MA | 261 | Multivariate Calculus | exam ...
MA | 261 | Multivariate Calculus | exam |MA 26100 EXAM 1 Form A September 29, 2016 NAME YOUR TA’S NAME STUDENT ID # RECITATION TIME 1. You must use a #2 pencil on the mark–sense sheet (answer sheet). 2. On the scantron, write 01 in the TEST/QUIZ NUMBER boxes and blacken in the ap- propriate spaces below. 3. On the scantron, ﬁll in your TA’s name and the course number . 4. Fill in your NAME and STUDENT IDENTIF
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MA | 261 | Multivariate Calculus | ...
MA | 261 | Multivariate Calculus | exam |MA 26100 EXAM 1 Form A September 29, 2016 NAME YOUR TA’S NAME STUDENT ID # RECITATION TIME 1. You must use a #2 pencil on the mark–sense sheet (answer sheet). 2. On the scantron, write 01 in the TEST/QUIZ NUMBER boxes and blacken in the ap- propriate spaces below. 3. On the scantron, ﬁll in your TA’s name and the course number . 4. Fill in your NAME and STUDENT IDENTIF
##### Page 5
5.
Find the length of the curve
~
r
(
t
)=
<t
-
sin
t,
1
-
cos
t>
on 0
t
π
. (Hint: use the
double-angle formula cos(2
x
)=1
-
2 sin
2
x
.)
A. 4
B. 8
C.
-
4
D.
-
5
E. 2
π
6.
The curvature of the curve
~
r
(
t
)=
<
9 cos
t,
9 sin
t>
at
t
=
π
is
A. 9
B. 3
C.
1
9
D.
1
3
E. 1
5

##### Page 6
7.
If a particle has the given acceleration
~a
(
t
)=
<
2
,
-
cos
t,
3
4
t
>
with initial position
~
r
(0) =
<
0
,
1
,
1
>
and initial velocity
~v
(0) =
<
1
,
0
,
0
>
, then its position at
t
=
π
is
A.
2
+
π,
-
1
3
/
2
+1
>
B.
2
+
π,
-
1
3
/
2
>
C.
2
,
-
1
3
/
2
+1
>
D.
2
+
π,
1
3
/
2
+1
>
E.
2
,
1
3
/
2
+1
>
8.
Consider the function
f
(
x, y
)=
2+
xy
3
1+
x
2
-
y
2
on its maximal domain of deﬁnition. Calcu-
late
lim
(
x,y
)
(0
,
0)
f
(
x, y
)
A. This limit does not exist.
B. This limit in not well deﬁned, since the function is not deﬁned at (0
,
0)
C. 0
D.
1
2
E. 2
6

##### Page 7
9.
Let
P
=
u
2
+
v
2
+
w
2
,
u
=
u
(
x, y
),
v
=
v
(
x, y
), and
w
=
w
(
x, y
).
If
u
(0
,
1) = 0,
u
x
(0
,
1) = 2,
w
(0
,
1) = 2,
w
x
(0
,
1) = 0, and
v
(
x, y
)=
ye
x
. Find
P
x
(0
,
1).
A. 2
·
5
-
1
/
2
B. 2
·
5
1
/
2
C. 5
1
/
2
D. 5
-
1
/
2
E. 4
·
5
1
/
2
10.
The total surface area of a cone having height
h
r
is
A
=
πr
r
2
+
h
2
+
πr
2
.
If
r
and
h
are measured as 3 in and 4 in, respectively, with an error in measurement of at
most 1 in each, use diﬀerentials to estimate the maximum error (in in
2
) in the calculated
surface area.
A. 2
.
4
π
B. 12
.
8
π
C. 15
.
2
π
D. 7
.
6
π
E. 3
.
8
π
7

##### Page 8
11.
Let
R
(
p, q
) = tan
-
1
(
pq
2
). Find
2
R
∂q∂p
.
A.
2
q
(1
-
p
2
q
4
)
3
/
2
B.
2
q
-
2
q
5
p
2
(1 +
p
2
q
4
)
2
C.
q
(1
-
p
2
q
4
)
3
/
2
D.
2
q
(1 +
p
2
q
4
)
2
E.
q
2
(1
-
p
2
q
4
)
3
/
2
12.
Find the directional derivative for
T
(
x, y
)=
y
-
1
x
-
2
at (3
,
-
2) in the direction toward the
origin.
A. 7
B.
-
7
C.
7
13
D.
-
7
13
E. 5
8

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