9.2 - MATH
1081
 Monday,
March
28
 


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Unformatted text preview: MATH
1081
 Monday,
March
28
 
 Chapter
9
–
Section
2
 
 PARTIAL
DERIVATIVES
 
 Homework
#10
(due
4/4):
 
 Section
9.2
#32,
40,
48,
60
 
 
 
 
 
 
 
 
 
 Section
9.3
#14,
36,
38
 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 As
mentioned
last
time,
when
considering
rates
of
change
of
a
 function
of
more
than
one
variable,
we
must
know
with
respect
 to
which
variable
we
are
measuring
the
change.
 
 In
particular,
for
a
function
of
more
than
one
variable,
we
have
 a
partial
derivative
for
each
variable
defining
that
function.

 Each
of
these
partial
derivatives
indicates
the
rate
of
change
of
 the
function
looking
at
that
particular
variable.
 
 We
will
typically
use
a
subscript
to
indicate
which
variable
is
 “changing”.

For
all
other
variables
of
the
function,
we
hold
 them
fixed
and
treat
them
like
constants
when
differentiating.
 
 The
partial
derivative
of
the
function
 f ( x, y ) 
with
respect
to
 x 
 is
written
with
the
subscript
 x 
as
 f x ( x, y ).


 In
this
case,
we
are
holding
the
other
variable
 y 
constant
and
 asking
how
does
 f 
change
as
 x 
does.
 
 The
partial
derivative
of
the
function
 f ( x, y ) 
with
respect
to
 y 
 is
written
with
the
subscript
 y 
as
 f y ( x, y ) .


 In
this
case,
we
are
holding
the
other
variable
 x 
constant
and
 asking
how
does
 f 
change
as
 y 
does.
 
 本页已使用福昕阅读器进行编辑。 福昕软件(C)2005-2009,版权所有, 仅供试用。 Alternatively,
we
may
see
the
following
notations.
 
 ∂f f x ( x, y ) = = Dx ⎡ f ( x , y ) ⎤ 
 ⎣ ⎦ ∂x 
 
 This
notation
and
concept
extends
in
the
obvious
way
to
 functions
of
3
or
more
variables
too.

So,
for
example,
we
have
 3
partial
derivatives
of
the
function
 f ( x, y, z ) .


 Namely,

 f x ( x, y, z ) 
where
 y 
and
 z 
are
held
constant,
 


 f y ( x, y, z )
where
 x 
and
 z 
are
held
constant,
and
 


 fz ( x, y, z )
where
 x 
and
 y 
are
held
constant.
 
 Just
as
with
the
derivative
of
a
single
variable,
it
is
defined
as
a
 limit.

In
the
case
of
multi‐variable
functions,
we
just
need
to
 define
such
a
limit
for
each
variable
separately,
like
 
 f ( x + h, y ) − f ( x, y ) 
 


 f x ( x, y ) = lim h→ 0 h 
 and
 
 f ( x, y + h ) − f ( x, y ) 
 f y ( x, y ) = lim h→ 0 h 
 
 However,
we
can
also
take
advantage
of
the
rules
of
 differentiation
that
we
now
know. Example:

Find
the
partial
derivatives
of
the
function.
 
 
 
 1.

 f ( x, y ) = 3x + 2 y 
 
 
 
 
 
 
 2.

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

 f ( x, y ) = 2 
 y Example:

Find
the
partial
derivatives
of
the
function.
 
 x y3 
 
 
 1.

 f ( x, y ) = e + e 
 
 
 
 
 
 
 
 
 
 2.

 f ( x, y ) = ln( xy ) − y ln( y )
 ( ) Just
as
with
functions
of
a
single
variable,
the
derivative
of
the
 derivative
(or
second
derivative)
has
meaning
and
application
 in
functions
of
more
than
one
variable.
 
 However,
because
for
each
variable
of
the
function
we
must
 take
a
separate
partial
derivative,
when
we
compute
a
second
 derivative,
we
must
take
the
derivative
of
each
first
partial
 with
respect
toeach
variable.
 
 Thus,
a
function
of
two
variables
has
2
first
partial
derivatives
 and
4
second
partial
derivatives.
 
 In
particular,
for
 f ( x, y ) 
we
get
 






 f ( x, y ) f x ( x, y ) f y ( x, y ) f ( x, y ) xx f ( x, y ) xy f ( x, y ) yx f ( x, y ) yy Example: Find the second partial derivatives of each function. 1. f ( x, y ) = x 2 y + xy 3 x 2. f ( x, y ) = x + + xye y + 4 y 3 本页已使用福昕阅读器进行编辑。 福昕软件(C)2005-2009,版权所有, 仅供试用。 These
examples
just
indicated,
the
mixed
second
partials
 always
seem
to
be
equal.

That
is,
 f xy = f yx .
 
 It
turns
out
that
for
all
functions
that
we
will
see,
this
is
true.
 
 So,
when
finding
second‐order
partial
derivatives,
we
really
 only
need
to
do
the
work
to
find
3
derivatives.

Whether
you
 prefer
to
find
 f xy 
or
you
prefer
to
find
 f yx 
is
up
to
you. Clicker
Check­out:

Choose
any
letter
to
check
out
now.
 
 
 Next
time
–
Section
9.3.
 
 ...
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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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