Derivative of a parametric line
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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
First, let’s discuss this function. Is the dependence of
x
on
t
linear, quadratic, cubic, or other?
Linear.
Is the dependence of
y
on
t
linear, quadratic, cubic, or other?
It also is linear.
What can we then expect for the shape of the graph of
y
vs.
x
?
A line.
Try to draw the graph from the parametric equations without using your graphing calculator. Then check by clicking “Next”.
alt="Graph of a line. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
From this graph, what can we expect for the value of
?
Since the slope is the same everywhere, we expect a constant value for this derivative.
What can we expect for the value of
?
Since the slope is constant, the second derivative is zero.
Now let’s determine
from the parametric equations. State the general relationship among
.
Determine
Determine
Substitute these results in
.
Does this agree with the graph? Add a triangle to your diagram to help with determining the slope of the line. Then check by clicking “Next”.
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What is the slope of the graph?
The slope of the line is 2, in agreement with our
Now for the second derivative.
State the relationship between
.
Apply this to our result for
We get
, as we expected.
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