MAT 1236Calculus III

Section 10.1

Curves Defined by Parametric equations

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Homework

WebAssign 10.1(9 problems, 45 min.)

Preview

Limitations of Cartesian Equations on describing curves

Parametric Equations

Example 1

:Equation

1

),( yx

Example 1

1

),( yx

We to do if we only want part of the circle?

Example 2

1

),( yx:Equation

Example 3

1

),( yx:Equation

Example 4

14

This curve cannot be defined as functions in or .

Definition

Parametric Equation is the parameter The curve defined by a parametric

equation is called a Parametric Curve

( ),

( )

x f ta t b

y g t

Example 1

1

),( yx

t

:Equation

Example 2

1

)sin,(cos tt:Equation

t increasing of direction

Example 2 (Terminology)

1

point initial

point final

:Equation

Example 3

1

:Equation

Example 4

14

:Equation

Interesting Property

The parametric representation of a curve is not unique.

Example 5a

t

ty

tx

0

sin

cos

1

)sin,(cos tt

Example 5a

t

ty

tx

0

sin

cos

Example 5b

s

sy

sx

0

sin

cos

1

)sin,cos( ss

Example 5b

s

sy

sx

0

sin

cos

Example 5c

1

)2sin,2(cos ss

20

2sin

2cos

s

sy

sx

Example 5c

20

2sin

2cos

s

sy

sx

Conversion

We can convert a parametric equation into the corresponding Cartesian equation by eliminating the parameter.

Pay attention to the range of the parameter.

There are no general “formula” to do this. Below are two examples.

Example 6

cos, 0

sin

x t

y tt

What is the domain of the function?

Example 7

2

1, 0 3

x t

y tt

What is the domain of the function?