alculus with Vector Functions
In this section we need to talk briefly about limits, derivatives and integrals of vector
functions. As you will see, these behave in a fairly predictable manner. We will be
doing all of the work in
but we can naturally extend the formulas/work in this
section to
(
i.e.
n
dimensional space).
Let’s start with limits. Here is the limit of a vector function.
So, all that we do is take the limit of each of the components functions and leave it as
a vector.
Example 1
Compute
where
.
Solution
There really isn’t all that much to do here.
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Notice that we had to use
L’Hospital’s Rule
on the
y
component.
Now let’s take care of derivatives and after seeing how limits work it shouldn’t be too
surprising that we have the following for derivatives.
Example 2
Compute
for
.
Solution
There really isn’t too much to this problem other than taking the derivatives.
Most of the basic facts that we know about derivatives still hold however, just to make
it clear here are some facts about derivatives of vector functions.
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 Fall '08
 prellis
 Calculus, Derivative, Integrals, Limits, vector functions

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