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Mathematical Expression Editor
[?]
Throughout this module, if something does not exist, write DNE in the
answer box.
Recap Video
Take a look at the following video which recaps the ideas from the section.
Example Video
Below is a video showing a worked example.
Example Video:
Problems
If is a function, the derivative, denoted or , is defined as
provided the limit exists.
If , evaluate using the limit definition above.
The limit definition says
If , find the equation of the tangent line to at .
In the previous problem, we
found . Plugging in gives . This is the slope of the line. To get a point, we
plug in into to get the -coordinate, and this gives . Using point-slope form,
we get
If , then the limit definition of the derivative tells us that .
Using the limit
definition, we get
From here we can
. In this case, the conjugate of the numerator is
Multiplying by this on the top and bottom of the fraction gives
In the previous problem, the domain of was:
We found a derivative of . The domain of is:
In particular, does not exist. If we go to the limit definition of the derivative
with , we would get (in the second to last step):
Since the graph of is only defined for , we need , so in reality we
have
What does this tell you about the slopes of the secant lines as the point gets
closer to ?
This means the lines themselves are:
The derivative can tell us useful information about a graph. For example, if ,
then we know the tangent line to the graph of at is
. We also know that if , then the tangent line to the graph of at has
slope. Similarly, if , then the tangent line to the graph of at has
slope.
Consider the following graph of a function .
We have and .
The limits and .
The limit .
Does have any points of discontinuity?
.
How many horizontal tangent lines does the graph of have?
.
List the points where in increasing order:
The number is
zero.
Which of the following is the graph of ?
Suppose .
The limit definition of the derivative tells us that .
The graph of has a horizontal tangent line at .
The point on the graph where the tangent line is parallel to is at
.
For the last part, parallel lines have the same slope. The slope of the tangent
line is given by , and the slope of the given line is . So set them equal and
solve for .
Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)
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Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)