Parametric Equations

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# Parametric Equations - PowerPoint PPT Presentation

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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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

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 9.42(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
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.

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