S E C T I O N
15.4
Differentiability, Linear Approximation, and Tangent Planes
(ET Section 14.4)
659
(c)
We verify, using a CAS, that for
(
x
,
y
)
=
(
0
,
0
)
the following derivatives hold:
f
xy
(
x
,
y
)
=
f
yx
(
x
,
y
)
=
x
6
+
9
x
4
y
2
−
9
x
2
y
4
−
y
6
(
x
2
+
y
2
)
3
To show that
f
xy
is not continuous at
(
0
,
0
)
, we show that the limit lim
(
x
,
y
)
→
(
0
,
0
)
f
xy
(
x
,
y
)
does not exist. We compute
the limit as
(
x
,
y
)
approaches the origin along the
x
axis. Along this axis,
y
=
0; hence,
lim
(
x
,
y
)
→
(
0
,
0
)
along the
x
axis
f
xy
(
x
,
y
)
=
lim
h
→
0
f
xy
(
h
,
0
)
=
lim
h
→
0
h
6
+
9
h
4
·
0
−
9
h
2
·
0
−
0
(
0
+
h
2
)
3
=
lim
h
→
0
1
=
1
We compute the limit as
(
x
,
y
)
approaches the origin along the
y
axis. Along this axis,
x
=
0, hence,
lim
(
x
,
y
)
→
(
0
,
0
)
along the
y
axis
f
xy
(
x
,
y
)
=
lim
h
→
0
f
xy
(
0
,
h
)
=
lim
h
→
0
0
+
0
+
0
−
h
6
(
0
+
h
2
)
3
=
lim
h
→
0
(
−
1
)
= −
1
Since the limits are not equal
f
(
x
,
y
)
does not approach one value as
(
x
,
y
)
→
(
0
,
0
)
, hence the limit
lim
(
x
,
y
)
→
(
0
,
0
)
f
xy
(
x
,
y
)
does not exist, and
f
xy
(
x
,
y
)
is not continuous at the origin.
(d)
The result of part (b) does not contradict Clairaut’s Theorem since
f
xy
is not continuous at the origin. The continuity
of the mixed derivative is essential in Clairaut’s Theorem.
15.4 Differentiability, Linear Approximation, and Tangent Planes
(ET Section 14.4)
Preliminary Questions
1.
How is the linearization of
f
(
x
,
y
)
at
(
a
,
b
)
defined?
SOLUTION
The linearization of
f
(
x
,
y
)
at
(
a
,
b
)
is the linear function
L
(
x
,
y
)
=
f
(
a
,
b
)
+
f
x
(
a
,
b
)(
x
−
a
)
+
f
y
(
a
,
b
)(
y
−
b
)
This function is the equation of the tangent plane to the surface
z
=
f
(
x
,
y
)
at
(
a
,
b
,
f
(
a
,
b
)
)
.
2.
Define local linearity for functions of two variables.
SOLUTION
f
(
x
,
y
)
is locally linear at
(
a
,
b
)
if the linear approximation
L
(
x
,
y
)
at
(
a
,
b
)
approximates
f
(
x
,
y
)
at
(
a
,
b
)
to first order. That is, if there exists a function
(
x
,
y
)
satisfying
lim
(
x
,
y
)
→
(
a
,
b
)
(
x
,
y
)
=
0 such that
f
(
x
,
y
)
−
L
(
x
,
y
)
=
(
x
,
y
)
(
x
−
a
)
2
+
(
y
−
b
)
2
for all
(
x
,
y
)
in an open disk
D
containing
(
a
,
b
)
.
In Questions 3–5, assume that
f
(
2
,
3
)
=
8
,
f
x
(
2
,
3
)
=
5
,
f
y
(
2
,
3
)
=
7
3.
Which of (a)–(b) is the linearization of
f
at
(
2
,
3
)
?
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 Winter '08
 GANGliu
 Approximation, Derivative, Linear Approximation, lim, S E V E R A L VA R

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