Parametric Equations

2 42

2 3

tx t and y

t

t -2 -1 0 1 2 3

x 0 -3 -4 -3 0 5

y -1 -.5 0 .5 1 1.5

Eliminating the Parameter

2 42

tx t and y

2 2(2 ) 4 4 4x y x y

2 4 2x t and t y

3cos 4sinx and y

2 22 2

2 22 2

2 2

cos sin9 16

cos sin9 16

19 16

x yand

x y

x y

cos sin3 4

x yand

1)

2)

11.2 Slope and Concavity

dydy dt

dxdxdt

2

2

d dyd y dt dx

dxdxdt

For the curve given byFind the slope and concavity at the point (2,3)

214

4x t and y t

3

21

2

12

12

tdyt

dxt

1

22

122

32 312

td yt

dxt

At (2, 3) t = 4 and the slope is 8. The second derivative is positive so graph is concave up

Horizontal and Vertical tangents dydy dt

dxdxdt

A horizontal tangent occurs when dy/dt = 0 but dx/dt 0.

A vertical tangent occurs when dx/dt = 0 but dy/dt 0.3 2 3 22 3 12 2 3 1x t t t and y t t

2 26 6 12 6 6dx dy

t t and t tdt dt

0 1, 2

0 1

dxat t and

dtdy

at tdt

Vertical tangents

Horizontal tangent

Arc Length 2 2dx dy

L dtdt dt

5 2 , 0 1t tx e e and y t t

2 22 2

2

2 4

t t

t t

dx dye e and

dt dt

dx dye e

dt dt

Arc Length 2 2dx dy

L dtdt dt

3

2 2

0

2 4t tL e e dt

3 32 2 2

0 0

333 3

0 0

2 ( )t t t t

t t t t

L e e dt e e dt

e e dt e e e e

543210

Polar Coordinate Plane

Figure 9.37.

Pole Polar axis

Polar Coordinates

Polar/Rectangular Equivalences

x2 + y2 = r2

tan θ = y/x

x = r cos θ y = r sin θ

θ)

Figure 9.40(a-c).

Symmetries

Figure 9.41(c).

Figure 9.42(a-b).Graph r2 = 4 cos θ

Figure 9.45.Finding points of intersection

1 2cos 1

1 1 2cos

3,

2 2

r and r

Third point does not show up.

On r = 1-2 cos θ, point is (-1, 0)

On r = 1, point is (1, π)

43210

Slope of a polar curve

• Where x = r cos θ = f(θ) cos θ

• And y = r sin θ = f(θ) sin θ

( )sin ( )cos

( )cos ( )sin

dydy d

dd fx

f fx

df

Horizontal tangent where dy/dθ = 0 and dx/dθ≠0

Vertical tangent where dx/dθ = 0 and dy/dθ≠0

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

43210

0

41 cos ( )

( )sin ( )cos

( )cos ( )sin

dydy d

dd fx

f fx

df

2

(1 cos )cos

cos cos

sin 2cos sin

x

x

dx

d

2 2

(1 cos )sin

sin sin cos

cos sin cos

y

y

dy

d

r = 1 – cos θ

2 2

6

1 3 1 1 3sin 2cos sin 2 *

6 6 6 2 2 2 2 2

3 1 3 3 1cos sin cos

6 6 6 2 4 4 2 2

at

dx

d

dy

d

6

1

at

dy

dx

Find Horizontal Tangents

2 2

2 2

cos sin cos

cos sin cos 0 ( 0)

dy

ddx

andd

2 2

2

cos 1 cos cos 0

0 2cos cos 1

0 (2cos 1)(cos 1)

2 4, ,0,2

3 3

2 40 , ,0,2

3 3

dyat

d

Find Vertical Tangents

sin 2cos sin

sin 2cos sin 0 0

sin ( 1 2cos ) 0

50, ,2 , ,

3 3

dx

ddy

andd

2 4, ,0,2

3 3

Horizontal tangents at:

Vertical tangents at:

2

0 0 2 2

cos 1 2cos sin 4cos sinlim lim 0

sin 2cos sin cos 2cos 2sin

5, ,

3 3

43210

2 40 , ,0,2

3 3

dyat

d

Figure 9.47.Finding Tangent Lines at the poler = 2 sin 3θ

( )sin ( ) cos

( )cos ( )sin

( )sintan

( )cos

f f

f f

dy f

f

d

d

dx

y

x

r = 2 sin 3θ = 03θ = 0, π, 2 π, 3 π θ = 0, π/3, 2 π/3, π 1tan

0

3

3

y x

y

y x

y x

Figure 9.48.

Area in the Plane

Figure 9.49.Area of region

221 1( )

2 2A r d f d

Figure 9.51.Find Area of region inside smaller loop2cos 1 0

2 4,

3 3

4

322

2 2

3 3

12cos 1

2A r d d

2

23

4cos 4cos 1A d

2 2

3 3

4cos 1 3 2cos2 4(2c 2 2 os co s)A d d

2

3

3 4 3 3 33 sin 2 4sin (3 ) (2 )

2 2 2

Figure 9.52.

Area between curves2 2

2 11

( )2

A r r d

Figure 9.53.1 cos 1

cos 0

,2 2

2 2 22 2 2

0 0 0

1 1((1) (1 cos ) ) (2cos cos ) (2cos cos2 )

2 2A d d d

2

0

1 12sin sin 2 2

2 4 4

Length of a Curve in Polar Coordinates

2 2( ) ( )L f f d

2

2 2

0

2

0

2 2cos 2sin

2 2 1 cos

L d

d

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

2 22

0 0

2 2 2sin 4 sin2 2

d d

2

0

8 cos 8(1 1) 162

sin2A =(1-cos2A)/22 sin2A =1-cos2A2 sin2 (1/2θ) =1-cosθ