Derivative of a parametric ellipse
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If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.
Find
directly from the parametric equations
and evaluate
at
First, let’s discuss this function. What does its graph look like?
An ellipse.
How long is its semi-major axis?
It is 5 units long.
How long is its semi-minor axis?
It is 2 units long.
Where is the center?
At the origin.
Try to draw the graph from the parametric equations without using your graphing calculator. Then check by clicking “Next”.
alt="Graph of an ellipse. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
Next let’s determine the point associated with
. Set up the determination of the
x
-coordinate of that point?
Evaluate this.
Set up the determination of the
y
-coordinate of that point?
Evaluate this.
Show this point on your graph and show the tangent line.
Then check by clicking “Next”.
alt="Graph of an ellipse with a tangent line. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
What is the algebraic sign of the slope of the tangent line?
When a line slopes down to the right, it has a negative “rise” over a positive “run”. Its slope is negative.
What will be the sign of
at this point?
It also is negative.
Now let’s determine
from the parametric equations.
State the general relationship among
.
Determine
Determine
Substitute these results in
.
Set up the evaluation of
.
How do we evaluate
on the calculator?
We determine
and take the reciprocal.
What “mode” does the calculator need to be in?
Radian mode.
Determine the decimal equivalent of
.
Is this consistent with the graph?
Yes, the tangent line is less steep than a line of slope = – 1..
Now let’s determine
from the parametric equations.
State the general relationship between
.
Apply this to our result for
We get
.
Simplify this result.
Using
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
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General Contents
Detailed Contents
Index