1 / 25

# 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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Parametric Equations' - azize

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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

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

Pole

Polar axis

Figure 9.37.

θ)

x2 + y2 = r2

tan θ = y/x

x = r cos θ

y = r sin θ

Figure 9.40(a-c).

Graph r2 = 4 cos θ

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

Horizontal tangents at:

Vertical tangents at:

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

Figure 9.48.

Figure 9.49.

Figure 9.51.

Figure 9.52.

Find the length of the arc for r = 2 – 2cosθ

sin2A =(1-cos2A)/2

2 sin2A =1-cos2A

2 sin2 (1/2θ) =1-cosθ