Midterm 1 of Math 254.01 (Au10)
October 18, 2010
The Ohio State University
12:30pm1:18pm
Including the cover sheet, this exam consists of seven (7) pages and six (6) problems and is
worth a total of 100 points. The point value of each problem is indicated.
To obtain full
credit, you must have the correct answers along with relevant supporting work to jus
tify them.
Partial credit will be given based on the work that is shown. However,
answers
without sufficient supporting work will receive no credit.
Also, you must complete the
exam in 48 minutes. Good luck!
Name:
Signature:
OSU Internet Username:
Problem #
Score
Problem #
Score
1
4
(15 pts)
(18 pts)
2
5
(12 pts)
(20 pts)
3
6
(15 pts)
(20 pts)
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1.
(a)
(5)
Calculate
lim
(
x,y
)
→
(0
,
0)
5
x
2
y
x
4
+ 4
y
2
, if it exists, or show that the limit does not exist.
Solution:
Along the curve
C
1
:
y
= 0
,
5
x
2
y
x
4
+ 4
y
2
= 0
for all
(
x, y
)
6
= (0
,
0)
. Hence
5
x
2
y
x
4
+ 4
y
2
→
0 =
L
1
as
(
x, y
)
→
(0
,
0)
along
C
1
. On the other hand, along the curve
C
2
:
y
=
x
2
,
5
x
2
y
x
4
+ 4
y
2
=
5
x
4
x
4
+ 4
x
4
= 1
for all
(
x, y
)
6
= (0
,
0)
. Hence
5
x
2
y
x
4
+ 4
y
2
→
1 =
L
2
as
(
x, y
)
→
(0
,
0)
along
C
2
. Since
L
1
6
=
L
2
,
lim
(
x,y
)
→
(0
,
0)
5
x
2
y
x
4
+ 4
y
2
does not
exist.
(b)
(2)
Is the function
f
(
x, y
) =
5
x
2
y
x
4
+ 4
y
2
if
(
x, y
)
6
= (0
,
0)
0
if
(
x, y
) = (0
,
0)
continuous? Justify your answer.
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 Fall '08
 SOMEGUY
 Math, Critical Point, Extreme value

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