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Practice Test 2
Question 1
2.
Find the domain of f (x, y) = √‾
x‾‾‾‾
− y‾
a)
{(x, y)|x > y2 }
b)
{(x, y)|x ≥ y2 }
c)
{(x, y)|x < y2 }
d)
all values on the xy-plane
e)
{(x, y)|x ≤ y2 }
Question 2
Identify the correct sketch of the surface x − 4y2 = 0 .
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a)
b)
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c)
Question 3
The surfaces x + 7 y + 2 z = 2 and x + y − 3 z = 2 intersect in a space curve C. Determine the
projection of C onto the xy-plane.
a)
8 x + 23 y = 10
b)
5 x + 23 y = 10
c)
23 x + 5 y = −10
d)
5 x + 26 y = 10
e)
−x + 23 y = 10
Question 4
Identify the level curves of the given surface: f (x, y) = ln(4 x2 + 4 y2 )
a)
parabolas
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b)
circles
c)
ellipses
d)
exponentials
e)
logarithms
Question 5
Find an equation for the the level curve of f (x, y) = 1 − 6x2 − 4y2 that contains the point P(0, 1) .
a)
6 x2 + 4 y2 − 1 = 0
b)
6 x2 + 4 y2 + 8 = 0
c)
6 x2 + 4 y2 − 4 = 0
d)
6 x2 + 4 y2 − 2 = 0
e)
6 x2 + 4 y2 − 5/4 = 0
Question 6
2 −y
‾.
Calculate fx given f (x, y) = 2 √‾
x‾‾‾‾
a)
−x
2 −y
‾
2 √x‾‾‾‾‾
b)
2x
2 −y
‾
√x‾‾‾‾‾
c)
2x − 1
2 −y
‾
√x‾‾‾‾‾
d)
1
2 −y
‾
4 √x‾‾‾‾‾
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e)
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−
1
2 −y
‾
√‾x‾‾‾‾
Question 7
Calculate fz given f (x, y, z) = ln( xy ) − 4 ye2 xz .
a)
b)
c)
d)
e)
Question 8
State whether the set is open, closed, or neither: {(x, y, z) : x2 + y2 < 5, 0 < z < 3} .
a)
The set is neither open nor closed.
b)
The set is closed.
c)
The set is open.
Question 9
Calculate fxy given f (x, y) = ln √‾
3‾‾‾‾‾‾‾‾
x2 + 3 y2‾ .
a)
−x2 + y2
(x 2 + y 2 )
2
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b)
c)
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−2 xy
(x 2 + y 2 )
2
2 (3 x2 − y2 )y
(x 2 + y 2 )
d)
y
x2 + y2
e)
x
x2 + y2
3
Question 10
Calculate
a)
0
b)
16
49
c)
16
343
d)
4
343
e)
16
7
lim
(x,y)→(0,0)
4 xy2
(3 x + 2 y)
3
along the path y = 2x .
Question 11
Find the gradient vector for f (x, y) = 2 x2 y + 4 xy2 at the point P(3, 2) .
a)
∇f (3, 2) = 4i + 24j
b)
∇f (3, 2) = 84i
c)
∇f (3, 2) = 66i + 40j
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d)
∇f (3, 2) = 40i + 66j
e)
∇f (3, 2) = 8i + 66j
Question 12
Find the directional derivative of
f (x, y, z) = x2 y + z2 x + y2 z at the point P(1, 0, 3) in the direction of j − 2k .
a)
√5‾
5
b)
−11√5‾
5
c)
−11
d)
27
e)
−6√5‾
5
Question 13
Suppose that the temperature at each point on a metal plate is given by the function
T(x, y) = 3 ex cos(y) + 4 ey cos(x) + 3 . In what direction does the temperature increase most rapidly at
the point (0,0) and what is this rate of increase?
a)
The temperature increases most rapidly in the direction 3i − 4j; rate = √26
‾‾
‾
b)
The temperature increases most rapidly in the direction 3i + 4j; rate = 5
c)
The temperature increases most rapidly in the direction 4i − 3j; rate = √7
‾
d)
The temperature increases most rapidly in the direction 4i + 3j; rate = 5
e)
The temperature increases most rapidly in the direction 3i + 4j; rate = √7
‾
Question 14
Find the rate of change of f (x, y, z) = xy − 2 yz with respect to t along the curve
r(t) = 2 t i + t2 j + t3 k .
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a)
−10 t4 + 6 t2
b)
10 t4 − 6 t2
c)
−2 t3 − t2 + 2 t
d)
3 t2 + 2 t + 2
e)
2 t3 + t2 − 2 t
Question 15
Find
du
t
for u = 5 ex sin(y) + 2 ey sin(x) where x =
and y = 2 t .
dt
2
a)
5/2 et/2 sin(2 t) + e2 t cos(t/2) − 10 et/2 cos(2 t) − 4 e2 t sin(t/2)
b)
1/2 tet/2 (2 e3/2 t cos(t/2) + 8 e3/2 t sin(t/2) + 5 sin(2 t) + 20 cos(2 t))
c)
5 et/2 sin(2 t) + 2 e2 t cos(t/2) + 5 et/2 cos(2 t) + 2 e2 t sin(t/2)
d)
5/2 et/2 sin(2 t) + e2 t cos(t/2) + 10 et/2 cos(2 t) + 4 e2 t sin(t/2)
e)
5 et/2 sin(2 t) + 2 e2 t cos(t/2) − 5 et/2 cos(2 t) − 2 e2 t sin(t/2)
Question 16
Find an equation for the tangent plane and scalar parametric equations for the normal line for
x3 + y3 = z2 at the point P(1, 2, −3) .
a)
Tangent plane: 4 x + y + z − 7 = 0 ; normal line:
b)
Tangent plane: x + 4 y + 2 z − 1 = 0 ; normal line:
c)
Tangent plane: x + 4 y + 2 z − 15 = 0 ; normal line:
{x(t) = 4 t + 1, y(t) = −2 t + 2, z(t) = 2 t − 3}
{x(t) = t + 1, y(t) = −4 t + 2, z(t) = 2 t − 3}
{x(t) = t + 1, y(t) = 4 t + 2, z(t) = −2 t − 3}
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d)
Tangent plane: x + 4 y + z + 10 = 0 ; normal line: {x(t) = t + 1, y(t) = t + 2, z(t) = t − 3}
e)
Tangent plane: x + 4 y + 2 z − 3 = 0 ; normal line: {x(t) = t + 1, y(t) = 4 t + 2, z(t) = 2 t − 3}
Question 17
Find the stationary points and the local extreme values for f (x, y) = 5 x3 + 5 y3 − 30 xy .
a)
(0, 0) gives a local max with f = 0 and (2, 2) gives a local min with f = −40
b)
(0, 0) is a saddle point and (2, 2) gives a local min with f = −40
c)
(0, 0) is a saddle point and (−2, −2) gives a local max with f = 40
d)
(2, 2) gives a local min with f = −40 and (−2, −2) gives a local max with f = 40
e)
(2, 2) gives a local min with f = −40
Question 18
Maximize 3x + 2y + 2z on the sphere x2 + y2 + z2 = 12 .
a)
The maximum is −2√51
‾‾
‾
b)
The maximum is
14√51
‾‾
‾
17
c)
The maximum is
288√51
‾‾
‾
289
d)
There is no maximum.
e)
The maximum is 2 √51
‾‾
‾
Question 19
3
Use differentials to find the approximate value of √78
‾‾
‾ ⋅ √10
‾‾
‾.
a)
97
6
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b)
115
6
c)
119
6
d)
575
36
e)
679
36
Question 20
Determine whether or not the vector function (4 ex + 2 xy) i + (x2 + sin(y)) j is the gradient ∇f (x, y)
of a function everywhere defined. If so, find all the functions with that gradient.
a)
f (x, y) = 4 ex + x2 y − cos(y) + C
b)
f (x, y) = 4 ex y + xy2 + C
c)
f (x, y) = 1/3 x3 + sin(y)x + C
d)
f (x, y) = 4 ex + 2 xy + C
e)
Not a gradient
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