applications of differentiation
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Applications of Differentiation. 3. 3.1. Maximum and Minimum Values. Maximum and Minimum Values. Some of the most important applications of calculus are optimization problems , which find the optimal way of doing something. Meaning: find the maximum or minimum values of some function. - PowerPoint PPT PresentationTRANSCRIPT
3 Applications of Differentiation
3.1 Maximum and Minimum Values
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Maximum and Minimum Values
Some of the most important applications of calculus are optimization problems, which find the optimal way of doing something. Meaning: find the maximum or minimum values of some function.
So what are maximum and minimum values?
Example: highest point on the graph of the function f shown is the point (3, 5) so the largest value off is f (3) = 5. Likewise, the smallest value is f (6) = 2.
Figure 1
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Absolute Maximum and Minimum
The maximum and minimum values of f are called extreme values of f.
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Local Maximum and Minimum
Something is true near c means that it is true on some open interval containing c.
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Example 1
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Figure 2 shows the graph of a function f with several extrema:
At a : absolute minimum is f(a)
At b : local maximum is f(b)At c : local minimum is f(c)At d : absolute (also local) maximum is f(d)
At e : local minimum is f(e)
Figure 2
Abs min f (a), abs max f (d)loc min f (c) , f(e), loc max f (b), f (d)
Example 2
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Practice! Define all local and absolute extrema of the graph below
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Maximum and Minimum Values
The following theorem gives conditions under which a
function is guaranteed to have extreme values.
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Maximum and Minimum Values
The Extreme Value Theorem is illustrated in Figure 7.
Note that an extreme value can be taken on more than
once.
Figure 7
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Maximum and Minimum Values
The Extreme Value Theorem says that a continuous function on a closed interval has a maximum value and a minimum value, but it does not tell us how to find these extreme values. We start by looking for local extreme values.
Figure 10 shows the graph of a function f with a local maximum at c and a local minimum at d.
Figure 10
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Counter examples:
f’ doesn’t exist but there is a local min or max
f’ exists but there is no local min or max
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Maximum and Minimum Values
To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local or it occurs at an endpoint of the interval.
Thus the following three-step procedure always works.