1. Parametric Equations t-20123 x0-3-4-305 y-.50.511.5

2. Eliminating the Parameter 1) 2)

3. 11.2 Slope and Concavity For the curve given by Find the slope and concavity at the point (2,3) At (2, 3) t = 4 and the slope is 8. The second derivative is positive so graph is concave up

4. Horizontal and Vertical tangents A horizontal tangent occurs when dy/dt = 0 but dx/dt  0. A vertical tangent occurs when dx/dt = 0 but dy/dt  0. Vertical tangents Horizontal tangent

5. Arc Length

7. Polar Coordinate Plane

8. Figure 9.37. Pole Polar axis Polar Coordinates

9. Polar/Rectangular Equivalences x 2 + y 2 = r 2 tan θ = y/x x = r cos θ y = r sin θ θ)

10. Figure 9.40(a-c). Symmetries

11. Figure 9.41(c).

12. Figure 9.42(a-b). Graph r 2 = 4 cos θ

13. Figure 9.45. Finding points of intersection Third point does not show up. On r = 1-2 cos θ, point is (-1, 0) On r = 1, point is (1, π)

14. Slope of a polar curve Where x = r cos θ = f(θ) cos θ And y = r sin θ = f(θ) sin θ Horizontal tangent where dy/dθ = 0 and dx/dθ≠0 Vertical tangent where dx/dθ = 0 and dy/dθ≠0

15. Finding slopes and horizontal and vertical tangent lines For r = 1 – cos θ (a) Find the slope at θ = π/6 (b) Find horizontal tangents (c) Find vertical tangents

16. r = 1 – cos θ

17. Find Horizontal Tangents

18. Find Vertical Tangents Horizontal tangents at: Vertical tangents at:

19. Figure 9.47. Finding Tangent Lines at the pole r = 2 sin 3θ r = 2 sin 3θ = 0 3θ = 0, π, 2 π, 3 π θ = 0, π/3, 2 π/3, π

20. Figure 9.48. Area in the Plane

21. Figure 9.49. Area of region

22. Figure 9.51. Find Area of region inside smaller loop

23. Figure 9.52. Area between curves

24. Figure 9.53.

25. Length of a Curve in Polar Coordinates Find the length of the arc for r = 2 – 2cosθ sin 2 A =(1-cos2A)/2 2 sin 2 A =1-cos2A 2 sin 2 (1/2θ) =1-cosθ