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Mathematical Expression Editor
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Here we make a connection between a graph of a function and its derivative and
higher order derivatives.
We say that a function is increasing on an interval if , for all pairs of numbers ,
in such that . We say that a function is decreasing on an interval if , for all pairs of numbers , in
such that .
Consider the graph of the function below:
On which of the following intervals is increasing?
The function is not increasing on the interval , because if we pick a pair of numbers
from , say, , and , then , but .
Which of the following famous functions are increasing on ?
The function is not increasing on , because if we take a pair of numbers in , say, ,
and , then , but , since , and .
Since the derivative gives us a formula for the slope of a tangent line to a curve, we
can gain information about a function purely from the sign of the derivative. In
particular, we have the following theorem
A function is increasing on any interval
where , for all in . A function is decreasing on any interval where , for all in .
Below we have graphed :
Is the function increasing or decreasing on the interval ?
From the graph of we can see that for all in . Then, the Theorem above implies
that the function is increasing on this interval.
We call the derivative of the derivative the second derivative, the derivative of the
second derivative (the derivative of the derivative of the derivative) the third
derivative, and so on. We have special notation for higher derivatives, check it
out:
First derivative:
.
Second derivative:
.
Third derivative:
.
We use the facts above in our next example.
Here we have unlabeled graphs of , , and :
Identify each curve above as a graph of , , or .
Here we see three curves, , , and .
Since is
when is positive and
when is negative, we see Since is increasing when is
and decreasing when is
, we see Hence , , and .
Here we have unlabeled graphs of , , and :
Identify each curve above as a graph of , , or .
Here we see three curves, , , and .
Since is
when is positive and
when is negative, we see Since is increasing when is
and decreasing when is
, we see Hence , , and .
Here we have unlabeled graphs of , , and :
Identify each curve above as a graph of , , or .
Here we see three curves, , , and .
Since is
when is positive and
when is negative, we see Since is increasing when is
and decreasing when is
, we see Hence , , and .
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.)