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Example Questions
Example Question #471 : Derivatives
Use the chain rule to differentiate the following function:
By the chain rule:
Differentiate using the product rule:
Substitute this derivative for in the first equation:
Factor the equation:
Example Question #771 : Ap Calculus Ab
Find .
is undefined.
Therefore:
Example Question #22 : Functions, Graphs, And Limits
Which of the following functions contains a removeable discontinuity?
A removeable discontinuity occurs whenever there is a hole in a graph that could be fixed (or "removed") by filling in a single point. Put another way, if there is a removeable discontinuity at , then the limit as
approaches
exists, but the value of
does not.
For example, the function contains a removeable discontinuity at
. Notice that we could simplify
as follows:
, where
.
Thus, we could say that .
As we can see, the limit of exists at
, even though
is undefined.
What this means is that will look just like the parabola with the equation
EXCEPT when
, where there will be a hole in the graph. However, if we were to just define
, then we could essentially "remove" this discontinuity. Therefore, we can say that there is a removeable discontinuty at
.
The functions
, and
have discontinuities, but these discontinuities occur as vertical asymptotes, not holes, and thus are not considered removeable.
The functions
and
are continuous over all the real values of
; they have no discontinuities of any kind.
The answer is
.
Example Question #1 : Approximate Rate Of Change From Graphs And Tables Of Values
Example Question #71 : Derivative At A Point
Example Question #72 : Derivative At A Point
Example Question #73 : Derivative At A Point
Example Question #74 : Derivative At A Point
Example Question #75 : Derivative At A Point
Example Question #76 : Derivative At A Point
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