Announcements
This
Week:
‐  Par5al
Deriva5ves
‐  Tangent
Planes,
Lineariza5on,
Differen5ability
Work
On:
‐
Prac5ce
Problem
Set
9
Lineariza5on
and
Linear
Approxima5on
Defini5on:
Assume
that
z=f(x,y)
has
con5nuous
par5al
deriva5ves
at
(a,b).
Lineariza(on
of
f
at
(a,b):
L(a,b ) (x, y) = f (a,b) + f x (a,b)(x − a) + f y (a,b)(y − b)
€
Linear
approxima(on
(or
tangent
plane
approxima(on)
of
f
at
(a,b):
.
€
f (x, y) ≈ f (a,b) + f x (a,b)(x − a) + f y (a,b)(y − b)
Lineariza5on
and
Linear
Approxima5on
Example:
Find
the
linear
approxima5on
of
f (x, y) = ln(x − 3y)
(7,2)
f (6.9, 2.06).
at
and
use
it
to
approximate
€
€
€
€
Lineariza5on
and
Linear
Approxima5on
What
if
fx
and
fy
are
not
con5nuous?
Example:
 xy
 2
f (x, y) =  x + y 2
0

if
(x, y) ≠ (0,0)
if
(x, y) = (0,0)
Lineariza5on
and
Linear
Approxima5on
Notes:
 xy

f (x, y) =  x 2 + y 2
0

€
if
(x, y) ≠ (0,0)
if
(x, y) = (0,0)
What
Makes
a
“Good”
Approxima5on?
A
good
approxima5on
of
a
func5on
f(x,y)
at
a
point
(a,b)
has
the
property
that
the
difference
between
the
actual
func5on
and
the
approxima5ng
func5on
goes
to
zero
very
fast
as
the
point
(x,y)
approaches
(a,b).
error
.
€
What
Makes
a
“Good”
Approxima5on?
Example:
 xy
 2
f (x, y) =  x + y 2
0

if
(x, y) ≠ (0,0)
if
(x, y) = (0,0)
Show
that
L(0,0)(x,y)=0
is
not
a
good
approxima5on
to
f
at
(0,0).
.
What
Makes
a
“Good”
Approxima5on?
When
the
par5als
fx
and
fy
are
con5nuous,
the
L(a,b ) (x, y)
lineariza5on
is
the
best
approxima5on
of
the
func5on
f
at
(a,b)
since
the
error
goes
to
zero
the
fastest
as
(x,y)
approaches
(a,b),
i.e.
€
f (x, y)
− L(a,b ) (x, y) → 0 faster than (x, y) − (a,b) → 0
Proof:
€
.
Differen5ability
for
a
Func5on
of
Two
Variables
Defini(on:
Assume
that
f
is
a
real‐valued
func5on
of
two
variables
and
let
L(a,b ) (x, y) = f (a,b) + f x (a,b)(x − a) + f y (a,b)(y − b)
We
say
that
f
is
differen(able
at
(a,b)
if:
€
(a) 
both
par5al
deriva5ves
fx
and
fy
exist
in
some
disk
around
(a,b).
(b) 
the
func5on
L(a,b)(x,y)
sa5sfies
lim
(x,y )→(a,b )
f (x, y) − L(a,b ) (x, y)
2
(x − a) + (y − b)
2
=0
Differen5ability
for
a
Func5on
of
Two
Variables
Note
that
this
defini5on
says
if
a
func5on
f
is
differen5able
at
a
point
(a,b)
then
its
lineariza5on
L(a,b)(x,y)
is
a
good
approxima5on
(actually
the
best
approxima5on)
to
f
at
(a,b).
Theorems
Sufficient
Condi(on
for
Differen(ability
Assume
that
f
is
defined
on
an
open
disk
Br(a,b)
centred
at
(a,b),
and
that
the
par5al
deriva5ves
fx
and
fy
are
con5nuous
on
Br(a,b).
Then
f
is
differen5able
at
(a,b).
r
(a,b)
Differen(ability
Implies
Con(nuity
Assume
that
a
func5on
f
is
differen5able
at
(a,b).
Then
it
is
con5nuous
at
(a,b).
Differen5ability
for
a
Func5on
of
Two
Variables
Example:
Verify
that
the
linear
approxima5on
2x + 3
≈ 3 + 2x −12y
4y +1
is
valid
for
(x,y)
near
(0,0).
€
Differen5ability
for
a
Func5on
of
Two
Variables
Example
#16.
Show
that
the
func5on
f (x, y) = x tan y
is
differen5able
at
(0,0).
What
is
the
largest
open
disk
centred
at
(0,0)
on
which
f
is
differen5able?
€