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 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

Arc Length

Polar Coordinate Plane

Polar Coordinates Pole Polar axis Figure 9.37.

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

Symmetries Figure 9.40(a-c).

Figure 9.41(c).

Figure 9.42(a-b). Graph r2 = 4 cos θ

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)

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

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

r = 1 – cosθ

Find Horizontal Tangents

Find Vertical Tangents Horizontal tangents at: Vertical tangents at:

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

Area in the Plane Figure 9.48.

Area of region Figure 9.49.

Find Area of region inside smaller loop Figure 9.51.

Area between curves Figure 9.52.

Figure 9.53.

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

Parametric Equations