Calculus 1.4

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1998 AB Exam
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1.4 Parametric Equations
Mt. Washington Cog Railway, NH
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There are times when we need to describe motion
(or a curve) that is not a function.
These are called parametric equations.
t is the parameter. (It is also the
independent variable)
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Example 1
Input formulas for x and y, the range for t, and
the size of the step between points.
(Your viewing window will probably be different.)
(You will need to use the delete key.)
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We can confirm this algebraically
parabolic function
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Circle
If we let t the angle, then
Since
We could identify the parametric equations as a
circle.
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Graph on your calculator
menu
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3
Change the window settings.
menu
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Now square it up.
menu
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You can watch the direction and relative velocity
of the graph by using the trace function
menu
Notice the x, y and t values displayed.
Use the right and left arrow keys to watch the
position change as t changes.
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Holding a key down makes the motion continuous.
Change the speed by changing the size of the
steps
menu
Smaller steps slow the graph down.
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The TI-nspire can also graph conic sections
directly without converting to parametric
equations.
To clear the screen, press
menu
enter
Now we can enter the Cartesian equation for a
circle.
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The TI-nspire can also graph conic sections
directly without converting to parametric
equations.
menu
The horizontal and vertical shifts are zero, and
the radius is 1.
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Ellipse
This is the equation of an ellipse.
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Converting Between Parametric and Cartesian
Equations
We have seen two techniques for converting from
parametric to Cartesian
The first method is called eliminating the
parameter. It requires solving one equation for
t and substituting into the other equation to
eliminate t. This is possible when the graph is
a function.
Both of these methods only work sometimes. There
are many curves that can only be described
parametrically.
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On the other hand, changing from the Cartesian
equation for a function to a parametric equation
always works and it is easy!
The steps are
1) Replace x with t in the original equation.
2) Let x t .
becomes
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In the special case where we want the
parametrization for a line segment between two
points, we could find the Cartesian equation
first and then convert it to parametric, but
there is an easier way. We will use an example
to illustrate
Find a parametrization for the line segment with
endpoints (-2,1) and (3,5).
Using the first point, start with
Notice that when t 0 you get the point (-2,1) .
Substitute in (3,5) and t 1 .
The equations become
p