Horizontal Tangent Line: Cubic-1
General Contents
Detailed Contents
Index
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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.
For the parametric equations
find the coordinates where the tangent line is horizontal.
How do we proceed?
We can use the derivative
, obtained from the parametric equations.
Can we get the points for a horizontal tangent in the usual way?
Yes, we’ll set the derivative to 0 and solve.
That will give us values of
t
. Will that solve the problem?
No, we’ll use such values of
t
in the parametric equations to find the rectangular coordinates.
State the general relationship among
.
Determine
Determine
Substitute these results in
.
What values of
t
will make this zero?
We see that we need
Determine the
x
and
y
-coordinates for
.
Determine the
x
and
y
-coordinates for
.
To check these results, plot the parametric equations for the interval
. Then check your graph by clicking “Next”.
alt="Graph of a cubic. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
Add the point for
. Then check your graph by clicking “Next”.
alt="Graph of a parametric cubic showing a tangent point. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
Add the point for
. Then check your graph by clicking “Next”.
alt="Graph of a parametric cubic showing two tangent points. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
Do the horizontal tangent lines go through the appropriate points on the graph?
Yes: through
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General Contents
Detailed Contents
Index