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Parametric Equations

Parametric Equations. Eliminating the Parameter. 1). 2). 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. Horizontal and Vertical tangents.

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Parametric Equations

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  1. Parametric Equations

  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

  6. Arc Length

  7. Polar Coordinate Plane

  8. Polar Coordinates Pole Polar axis Figure 9.37.

  9. Polar/Rectangular Equivalences θ) x2 + y2 = r2 tan θ = y/x x = r cos θ y = r sin θ

  10. Symmetries Figure 9.40(a-c).

  11. Figure 9.41(c).

  12. Figure 9.42(a-b). Graph r2 = 4 cos θ

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

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

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

  16. r = 1 – cosθ

  17. Find Horizontal Tangents

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

  19. Finding Tangent Lines at the pole Figure 9.47. 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. Area in the Plane Figure 9.48.

  21. Area of region Figure 9.49.

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

  23. Area between curves Figure 9.52.

  24. Figure 9.53.

  25. Length of a Curve in Polar Coordinates Find the length of the arc for r = 2 – 2cosθ sin2A =(1-cos2A)/2 2 sin2A =1-cos2A 2 sin2 (1/2θ) =1-cosθ

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