past calc 3 exam 1
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Course
2433
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
10
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wait=-4479757, due=1697259600, due2=1697864400 Question number 1.
Your answer was D. Correct. Which of the following is a representation for line which is orthogonal (perpendicular) to the the plane
:
− 8
x
− 8
y
=
− 2 and contains the point (1, 1, 2)?
A
x
(
t
) = 2 +
t
,
y
(
t
) = 3 +
t
,
z
(
t
) = − 1 + 2
t
B
x
(
t
) = 1,
y
(
t
) = 1 + 2
t
,
z
(
t
) = 2 + 6
t
C
x
(
t
) = 2 +
t
,
y
(
t
) = 2 +
t
,
z
(
t
) = 2
t
D
x
(
t
) = 1 + 2
t
,
y
(
t
) = 1 + 2
t
,
z
(
t
) = 2
E
x
(
t
) = 1 + 2
t
,
y
(
t
) = 1 + 3
t
,
z
(
t
) = 2 −
t
F
None of the above.
Question number 2.
Your answer was E. Correct. Which of the following is a description for the plane which includes the point P
(3,
− 3, 2), while also being
perpendicular to the line below?
x
− 4
4
=
y
3
=
z
− 1
3
A
4
x
− 3
y
+ 3
z
+ 27 = 0
B
4
x
+ 3
y
+ 3
z
− 15 = 0
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2/10
C
4
x
+ 3
y
+ 3
z
+ 15 = 0
D
−4
x
+ 3
y
− 3
z
+ 27 = 0
E
4
x
+ 3
y
+ 3
z
− 9 = 0
F
None of the above.
Question number 3.
Your answer was E. Wrong. Find a vector-valued function r
(
t
) for the line passing through the points (3, 2, 3) and (4, 3,
− 3).
A
r
(
t
) = (3 −
t
)
i
+ (2 −
t
)
j
+ (3 + 12
t
)
k
B
r
(
t
) = (3 +
t
)
i
+ (2 +
t
)
j
+ (3 − 6
t
)
k
C
r
(
t
) = (3 − 4
t
)
i
+ (2 − 3
t
)
j
+ (3 + 3
t
)
k
D
r
(
t
) = (3 + 8
t
)
i
+ (2 − 3
t
)
j
+ (3 − 6
t
)
k
E
r
(
t
) = (3 + 4
t
)
i
+ (2 + 3
t
)
j
+ (3 − 3
t
)
k
F
None of the above.
Question number 4.
Your answer was D. Wrong. Loading [MathJax]/jax/output/HTML-CSS/fonts/TeX/fontdata.js
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3/10
Given the function
f
(
x
,
y
) = − e
2
x
2
+ 4
y
2
+ e
5
y
2
+ 1
identify the shape of the level curve which contains the point √
2
2
, 0 .
A
exponential
B
hyperbola
C
ellipse
D
line
E
logarithm
F
circle
G
None of the above.
Question number 5.
Your answer was E. Correct. Given the points P
( − 4, 4,
− 3) and Q
(0, 6,
− 3), find the vector of length 6 in the direction opposite →
QP
.
A
−12
√
5
i
−
6
√
5
j
B
−2
√
5
i
−
1
√
5
j
(
)
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4/10
C
−5
2
√
5
i
+
5
√
5
j
−
5
2
√
5
k
D
2
√
5
i
+
1
√
5
j
E
12
√
5
i
+
6
√
5
j
F
None of the above.
Question number 6.
Your answer was A. Correct. Suppose a particle moving with constant speed along a curve C
is parameterized by the function r
(
t
). Given that
the particle moving along this curve from t
= 4 to t
= 8 travels 20 units, find | |
r
′
(
t
) | | .
A
5
B
40
C
4
D
80
E
8
F
None of the above.
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5/10
Question number 7.
Your answer was C. Wrong. Find the measure of the angle of intersection between the lines:
r
1
(
t
) =
i
+ 2
j
+ 3
k
+
t
(2
i
−
j
+
k
) and
r
2
(
u
) =
i
+ 2
j
+ 3
k
+
u
( −
i
−
k
)
A
θ
=
π
4
B
θ
=
π
6
C
θ
=
2
π
3
D
θ
=
π
3
E
θ
=
5
π
6
F
None of the above.
Question number 8.
Your answer was E. Correct. Scalar parametric equations for the line tangent to the graph of
r
(
t
) = (2
t
+ 1)
i
+
t
2
+ 3
t
+ 2
j
+
t
3
+ 3
t
− 3
k
at the point P
(1, 2,
− 3) are:
(
)
(
)
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6/10
A
x
(
u
) = 1 + 2
u
,
y
(
u
) = 2 + 5
u
,
z
(
u
) = − 3 + 6
u
B
x
(
u
) = 2 +
u
,
y
(
u
) = 5 + 2
u
,
z
(
u
) = 6 − 3
u
C
x
(
u
) = 2 +
u
,
y
(
u
) = 3 + 2
u
,
z
(
u
) = 3 − 3
u
D
x
(
u
) = 1 + 2
u
,
y
(
u
) = 2 + 7
u
,
z
(
u
) = − 3 + 30
u
E
x
(
u
) = 1 + 2
u
,
y
(
u
) = 2 + 3
u
,
z
(
u
) = − 3 + 3
u
F
None of the above.
Question number 9.
Your answer was A. Wrong. Given vectors a
= 3
i
+ 2
j
+
k
and b
=
i
+ 3
j
− 2
k
, let c
= − 4
a
and compute: proj
c
b
A
7
√
14
i
−
21
√
14
j
+
14
√
14
k
B
−7
√
14
i
+
7
√
14
j
+
21
√
14
k
C
−
3
2
i
+
3
2
j
−
1
2
k
D
3
2
i
+
j
+
1
2
k
E
21
√
14
i
+
14
√
14
j
+
7
√
14
k
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7/10
F
None of the above.
Question number 10.
Your answer was B. Correct. Determine if the point P
is an interior point of the set D
, boundary point of the set D
, or neither.
D
= {(
x
,
y
): 1 <
x
2
+
y
2
≤ 9}
and the given point is P
(1, 4).
A
Boundary point
B
Neither
C
Interior point
Question number 11.
Your answer was B. Wrong. Determine if the point P
is an interior point of the set D
, boundary point of the set D
, or neither.
D
= {(
x
,
y
): 1 ≤
x
2
+
y
2
< 9}
and the given point is P
(0, 3).
A
Interior point
B
Neither
C
Boundary point
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8/10
Question number 12.
Your answer was B. Correct. Determine if the point P
is an interior point of the set D
, boundary point of the set D
, or neither.
D
= {(
x
,
y
): 1 <
x
2
+
y
2
≤ 81}
and the given point is P
(0, 9).
A
Neither
B
Boundary point
C
Interior point
Question number 13.
Your answer was A. Correct. This is a written question, worth 13 points. DO NOT place the problem code on the answer sheet. A
proctor will fill this out after exam submission. Show all steps (work) on your answer sheet for full credit.
Problem Code: 1362
Given f
(
x
,
y
) =
(
x
+ 3
y
)
2
3
x
2
+ 2
y
2
,
a. [3 points]
Using proper notation, give the domain of f
(
x
,
y
).
b. [2 points]
Find lim
(
x
,
y
) → ( 0 , 0 )
f
(
x
,
y
) if (
x
,
y
) → (0, 0) along the y
− axis. (Proper limit notation must be used
to get credit.)
c. [3 points]
Find lim
(
x
,
y
) → ( 0 , 0 )
f
(
x
,
y
) if (
x
,
y
) → (0, 0) along the line y
= − 3
x
. (Proper limit notation must be
used to get credit.)
d. [4 points]
Find lim
(
x
,
y
) → ( 0 , 0 )
f
(
x
,
y
) if (
x
,
y
) → (0, 0) along the curve y
=
x
2
. (Proper limit notation must be
used to get credit.)
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9/10
e. [1 point]
What can you conclude about lim
(
x
,
y
) → ( 0 , 0 )
f
(
x
,
y
)? Why?
A
I have placed my work and my answer on my answer sheet.
B
I want to have points deducted from my test for not working this problem.
Question number 14.
Your answer was A. Correct. This is a written question, worth 14 points. DO NOT place the problem code on the answer sheet. A
proctor will fill this out after exam submission. Show all steps (work) on your answer sheet for full credit.
Problem Code: 1462
The vector function r
(
t
) = cos
t
2
i
− 2 sin
t
2
j
+
√
3cos
t
2
k
determines a curve C
in space. Assume t
> 0.
a. [4 points]
Find the unit tangent vector T
(
t
).
b. [3 points]
Find the principle normal vector N
(
t
).
c. [3 points]
Find the curvature, κ
.
d. [4 points]
Find the tangential and normal components of acceleration and express the acceleration vector, a
(
t
)
, in terms of the unit tangent vector T
and the principal normal vector N
.
A
I have placed my work and my answer on my answer sheet.
B
I want to have points deducted from my test for not working this problem.
Question number 15.
Your answer was A. Correct. This is a written question, worth 13 points. DO NOT place the problem code on the answer sheet. A
proctor will fill this out after exam submission. Show all steps (work) on your answer sheet for full credit.
Problem Code: 1555
Given the lines:
ℓ
1
:
x
(
t
) = 2 − 2
t
,
y
(
t
) = 5 +
t
,
z
(
t
) = − 7 − 2
t
ℓ
2
:
x
(
u
) = 6 −
u
,
y
(
u
) = 6 −
u
,
z
(
u
) = − 11 + 3
u
( )
( )
( )
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10/10
a. [8 points]
Determine whether ℓ
1
and ℓ
2
are parallel, skew or intersect. If the lines intersect, find the point of
intersection of ℓ
1
and ℓ
2
.
b. [5 points]
If the lines intersect or are parallel, give an equation for the plane which contains both lines. If the
lines are skew, find a pair of parallel planes with each plane containing one of the lines.
A
I have placed my work and my answer on my answer sheet.
B
I want to have points deducted from my test for not working this problem.
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