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Parametric Equations Dr. Dillon Calculus II Spring 2000

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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...