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?