Equations of Tangent Lines: Cubic-1
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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 equation of the tangent line at
How do we proceed?
We can use the point-slope form of the equation of the tangent line because we can find the coordinates of the points from the values of
t
and can find the slope at those points.
As a review, state the point-slope form for a line of slope
m
and passing through the point (
x
o
,
y
o
).
How do we find the slope?
From
, which we can obtain from the parametric equations.
State the general relationship among
.
Determine
Determine
Substitute these results in
.
So we need to fill in this table:
t
x
y
m
-1
3
Do that, and then check your results by clicking “Next”.
We get
t
x
y
m
-1
-2
11
-9
3
2
-9
15
Use this data in the point-slope equation to find the equation of the tangent line at
t
= -1.
At
t
= -1,
Simplify this equation.
The equation of the tangent line at
t
= -1 is
Determine the equation of the tangent line at
t
= 3.
At
t
= 3,
Simplify this equation.
We get
To check these results, plot the parametric equations for the interval – -4 <
t
< 4. Then check your graph by clicking “Next”.
alt="Graph of a parametric 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
t
= -1. 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 tangent line at that point. Then check your graph by clicking “Next”.
alt="Graph of a parametric cubic showing a tangent. 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
t
= 3. 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 tangent line at that point. Then check your graph by clicking “Next”.
alt="Graph of a parametric cubic showing a both tangents. 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 tangent lines go through the appropriate points on the graph?
Yes: through (-2, 11) and (2, -9).
Are the slopes correct?
Yes, one is negative and steep (-9),
and the other is positive and steeper (15).
Are the
y
-intercepts correct?
Yes, one is at
y
= – 7
and the other is at
y
= – 39.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
Email to John Taylor
General Contents
Detailed Contents
Index