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MAC2311, Calculus I 4.1 Maximum and Minimum Values
Some of the most important applications of differential calculus are
optimization
problems – finding the maximum and minimum values of functions. Here are
some examples:
1. What is the shape of a can that minimizes manufacturing costs?
2. What is the maximum acceleration of a space shuttle?
3. What is the radius of a contracted windpipe that expels air most rapidly
during a cough?
4. At what angle should blood vessels branch so as to minimize the energy
expended by the heart in pumping blood?
Definition:
A function
f
has an
absolute maximum
(or global maximum) at
c
if
( )
( )
f c
f
x
≥
for all
x
in
D
, where
D
is the domain of
f
. The number
( )
f c
is called
the maximum value of
f
on
D
. Similarly, a function
f
has an
absolute minimum
(or global minimum) at
c
if
( )
( )
f c
f
x
≤
for all
x
in
D
, where
D
is the domain of
f
.
The number
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 Spring '11
 Teague
 Calculus, Differential Calculus

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