9.1 - MATH
1081
 Wednesday,
March
16
 
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Unformatted text preview: MATH
1081
 Wednesday,
March
16
 
 Chapter
9
–
Section
1
 
 FUNCTIONS
OF
SEVERAL
VARIABLES
 
 Homework
#9
(due
3/28):
 
 Section
6.3
#10,
18,
22
 
 
 
 
 
 
 
 
 
 Section
9.1
#28,
46,
48
 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 In Chapter 5 and 6, we used the first derivative to determine the relative extrema of a function of a single variable. In Chapter 9, we want to find extrema of a function of two variables. We will again use the derivative of the function to help us determine the location of extrema, but to classify an extrema as a maximum or a minimum, we will use a test involving the second derivative, rather than the first derivative. So, to begin, let’s look at the Second Derivative Test for a function of a single variable from Section 5.3. Notice that, unlike the First Derivative Test, this text can only be applied to critical numbers where f ' ( c ) = 0 , not where the derivative does not exist. Further, there is a case (3) where the test is inconclusive. So, it made more sense in the case of the function of a single variable to just use the First Derivative Test, since it always gives a result. The idea behind why the Second Derivative Test works for a function of a single variable can be seen in the following graph. So
far
this
semester,
we
have
considered
only
functions
of
a
 single
variable.

However,
many
(perhaps
even
most)
models
 of
applications
require
more
than
one
variable.
 
 For
example,
if
you
want
to
compute
the
total
interest
earned
 from
a
CD
investment,
you
would
need
to
know
the
amount
 invested,
the
length
of
time
of
the
investment,
and
the
percent
 rate
of
interest
paid
on
that
investment.
 
 As
another
example,
if
you
are
a
scuba
diver,
there
are
dangers
 from
too
much
nitrogen
entering
your
system.

The
amount
 that
enters
your
system
depends
both
on
the
depth
of
your
 dive
and
the
length
of
time
of
your
dive.
 
 In
such
cases,
we
need
to
analyze
functions
of
more
than
one
 variable
(or
multi‐variable
functions).

The
notation
that
we
 use
for
such
functions
is
similar
to
that
for
single‐variable
 functions.
 
 We
will
write
 f ( x1 , x2 , x3 , ..., xn ) 
for
a
function
with
 n 
variables.

 As
with
a
function
of
a
single
variable,
there
is
exactly
ONE
 output
value
from
each
 n ‐tuple
input
of
real
numbers
in
the
 domain
of
 f .
 
 So,
for
example,
we
might
write
the
total
amount
of
interest
 earned
from
a
CD
investment
as
 I = f ( P, t , r ) 
where
 P 
 represents
the
amount
invested,
 t 
represents
the
length
of
 time,
and
 r 
represents
the
percent
rate
of
interest
paid.
 For
the
sake
of
being
able
to
visualize
a
graph,
we
will
 primarily
restrict
ourselves
to
analyzing
only
functions
of
two
 variables,
which
we
will
write
as
 z = f ( x, y ) .
 
 The
domain
of
this
function
is
the
set
of
all
ordered
pairs
( x, y ) 
 that
are
allowed
as
inputs
in
the
function.
 
 The
range
of
this
function
is
the
set
of
all
 z 
that
are
outputs
of
 the
function.
 
 The
graph
can
be
drawn
in
3‐dimensions
with
the
3
axes:
one
 for
 x ,
one
for
 y ,
and
one
for
 z .

 
 Such
a
set
of
axes
would
look
like
 
 
 
 An
example
of
the
graph
of
a
function
of
2
variables
is
 
 
 
 
 For
this
function,
if
look
at
the
 cross­section
for
any
fixed
 z 
we
 will
see
a
circle,
 k = x 2 + y 2 .
 
 If
we
look
at
a
cross‐section
for
 any
fixed
 x 
we
will
see
an
 upward
parabola,
 z = k + y 2 .
 
 If
we
look
at
a
cross‐section
for
 any
fixed
 y 
we
will
see
an
 upward
parabola,
 z = x 2 + k .

 

 


 
 You
will
not
be
asked
to
graph
any
such
functions
in
this
class,
 but
we
will
be
evaluating
and
examining
the
rate
of
change
of
 such
functions
at
various
points.
 
 To
evaluate
such
a
function,
we
do
much
like
we
do
with
a
 single
variable.

We
simply
plug
in
the
indicated
value
for
the
 appropriate
variable
and
complete
the
arithmetic
operations
 shown.
 
 For
example,

 if
 f ( x, y ) = 3x 2 − xy + 8 y ,

 
 2 then
 f ( 2, −4 ) = 3( 2 ) − ( 2 ) ( −4 ) + 8 ( −4 ) = −12 .
 In
order
to
examine
the
rate
of
change
of
such
a
function,
we
 will
focus
on
only
one
variable
at
a
time.

In
other
words,
we
 compute
the
derivative
with
respect
to
one
variable
by
 considering
all
the
other
variables
in
the
function
to
be
 constant
at
that
moment.
 
 Looking
at
the
paraboloid,
 z = x 2 + y 2 ,
this
means
that
we
will
 actually
have
2
derivatives
to
examine:

 
 one
when
we
consider
 x 
to
be
constant
and
have
 z = k + y 2 
and

 
 one
when
we
consider
 y 
to
be
constant
and
have
 z = x 2 + k .
 
 We
call
these
partial
derivatives. In
the
end,
our
analysis
is
much
the
same,
in
that
a
positive
 derivative
indicates
the
values
of
the
function
are
increasing
 (as
that
particular
variable
increases
and
the
others
stay
 constant)
and
a
negative
derivative
indicates
the
values
of
the
 function
are
decreasing
(as
that
particular
variable
increases
 and
others
stay
constant).
 
 At
points
where
the
direction
of
the
function
changes
(from
 increasing
to
decreasing
or
vice
versa),
we
will
have
potential
 maxima
and
minima.


 
 Look
at
the
point
at
( 0, 0 ) 
on
both
the
paraboloid
and
 hyperbolic
paraboloid
and
think
about
where
the
function
is
 increasing
and
where
it
is
decreasing.
 Clicker
Check­out:

Choose
any
letter
to
check
out
now.
 
 
 Tomorrow
in
recitation:
Workshop
#5
 
 Next
time
–
Section
9.2
 
 
 HAVE
A
FUN
AND
SAFE
SPRING
BREAK!!
 ...
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This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

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