9.3 - 
 MATH
1081
 Wednesday,
March
30
 


Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 
 MATH
1081
 Wednesday,
March
30
 
 Chapter
9
–
Section
3
 
 MAXIMA
AND
MINIMA
 
 Homework
#10
(due
April
4):
 
 Section
9.2
#32,
40,
48,
60
 
 
 
 
 
 
 
 
 
 
 Section
9.3
#14,
36,
38
 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 When
finding
the
relative
extrema
of
a
function
of
a
single
 variable,
we
began
by
finding
the
critical
numbers
of
the
 function
(points
in
the
domain
where
the
derivative
is
zero
or
 the
derivative
does
not
exist).

This
gave
us
a
list
of
potential
 maxima
and
minima.
 
 We
then
had
to
use
some
kind
of
test
to
classify
each
of
those
 critical
numbers.

Depending
on
the
result
of
the
test,
we
could
 conclude
that
there
was
a
maximum,
a
minimum,
or
neither
at
 each
of
those
critical
numbers.
 
 In
particular,
there
was
the
Second
Derivative
Test
for
a
 function
of
a
single
variable.
 
 
 So,
the
sign
of
the
second
derivative
at
the
critical
number
 helped
us
classify
it
as
a
maximum
or
a
minimum.
 
 
 For
a
function
of
two
variables,
 f ( x, y ) ,
we
first
find
the
critical
 points.

In
this
case,
the
critical
points
are
the
points
at
which
 BOTH
 f x ( x, y ) = 0 
and
 f y ( x, y ) = 0 ,
simultaneously.
 
 We
then
will
test
each
of
these
critical
numbers
that
we
found
 2 by
computing
the
value
 D = f xx ⋅ f yy − ⎡ f xy ⎤ 
at
the
point.

The
 ⎣⎦ sign
of
 D 
(a
mix
of
the
second
partial
derivatives)
helps
us
 classify
our
critical
number
as
a
maximum,
a
minimum,
or
 neither.
 
 

 
 Example:

Find
all
points
where
the
function
has
a
relative

 
extrema.
 
 
 
 
 1.

 f ( x, y ) = 2 x 2 + 3xy + 2 y 2 − 5 x + 5 y 
 
 
 
 
 
 
 
 
 
 2.

 z = − y 4 + 4 xy − 2 x 2 + 1
 Example:

Section
9.3
#33:

Suppose
that
the
profit
(in

 hundreds
of
dollars)
of
a
certain
firm
is
 approximated
by
the
function
 P ( x, y ) = 1500 + 36 x − 1.5 x 2 + 120 y − 2 y 2 ,
 where
 x 
is
the
cost
of
a
unit
of
labor
and
 y 
is
the
cost
 of
a
unit
of
goods.

Find
the
values
of
 x 
and
 y 
that
 will
maximize
the
profit
for
the
firm.
 
 Clicker
Check­out:

Choose
any
letter
to
check
out
now.
 
 
 Tomorrow
in
recitation:
Quiz
on
Sections
6.1,
6.2,
6.3,
9.1
 
 Next
time
–
Section
7.1
 
 
 
 ...
View Full Document

This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

Ask a homework question - tutors are online