Parametric Equations

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Parametric Equations
t -2 -1 0 1 2 3
x 0 -3 -4 -3 0 5
y -1 -.5 0 .5 1 1.5
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Eliminating the Parameter
1)
2)
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11.2 Slope and Concavity
At (2, 3) t 4 and the slope is 8. The second
derivative is positive so graph is concave up
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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
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Arc Length
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Arc Length
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Polar Coordinate Plane
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Polar Coordinates
Figure 9.37.
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Polar/Rectangular Equivalences
x2 y2 r2 tan ? y/x
x r cos ? y r sin ?
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Symmetries
Figure 9.40(a-c).
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Figure 9.41(c).
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Figure 9.42(a-b).
Graph r2 4 cos ?
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Finding points of intersection
Figure 9.45.
Third point does not show up.
On r 1, point is (1, p)
On r 1-2 cos ?, point is (-1, 0)
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Slope of a polar curve

• Where x r cos ? f(?) cos ?
• And y r sin ? f(?) sin ?

Horizontal tangent where dy/d? 0 and dx/d??0
Vertical tangent where dx/d? 0 and dy/d??0
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Finding slopes and horizontal and vertical
tangent lines

• For r 1 cos ?
• (a) Find the slope at ? p/6
• (b) Find horizontal tangents
• (c) Find vertical tangents

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r 1 cos ?
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Find Horizontal Tangents
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Find Vertical Tangents
Horizontal tangents at
Vertical tangents at
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Finding Tangent Lines at the pole
Figure 9.47.
r 2 sin 3?
r 2 sin 3? 0 3? 0, p, 2 p, 3 p ? 0,
p/3, 2 p/3, p
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Area in the Plane
Figure 9.48.
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Area of region
Figure 9.49.
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Find Area of region inside smaller loop
Figure 9.51.
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Area between curves
Figure 9.52.
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Figure 9.53.
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Length of a Curve in Polar Coordinates
Find the length of the arc for r 2 2cos?
sin2A (1-cos2A)/2 2 sin2A 1-cos2A 2 sin2 (1/2?)
1-cos?