Calculus III-Final review
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the corresponding position vector.
1) Define the points P = (-2, 4) and Q = (-9, -4). Find the position vector corresponding to PQ .
B) -11, 0
C) 2, -4
D) -7, -8
A) 7, 8
1)
Express the vector in the form ai + bj.
2) P1 P2 if P1 is the point (1, -1) and P2 is the point (3, -4)
A) v = 2i - 3j
B) v = -i + 3j
C) v = -2i + 3j
Find the indicated vector.
3) Let u = 9, 9 , v = 1, 4 . Find u - 4v.
B) 13, 25
A) 5, -7
2)
D) v = -2i - 3j
3)
C) 18, -20
D) 10, -52
Express the vector in the form ai + bj + ck.
4) P1 P2 if P1 is the point (3, 3, 4) and P2 is the point (5, 0, 0)
A) v = 2i - 3j - 4k
B) v = -2i + 3j - 4k
C) v = -2i + 3j + 4k
5) 3u - 5v if u = 1, 1, 0 and v = 3, 0, 1
A) v = -12i + 3j - 5k
C) v = 3i + 3j - 5k
4)
D) v = -2i - 3j + 4k
5)
B) v = -12i + 8j - 5k
D) v = 18i + 3j - 5k
Find an equation for the sphere with the given center and radius.
6) Center (-7, 5, 0), radius = 4
A) x2 + y2 + z 2 + 14x + 10y = -58
B) x2 + y2 + z 2 + 14x - 10y = -58
C) x2 + y2 + z 2 - 14x - 10y = -58
Find v · u.
7) v = 5i - 2j and u = -8i - 7j
A) -3i - 9j
6)
D) x2 + y2 + z 2 - 14x + 10y = -58
7)
B) -54
C) -26
D) -40i + 14j
Find the angle between u and v in radians.
8) u = 10i + 3j + 5k, v = 7i + 6j + 9k
A) 1.11
B) 1.49
C) 0.47
D) 1.10
9) u = 7j - 9k, v = 3i - 5j - 2k
A) -0.24
C) 1.82
D) 1.57
8)
9)
B) 1.67
Find proj v u.
10) v = i + j + k, u = 4i + 12j + 3k
19
19
19
A)
i + j + k
13
13
13
C)
10)
19
19
19
B)
i + j + k
169
169
169
19
19
19
i + j + k
3
3
3
D)
1
20
20
20
i + j + k
3
3
3
11)
11) v = k, u = 2i + 6j + 9k
18
54
81
A)
i + j + k
121
121
121
18
54
81
B)
i + j + k
11
11
11
C) 9k
D)
9
k
121
Solve the problem.
12) How much work does it take to slide a box 47 meters along the ground by pulling it with a 196 N
force at an angle of 11° from the horizontal?
A) 192.4 joules
B) 9043 joules
C) 40.77 joules
D) 9212 joules
Find the magnitude and direction (when defined) of u × v.
13) u = 2i, v = 8j
A) 16; -k
B) 2; 8k
14) u = 2i + 2j - k, v = -i + k
1
2
2
A) 3; i + j - k
3
3
3
12)
13)
C) 16; k
D) 16; 16k
2
1
2
C) 9; i - j + k
9
9
9
2
1
2
D) 3; i - j + k
3
3
3
14)
2
1
2
B) 9; i + j - k
9
9
9
Solve the problem.
15) Find the area of the parallelogram determined by the points P(7, -5, 5), Q(-1, -2, 8), R(6, 7, -2) and
S(-2, 10, 1).
12,835
13 91
A)
B)
C) 13 91
D) 12,835
2
2
Find the triple scalar product (u x v) · w of the given vectors.
16) u = i + j + j; v = 3i + 7j + 5k; w = 8i + 6j + 2k
A) -88
B) 72
17) u = 4i + 2j - j; v = 10i + 5j - 5k; w = 1i + 6j - 3k
A) 55
B) 95
15)
16)
C) 56
D) -20
C) -175
D) -45
17)
Find parametric equations for the line described below.
18) The line through the points P(-1, -1, -7) and Q(5, 2, -3)
A) x = t - 6, y = t - 3, z = -7t - 4
B) x = 6t + 1, y = 3t + 1, z = 4t + 7
C) x = 6t - 1, y = 3t - 1, z = 4t - 7
D) x = t + 6, y = t + 3, z = -7t + 4
18)
Find a parametrization for the line segment beginning at P1 and ending at P2 .
19)
19) P1 (6, -5, 3) and P2 (0, -5, -4)
B) x = 6t, y = -5t, z = 7t - 4, 0 ≤ t ≤ 1
D) x = -6t + 6, y = -5, z = -7t + 3, 0 ≤ t ≤ 1
A) x = 6t, y = -5, z = 7t - 4, 0 ≤ t ≤ 1
C) x = -6t + 6, y = -5t, z = -7t + 3, 0 ≤ t ≤ 1
Solve the problem.
20) Find symmetric equations for the line through the points P(-1, -1, -3) and Q(2, 3, 4).
x - 1 y + 1 z + 3
x + 1 y + 1 z + 3
A)
= = B)
= = 3
4
7
3
4
7
C)
x + 1 y - 1 z + 3
= = 3
4
7
D)
2
x - 1 y - 1 z + 3
= = 3
4
7
20)
21) Find the symmetric equations of the line through (3, 2, 1) and perpendicular to the plane x + 2y 2z = 2.
x + 3 y + 2 z + 1
x - 3 y - 2 z - 1
= = B)
= = A)
2
1
2
1
-2
-2
C)
x - 3 y - 2 z - 1
= = 2
-1
-2
D)
x + 3 y + 2 z + 1
= = 2
-1
-2
22) Find the equation of the plane containing the line x = 2t, y = 6 - t, z = 4 and the point A) 2x - y - 2z = 10
C) 2x + y + 2z = 10
16 22
, , 4 .
5 5
22)
B) 2x + y - 2z = 10
D) 2x - y + 2z = 10
23) Find the equation of the plane through the point P(-2, 1, 3) and perpendicular to the line
x = 10 + 8t, y = -5 + 8t, z = 2 - t.
A) 8x + 8y - z = 11
B) 8x + 8y + z = -11
C) 8x + 8y - z = 19
D) 8x + 8y - z = -11
Find the following.
24) If v = -1, 6, 0 , find v .
A) 37
21)
23)
24)
B)
37
C) 2 37
D) 5
Find the distance between points P1 and P2 .
25) P1 (5, -8, -1) and P2 (10, 2, 9)
A) 15
25)
B) 14
C) 10
D) 18
Solve the problem.
26) Find the center and radius of the sphere with the equation 2x2 + 2y2 + 2z 2 - x + y - z - 9 = 0.
1
1 1
5 6
1 1
1
75
A) center = , - , ; radius = B) center = - , , - ; radius = 4
4
4 4
4 4
4
8
C) center = 1
1 1
5 6
, - , , radius = 2
4
4 4
D) center = 26)
1
1 1
5 3
, - , ,; radius = 4
4
4 4
27) Find the equation of the sphere with center = (0, -6, 1) and radius = 5.
A) x2 + (y + 6)2 + (z - 1)2 = 25
B) x2 + (y + 6)2 + (z + 1)2 = 25
27)
D) x2 - (y + 6)2 - (z - 1)2 = 25
C) x2 + (y - 6)2 + (z - 1)2 = 25
Identify the type of surface represented by the given equation.
x2 y2 z 2
+ = 28)
8
9
7
A) Hyperbolic paraboloid
B) Paraboloid
C) Elliptic cone
D) Ellipsoid
3
28)
29) x2 z 2 y
+ = 8
7
7
29)
A) Hyperbolic paraboloid
C) Ellipsoid
B) Elliptic cone
D) Paraboloid
Solve the problem.
30) The Cartesian coordinates of a point are ( -1, -1, -7). Find the cylindrical coordinates.
5π
5π
5π
5π
, -7
B) 2, , -7
C) 2, , -7
D) 2, , -7
A) 2, 4
2
2
4
3
31) The cylindrical coordinates of a point are 5, π, 10 . Find the Cartesian coordinates.
2
A) (0, -5, 10)
B) (0, 5, 10)
C) 0, - 5
2
2, 10
A) (0, 10, 0)
31)
D) (-5, 0, 10)
32) The Cartesian coordinates of a point are ( -1, -1, 0). Find the spherical coordinates.
5π π
5π π
5π π
, B) 2, , C) 2, , D)
A) 3, 4 2
8 4
4 2
33) The spherical coordinates of a point are 10, π, 32)
2, 5π
, 0
4
π
. Find the cylindrical coordinates.
2
B) (0, 0, -10)
C) (10, π, 0)
D)
33)
20
3
3, π, 0
34) The cylindrical coordinates of a point are (0, 0, 9). Find the spherical coordinates.
π
π
π
A) 9, , 0
B) (9, 0, 0)
C) 9, 0, D) 9, 0, 4
2
2
Make the required change in the given equation.
35) z = x2 + y2 to cylindrical coordinates
35)
D) r = cot φ csc φ
36) z = cot θ to Cartesian coordinates
B) z = 36)
x
y
C)
z x
= r y
37) r = 7 to spherical coordinates
x
r2
B) ρ2 (1 - cos φ) = 49
ρ
7
D) ρ = 7 sin φ
38) ρ = 2 cos θ sin φ to Cartesian coordinates
A) x2 + y2 + z 2 = 2y
C)
D) z = 37)
A) ρ = 7 csc φ
C) cos φ = 34)
B) z = r2
A) r tan θ = 1
C) z = r2 (cos θ + sin θ)
A) x2 + y2 = z
30)
38)
B) x2 + y2 + z 2 - 2z = 0
x2 + y2 + z 2 = 2x
D) x2 + y2 + z 2 - 2x = 0
4
Write the equation for the plane.
39) The plane through the point P(2, 4, 2) and normal to n = 3i + 7j + 6k.
A) -3x - 7y - 6z = 46
B) 2x + 4y + 2z = 46
C) 3x + 7y + 6z = 46
D) -2x - 4y - 2z = 46
40) The plane through the points P(5, -7, -18) , Q(-3, 8, 1) and R(-1, -5, 24).
A) 8x + 3y + z = -1
B) 3x + y + 8z = 1
C) 8x + 3y + z = 1
39)
40)
D) 3x + y + 8z = -1
Solve the problem.
41) Find an equation for the level curve of the function f(x, y) = 36 - x2 - y2 that passes through the
point 3, 5 .
A) x2 - y2 = 8
B) x2 + y2 = - 8
C) x2 + y2 = 8
D) x2 + y2 = 44
41)
42) Find an equation for the level surface of the function f(x, y, z) = x2 + y2 + z 2 that passes through
the point 4, 12, 3 .
A) x2 + y2 + z 2 = 13
B) x + y + z = ± 13
42)
C) x2 + y2 + z 2 = 169
D) x + y + z = 13
43) Find an equation for the level surface of the function f(x, y, z) = ln point e12, e3 , e4 .
1
xy
= A) ln 9
z
B) xy
= e11
z
C) xy
= e9
z
xy
that passes through the
z
D) ln Find all the first order partial derivatives for the following function.
y8
44) f(x, y) = ln x2
A)
2 ∂f
8
∂f
= - ; = x ∂y y
∂x
B)
2y8
8y7
∂f
∂f
= -ln ; = ln ∂x
∂y
x3
x2
C)
8 ∂f
2
∂f
= ; = ∂x y ∂y x
D)
2
8
∂f
∂f
= -ln ; = ln x ∂y
y
∂x
45) f(x, y) = e-x
x2 + y2
A)
2xe-x
2ye-x
∂f
∂f
= - ; = - 2
∂x
∂y
(x2 + y2 )
(x2 + y2 )2
B)
e-x(x2 + y2 + 2x) ∂f
2ye-x
∂f
= ; = ∂x
∂y (x2 + y2 )2
(x2 + y2 )2
C)
e-x(x2 + y2 + x) ∂f
ye-x
∂f
= - ; = - 2
∂x
∂y
(x2 + y2 )
(x2 + y2 )2
D)
e-x(x2 + y2 + 2x) ∂f
2ye-x
∂f
= - ; = - 2
∂x
∂y
(x2 + y2 )
(x2 + y2 )2
43)
xy
= 9
z
44)
45)
5
Find all the second order partial derivatives of the given function.
46) f(x, y) = x2 + y - ex+y
46)
A)
∂2 f
∂2 f
∂2 f
∂2 f
= 1 - ex+y; = - ex+y; = = -ex+y
2
2
∂y∂x
∂x∂y
∂x
∂y
B)
∂2 f
∂2 f
∂2 f
∂2 f
= 2 - ex+y; = - ex+y; = = -ex+y
∂y∂x ∂x∂y
∂x2
∂y2
C)
∂2 f
∂2 f
∂2 f
∂2 f
= 2 - y2 ex+y; = -x2 ex+y; = = - y2 ex+y
∂y∂x ∂x∂y
∂x2
∂y2
D)
∂2 f
∂2 f
∂2 f
∂2 f
= 2 + ex+y; = ex+y; = = ex+y
∂y∂x ∂x∂y
∂x2
∂y2
47) f(x, y) = ln (x2 y - x)
2xy - 2x2 y2 - 1 ∂2 f
x4
x2
∂2 f
∂2 f
∂2 f
A)
= ; = ; = = ∂x2
∂y2 (x2 y - x)2 ∂y∂x ∂x∂y
(x2 y - x)2
(x2 y - x)2
B)
2xy - 2x2 y2 - 1 ∂2 f
x2
x2
∂2 f
∂2 f
∂2 f
= ; = - ; = = - 2
2
∂y∂x ∂x∂y
∂x2
∂y2
(x2 y - x)
(x2 y - x)
(x2 y - x)2
C)
xy - x2 y2 - 1 ∂2 f
x4
x2
∂2 f
∂2 f
∂2 f
= ; = - ; = = - ∂x2
∂y2
(x2 y - x)2
(x2 y - x)2 ∂y∂x ∂x∂y
(x2 y - x)2
D)
2xy - 2x2 y2 - 1 ∂2 f
x4
x2
∂2 f
∂2 f
∂2 f
= ; = - ; = = - 2
2
2
2
∂y∂x
∂x∂y
∂x
∂y
(x2 y - x)
(x2 y - x)
(x2 y - x)2
47)
Solve the problem.
48) Evaluate A) - 2
dw
3
xy
at t = π for the function w(x, y, z) = ; x = sin t, y = cost, z = t2 .
dt
2
z
1
π2
B) - 2
1
π
C) - 4 1
9 π2
48)
D) 2
1
π2
Use implicit differentiation to find the specified derivative at the given point.
dy
49) Find at the point (1, 1) for 3x2 + 3y3 + 4xy = 0.
dx
A) - 50) Find A)
10
7
B)
10
13
C) - 1
49)
D) - 10
13
yz
∂z
at the point (8, 1, -1) for ln - exy+z 2 = 0.
x
∂y
2e9 - 1
1 - 8e9
B)
1 - 8e9
1 - 2e9
50)
C)
8e9 - 1
1 - 2e9
Find the derivative of the function at the given point in the direction of A.
51) f(x, y, z) = 9x - 2y - 4z, (4, -9, -6), A = 3i - 6j - 2k
53
47
41
B)
C)
A)
7
7
7
6
D)
1 - 2e9
1 - 8e9
51)
39
D)
7
Compute the gradient of the function at the given point.
52) f(x, y) = ln(-6x - 8y), (-9, -4)
9
2
3
4
A) - i - j
B) - i - j
86
43
43
43
52)
2
9
C) - i - j
43
86
1
1
D)
i + j
86
86
Provide an appropriate response.
53) Find the direction in which the function is increasing most rapidly at the point P0 .
53)
f(x, y, z) = xy - ln(z), P0 (1, 2, 2)
A)
21
(4i + 2j - k)
21
B)
1
(4i + 2j - k)
21
C)
1
(4i - 2j + k)
21
D)
1
(4i + 2j - k)
21
Solve the problem.
54) Write an equation for the tangent line to the curve x2 - 4xy + y2 = 6 at the point (-1, 1).
A) y = x - 2
B) y = x + 1
C) x + y = 1
D) x - y + 2 = 0
54)
55) Find parametric equations for the normal line to the surface -4x - 6y + 5z = 12 at the point
(1, -1, 2).
A) x = -4t + 1, y = -6t - 1, z = 5t + 2
B) x = -4t - 1, y = -6t + 1, z = 5t - 2
D) x = t - 4, y = -t - 6, z = 2t + 5
C) x = -t - 4, y = t - 6, z = -2t + 5
55)
56) Write parametric equations for the tangent line to the curve of intersection of the surfaces
z = 5x2 + 4y2 and z = x + y + 7 at the point (1, 1, 9).
56)
B) x = -9t + 1, y = 11t + 1, z = 2t + 9
D) x = -9t + 1, y = 9t + 1, z = 2t + 9
A) x = -7t + 1, y = 11t + 1, z = 2t + 9
C) x = -7t + 1, y = 9t + 1, z = 2t + 9
57) If the length, width, and height of a rectangular solid are measured to be 6, 3, and 5 inches
respectively and each measurement is accurate to within 0.1 inch, estimate the maximum
percentage error in computing the volume of the solid.
A) 5.60%
B) 7.00%
C) 8.40%
D) 6.30%
57)
Find all local extreme values of the given function and identify each as a local maximum, local minimum, or saddle
point.
58)
58) f(x, y) = x2 - 18x + y2 + 12y - 6
B) f(-9, 6) = 345, local maximum
D) f(9, 6) = 21, saddle point
A) f(9, -6) = -123, local minimum
C) f(-9, -6) = 201, saddle point
Find the extreme values of the function subject to the given constraint.
59) f(x, y) = x2 + 4y3 , x2 + 2y2 = 2
59)
A) Maximum: 4 at (0, 1); minimum: -4 at (0, -1)
B) Maximum: 8 at (2, 1); minimum: -4 at (0, -1)
C) Maximum: 8 at (2, 1); minimum: -31 at (1, -2)
D) Maximum: 4 at (0, 1); minimum: -31 at (1, -2)
Solve the problem.
60) A rectangular box with square base and no top is to have a volume of 32 ft 3 . What is the least
amount of material required?
A) 48 ft 2
B) 40 ft 2
C) 36 ft 2
D) 42 ft 2
7
60)
61) What is the largest possible volume of an open metal box made from 75 ft 2 of tin? Ignore the
thickness of the tin.
A) 31.2 ft 3
B) 62.5 ft 3
C) 12.5 ft 3
D) 125.0 ft 3
Find the indicated limit or state that it does not exist.
x2
62)
lim
cos 2
x + y2
(x, y) → (0, 0)
A)
π
2
61)
62)
B) 0
C) 1
D) No limit
Solve the problem.
63) Find the mass of a thin triangular plate bounded by the coordinate axes and the line x + y = 10 if
δ(x, y) = x + y.
2000
4000
1000
500
B)
C)
D)
A)
3
3
3
3
Find the area of the region specified by the integral(s).
4
4 - x
64)
dy dx
0 0
8
A)
B) 8
3
∫ ∫
63)
64)
C)
16
3
D) 4
Solve the problem.
65) Integrate f(x, y) = A) π(ln 4)2
ln(x2 + y2 )
over the region 1 ≤ x2 + y2 ≤ 16.
x2 + y2
C) 2π(ln 4)2
B) 2π ln 4
65)
D) π ln 4
66) Find the mass of the rectangular solid of density δ(x, y, z) = xyz defined by 0 ≤ x ≤ 7, 0 ≤ y ≤ 5,
0 ≤ z ≤ 2.
1225
2450
980
1225
B)
C)
D)
A)
2
3
3
3
66)
Find the volume of the indicated region.
67) the region bounded by the paraboloid z = 81 - x2 - y2 and the xy-plane
729
A) 243π
B)
π
C) 2187π
2
67)
6561
D)
π
2
Change the Cartesian integral to an equivalent polar integral, and then evaluate.
5
25 - x2
68)
dy dx
-5 0
∫ ∫
A)
25π
2
B)
125π
2
C)
8
π
2
68)
D)
5π
2
Evaluate the cylindrical coordinate integral.
6
9
7r
69)
z dz r dr dθ
2 2
0
320,705
A)
B) 641,410
2
∫ ∫ ∫
69)
C)
320,705
3
D)
1,603,525
4
Solve the problem.
70) Find the center of mass of the thin semicircular region of constant density δ = 7 bounded by the
x-axis and the curve y = 25 - x2 .
A) x = 0, y = 20
3π
B) x = 0, y = 40
3π
C) x = 0, y = 10
3π
D) x = 0, y = 5
3π
71) Let D be the region bounded below by the cone z = x2 + y2 and above by the sphere
z = 25 - x2 - y2 . Set up the triple integral in cylindrical coordinates that gives the volume of
using the order of integration dz dr dθ.
π/2 5
25 - r2
r dz dr dθ
A)
0
0 0
∫
C)
∫
∫ ∫
π/2
0
∫
5/ 2
0
∫
25 - r2
B)
∫
2π
∫
2π
0
D)
r dz dr dθ
∫ ∫
0
0
0
5
∫
0
25 - r2
71)
r dz dr dθ
0
5/ 2
∫
25 - r2
r dz dr dθ
0
Find the volume of the indicated region.
72) the region bounded below by the xy-plane, laterally by the cylinder r = 8 cos θ, and above by the
plane z = 7
A) 112π
B) 14π
C) 98π
D) 784π
Find the Jacobian 70)
72)
∂(x, y)
∂(x, y, z)
or (as appropriate) using the given equations.
∂(u, v)
∂(u, v, w)
73) x = 7u cos 3v, y = 7u sin 3v
A) 147v
B) 63v
73)
C) 147u
D) 63u
Use the given transformation to evaluate the integral.
74) u = x + y, v = -2x + y;
74)
∫ ∫ 4x dx dy,
R
where R is the parallelogram bounded by the lines y = -x + 1, y = -x + 4, y = 2x + 2, y = 2x + 5
A) -4
B) 4
C) 8
D) -8
Evaluate the line integral of f(x,y) along the curve C.
75) f(x, y) = x2 + y2 , C: y = 2x + 2, 0 ≤ x ≤ 3
A) 93 5
75)
B) 93
C) 31 5
9
D) 183 5
Find the work done by F over the curve in the direction of increasing t.
76) F = -5yi + 5xj + 8z 3 k; C: r(t) = cos ti + sin tj, 0 ≤ t ≤ 9
A) W = 90
B) W = 405
2
C) W = 45
76)
D) W = 0
Test the vector field F to determine if it is conservative.
77) F = 4x3 y4 z 4 i + 4x4 y3 z 4 j + 4x4 y4 z 3 k
77)
A) Conservative
B) Not conservative
Find the potential function f for the field F.
x
1
78) F = i - 5j - k
z
z2
78)
x
B) f(x, y, z) = + C
z
x
A) f(x, y, z) = - 5 + C
z
C) f(x, y, z) = x
D) f(x, y, z) = - 5y + C
z
2x
- 5y + C
z
Find the divergence of the field F.
79) F = -6x6 i + 5xyj + 5xzk
79)
A) -36x5 + 10x - 26
C) -36x5 + 5y + 5z
B) -36x5 + 10x
D) -26
Evaluate the work done between point 1 and point 2 for the conservative field F.
80) F = (y + z)i + xj + xk; P1 (0, 0, 0), P2 (9, 10, 8)
A) W = 0
B) W = 90
C) W = 162
Find the mass of the wire that lies along the curve r and has density δ.
81) r(t) = 6i + (5 - 3t)j + 4tk, 0 ≤ t ≤ 2π ; δ = 5(1 + sin 9t)
100
50
+ 50π units
B)
+ 50π units
C) 50π units
A)
9
9
Evaluate. The differential is exact.
(4, 4, 2)
82)
(2xy2 - 2xz 2 ) dx + 2x2 y dy - 2x2 z dz
(0, 0, 0)
A) 0
B) 320
80)
D) W = 18
81)
D) 10π units
∫
82)
C) 192
D) 384
Apply Greenʹs Theorem to evaluate the integral.
83)
C
(y2 + 6) dx + (x2 + 1) dy
83)
C: The triangle bounded by x = 0, x + y = 1, y = 0
2
B) -3
A)
3
C) 0
10
D) 4
84)
C
(6y dx + 5y dy)
84)
C: The boundary of 0 ≤ x ≤ π, 0 ≤ y ≤ sin x
A) -1
B) 2
C) 0
Find the potential function f for the field F.
85) F = (y - z)i + (x + 2y - z)j - (x + y)k
A) f(x, y, z) = x + y2 - xz - yz + C
D) -2
85)
B) f(x, y, z) = xy + y2 - x - y + C
D) f(x, y, z) = x(y + y2 ) - xz - yz + C
C) f(x, y, z) = xy + y2 - xz - yz + C
11
Answer Key
Testname: CAL3‐FINAL‐REVIEW
1) D
2) A
3) A
4) A
5) A
6) B
7) C
8) C
9) C
10) C
11) C
12) B
13) C
14) D
15) C
16) D
17) A
18) C
19) D
20) B
21) A
22) D
23) D
24) B
25) A
26) D
27) A
28) C
29) D
30) D
31) A
32) C
33) C
34) B
35) B
36) B
37) A
38) D
39) C
40) C
41) C
42) C
43) B
44) A
45) D
46) B
47) D
48) C
49) D
50) B
12
Answer Key
Testname: CAL3‐FINAL‐REVIEW
51) C
52) B
53) B
54) D
55) A
56) C
57) B
58) A
59) A
60) A
61) B
62) D
63) D
64) B
65) C
66) B
67) D
68) A
69) A
70) A
71) D
72) A
73) C
74) A
75) A
76) C
77) A
78) D
79) B
80) C
81) C
82) C
83) C
84) D
85) C
13