parametric equations t-20123 x0-3-4-305 y-.50.511.5
TRANSCRIPT
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
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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)
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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
( )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
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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