Download Calculus, Section 4.1

Section 4.1
Maximum and
Minimum Values
AP Calculus
October 20, 2009
Berkley High School, D1B1
[email protected]
Max and Min
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Of course they can’t be too easy; this IS
calculus.
Absolute Maximum: f has an absolute
maximum if there exists a c such that f(c)≥f(x),
for x in D. f(c) is called the maximum value.
Absolute Minimum: f has an absolute minimum
if there exists a c such that f(c)≤f(x), for x in D.
f(c) is called the minimum value.
The absolute max and absolute min are called
the extremes.
Calculus, Section 4.1
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Max and Min
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Local Maximum: f has a local maximum if there
exists a c such that f(c)≥f(x), for all x near c.
 How
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close to c? Calculus close.
Local Minimum: f has a local minimum if there
exists a c such that f(c)≤f(x), for all x near c.
Endpoints of closed intervals can not be local
maximums or minimums.
Calculus, Section 4.1
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Example
Local Minimum and
Absolute Minimum
Local Maximum and
Absolute Maximum
Local Maximum
Local Minimum
Calculus, Section 4.1
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Local
Minimum
Example
Local
Minimums
Local and Absolute
Maximum
Local
Maximum
Absolute
Minimum
Calculus, Section 4.1
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Example
Local
Maximum
Local and
Absolute
Maximum
Absolute
Minimum
Local
Minimum
Local
Minimum
Calculus, Section 4.1
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Critical Numbers
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Definition: “A critical number of a function f is a
number c in the domain of f such that either
f’(c) = 0 or
f’(c) does not exist.”
Theorem: “If f has a local maximum or minimum
at c, then c is a critical number of f.”
Translation: Any critical number has the potential
of being a local maximum of minimum. Only at
critical numbers can a local max or a local min
exist.
Calculus, Section 4.1
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Example
x=0 is a
critical value,
but not a local
maximum or
minimum
Calculus, Section 4.1
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Example
f(0)=0 is a local
minimum.
Because the
derivative at 0 is
undefined, 0 is a
critical value.
Calculus, Section 4.1
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Closed Interval Method
To find the absolute maximum and minimum
values of a continuous function f on a closed
interval [a, b]:
1. Find the values of f at the critical numbers of f
on (a,b)
2. Find the values of f at the endpoints of the
interval.
3. The largest of the values for steps 1 & 2 is the
absolute maximum value; the smallest of these
values is the absolute minimum value.
Calculus, Section 4.1
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f ( x)  3 x 2  6 x
f ( x)  3 x( x  2)
Example
Find the absolute
maximum and minimum
values of the function:
1
f ( x)  x3  3x 2  1,   x  4
2
0  3 x( x  2)
0  3 x, 0  x  2
x  0, x  2
 1 1
f   
 2 8
f 0  1
f  2   3
f  4   17
Calculus, Section 4.1
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f ( x)  3 x 2  6 x
f ( x)  3 x( x  2)
Example
Find the absolute
maximum and minimum
values of the function:
1
f ( x)  x3  3x 2  1,   x  4
2
Absolute
Minimum
0  3 x( x  2)
0  3 x, 0  x  2
x  0, x  2
 1 1
f   
 2 8
f 0  1
f  2   3
f  4   17
Absolute
Maximum
Calculus, Section 4.1
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Assignment
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Section 4.1, 15-55, odd
Calculus, Section 4.1
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