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Math 31BH  Homework 5. Due Tuesday, February 15.
1.
(
Wednesday, February 9.
)
Pathological functions.
From the textbook, solve
1
.
9
.
1
and
1
.
9
.
2
.
2.
(
Wednesday, February 9.
)
Pathological functions and second order derivatives.
Put
f
(
x,y
) =
(
xy
·
x
2

y
2
x
2
+
y
2
if
(
x,y
)
6
= (0
,
0)
0
if
(
x,y
) = (0
,
0)
.
(i) Calculate the ﬁrst derivative
f
x
at
(0
,
0)
directly from the deﬁnition.
(ii) Calculate the ﬁrst derivative
f
x
at any other point
(
x,y
)
6
= (0
,
0)
.
(iii) Differentiate once more i.e. calculate
f
xy
(0
,
0)
using the deﬁnition. Conﬁrm that
f
xy
(0
,
0) =
1
.
(iv) Repeat for the derivative
f
y
(0
,
0)
then
f
yx
(0
,
0)
. Conﬁrm that
f
yx
(0
,
0) =

1
. Observe that
f
xy
(0
,
0)
6
=
f
yx
(0
,
0)
.
3.
(
Monday, February 14.
) From the textbook solve
3
.
6
.
1
,
3
.
6
.
2
,
3
.
6
.
7
.
4.
(
Monday, February 14.
) Two space shuttles are at the points
(1
,
0
,

1)
and
(6
,
1
,
0)
at time
0
and are travelling in straight lines parallel to the vectors

2
i
+
j
and
4
i

j

k
. What is the
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This note was uploaded on 03/28/2011 for the course MATH 31B taught by Professor Dragosoprea during the Winter '11 term at UCSD.
 Winter '11
 DragosOprea
 Logic, Derivative

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