1579 parametric equations
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Parametric Equations
Dr. DillonCalculus IISpring 2000
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Introduction
Some curves in the plane can be described as functions.
)(xfy
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Others...
cannot be described as functions.
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Ways to Describe a Curve in the Plane
An equation in two variables
This equation describes a
circle.
086222 yxyxExample:
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A Polar Equationr
This polar equation describes a double spiral.
We’ll study polar curves later.
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Parametric Equations
Example:
122
tyttx
The “parameter’’ is t.
It does not appear in the graph of the curve!
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Why?
The x coordinates of points on the curve are given by a function.
ttx 22 The y coordinates of points on the
curve are given by a function.
1ty
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Two Functions, One Curve?
Yes.
then in the xy-plane the curve looks like this, for values of t from 0 to 10...
1 and 22 tyttxIf
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Why use parametric equations?
• Use them to describe curves in the plane when one function won’t do.
• Use them to describe paths.
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Paths?
A path is a curve, together with a journey traced along the curve.
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Huh?
When we write
122
tyttx
we might think of x as the x-coordinate of the position on the path at time t
and y as the y-coordinate of the position on the path at time t.
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From that point of view...
The path described by
122
tyttx
is a particular route along the curve.
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As t increases from 0, x first decreases,
Path moves left!
then increases. Path moves right!
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More Paths
To designate one route around the unit circle use
)sin()cos(
tytx
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counterclockwise from (1,0).
That Takes Us...
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Where do you get that?
Think of t as an angle.
2 ,If it starts at zero, and increases to
then the path starts at t=0, where
cos(0) 1, and sin(0) 0.x y
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To start at (0,1)...
Use
)cos()sin(tytx
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That Gives Us...
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How Do You Find The Path
• Plot points for various values of t, being careful to notice what range of values t should assume
• Eliminate the parameter and find one equation relating x and y
• Use the TI82/83 in parametric mode
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Plotting Points
• Note the direction the path takes• Use calculus to find
– maximum points– minimum points– points where the path changes direction
• Example: Consider the curve given by
2 1, 2 , 5 5x t y t t
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Consider
• The parameter t ranges from -5 to 5 so the first point on the path is (26, -10) and the last point on the path is (26, 10)
• x decreases on the t interval (-5,0) and increases on the t interval (0,5). (How can we tell that?)
• y is increasing on the entire t interval (-5,5). (How can we tell that?)
2 1, 2 , 5 5x t y t t
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Note Further
• x has a minimum when t=0 so the point farthest to the left on the path is (1,0).
• x is maximal at the endpoints of the interval [-5,5], so the points on the path farthest to the right are the starting and ending points, (26, -10) and (26,10).
• The lowest point on the path is (26,-10) and the highest point is (26,10).
2 1, 2 , 5 5x t y t t
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Eliminate the Parameter
2 2( / 2) 1 or ( / 4) 1x y x y
2 1, 2 , 5 5x t y t t Still useSolve one of the equations for t
Here we get t=y/2Substitute into the other equation
Here we get
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Using the TI82
• Change mode to PAR (third row)• Mash y= button• Enter x as a function of t, hit enter• Enter y as a function of t, hit enter• Check the window settings, after
determining the maximum and minimum values for x and y
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Some Questions
• What could you do in the last example to reverse the direction of the path?
• What could you do to restrict or to enlarge the path in the last example?
• How can you cook up parametric equations that will describe a path along a given curve? (See the cycloid on the Web.)
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Web Resources
• MathView Notebook on your instructor’s site (Use Internet Explorer to avoid glitches!)
• IES Web
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Summary
• Use parametric equations for a curve not given by a function.
• Use parametric equations to describe paths.• Each coordinate requires one function.• The parameter may be time, angle, or
something else altogether...