curve y = f (x), a ≤ x ≤ b, about the x-axis as Erickson

14. Surface Area Erickson10.2 Calculus with Parametric Curves14  If the curve given by the parametric equations x = f(t), y = g(t),   t   is rotated about the x-axis, where f ′ and g′ are continuous and g(t) ≥ 0, then the resulting surface is given by

15. Surface Area Erickson10.2 Calculus with Parametric Curves15  The general symbolic formulas from section 8.2 are still valid, but for parametric curves we use

16. Example 5 – pg. 652 #44 10.2 Calculus with Parametric Curves16  Find the exact length of the curve. Erickson

17. Book Resources Erickson10.2 Calculus with Parametric Curves17  Video Examples  Example 2 – pg. 646 Example 2 – pg. 646  Example 3 – pg. 647 Example 3 – pg. 647  Example 5 – pg. 649 Example 5 – pg. 649  More Videos  Arc length parameter Arc length parameter  Surface area of a plane region Surface area of a plane region  Wolfram Demonstrations  Arc Length of a Cycloid Arc Length of a Cycloid