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

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Eliminating the Parameter 1) 2)

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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

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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

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Arc Length

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Polar Coordinate Plane

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Figure 9.37. Pole Polar axis Polar Coordinates

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

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Figure 9.40(a-c). Symmetries

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Figure 9.41(c).

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Figure 9.42(a-b). Graph r 2 = 4 cos θ

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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)

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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

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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

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r = 1 – cos θ

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Find Horizontal Tangents

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Find Vertical Tangents Horizontal tangents at: Vertical tangents at:

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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

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Figure 9.48. Area in the Plane

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Figure 9.49. Area of region

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Figure 9.51. Find Area of region inside smaller loop

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Figure 9.52. Area between curves

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Figure 9.53.

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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

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P OLAR E QUATIONS Section 10-4. Polar Coordinates Given: r: Directed distance from the Polar axis (pole) to point P Ɵ: Directed angle from the Polar axis.

P OLAR E QUATIONS Section 10-4. Polar Coordinates Given: r: Directed distance from the Polar axis (pole) to point P Ɵ: Directed angle from the Polar axis.

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