Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the graph to evaluate the limit.
1) lim f(x)
x→0
1)
y
4
3
2
1
-4
-3
-2
-1
1
2
4 x
3
-1
-2
-3
-4
A) -3
B) 0
C) does not exist
D) 3
Find all points where the function is discontinuous.
2)
2)
A) x = 0
C) x = -2, x = 2
B) x = -2, x = 0, x = 2
D) None
Find the limit.
3)
x
lim
3x
+2
x→-1
A) -
1
5
3)
B) does not exist
C) 1
D) 0
Find the average rate of change of the function over the given interval.
7
4) y = 4x2 , 0,
4
A) 2
B) -
3
10
C)
1
1
3
4)
D) 7
Find the limit.
5) lim
x→3
A)
9x + 92
5)
B) - 119
119
C) -119
D) 119
Provide an appropriate response.
6) Is f continuous at f(2)?
-x2 + 1,
3x,
f(x) = -3,
-3x + 6
5,
6)
6
-1 ≤ x < 0
0<x<1
x=1
1<x<3
3<x<5
d
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
t
-2
-3
-4
(1, -3)
-5
-6
A) Yes
B) No
Find the limit.
7)
lim
x→π/3
A) 0
9 2 + sin(π sec x)
7)
B) 1
C)
2
92 + 1
D) 9
The graph of a function is given. Choose the answer that represents the graph of its derivative.
8)
8)
y
15
10
5
-15 -10
-5
5
15 x
10
-5
-10
-15
A)
B)
y
-15 -10
y
15
15
10
10
5
5
-5
5
10
15 x
-15 -10
-5
-5
-5
-10
-10
-15
-15
C)
5
10
15 x
5
10
15 x
D)
y
-15 -10
y
15
15
10
10
5
5
-5
5
10
15 x
-15 -10
-5
-5
-5
-10
-10
-15
-15
Find the derivative of the function.
θ-9
9) r =
θ+9
A) r ′ =
C) r ′ = -
9)
9
B) r ′ =
θ(θ + 9)2
9
D) r ′ =
θ(θ + 9)2
3
18
(θ + 9) θ2 - 81
9
θ+9
Find
d2 y
for the given function.
dx 2
x
10
10) y = 3 cot
A) -
10)
3
x
csc2
10
10
C) 6 csc2
B) -6 csc
x
x
cot
10
10
D)
x
10
3
x
x
csc2
cot
50
10
10
Find y ′.
11) y =
2
+x
x
A) -
4
x3
2
-x
x
- 2x
11)
B) -
8
+ 2x
x
C) -
8
x3
- 2x
D)
8
x3
+ 2x
Find the value(s) of x for which the slope of the curve y = f(x) is 0.
8x2
12) f(x) =
x2 + 1
A) x =
1
8
B) x = 0
12)
C) x = -8
D) x = -
1
8
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the
value of the indicated derivative.
13) u(1) = 4, u ′(1) = -6, v(1) = 7, v ′(1) = -3.
13)
d
(2u - 4v) at x = 1
dx
A) 36
C) -20
B) 0
D) -24
Find the derivative of the function.
14) r = (sec θ + tan θ)-3
14)
A) -3(sec θ tan θ + sec2 θ)-4
B) -3(sec θ + tan θ)-4
C) -3(sec θ + tan θ)-4 (tan2 θ + sec θ tan θ)
D)
-3 sec θ
(sec θ + tan θ)3
Find the extreme values of the function and where they occur.
15) f(x) = x3 - 12x + 2
A) Local maximum at (0, 0).
B) None
C) Local maximum at (2, -14), local minimum at (-2, 18).
D) Local maximum at (-2, 18), local minimum at (2, -14).
4
15)
Provide an appropriate response.
16) Suppose the velocity of a body moving along the s-axis is
ds
= 9.8t - 8.
dt
16)
Is it necessary to know the initial position of the body to find the body's displacement over some
time interval? Justify your answer.
A) No, displacement has nothing to do with the position of the body.
B) Yes, integration is not possible without knowing the initial position.
C) No, the initial position is necessary to find the curve s= f(t) but not necessary to find the
displacement. The initial position determines the integration constant. When finding the
displacement the integration constant is subtracted out.
D) Yes, knowing the initial position is the only way to find the exact positions at the beginning
and end of the time interval. Those positions are needed to find the displacement.
Find the largest open interval where the function is changing as requested.
17) Increasing y = (x2 - 9)2
A) (3, ∞)
B) (-∞, 0)
C) (-3, 0)
17)
D) (-3, 3)
Solve the problem.
18) A private shipping company will accept a box for domestic shipment only if the sum of its length
and girth (distance around) does not exceed 120 in. What dimensions will give a box with a square
end the largest possible volume?
A) 20 in. × 20 in. × 100 in.
C) 20 in. × 40 in. × 40 in.
18)
B) 20 in. × 20 in. × 40 in.
D) 40 in. × 40 in. × 40 in.
19) At noon, ship A was 12 nautical miles due north of ship B. Ship A was sailing south at 12 knots
(nautical miles per hour; a nautical mile is 2000 yards) and continued to do so all day. Ship B was
sailing east at 6 knots and continued to do so all day. The visibility was 5 nautical miles. Did the
ships ever sight each other?
A) No. The closest they ever got to each other was 6.4 nautical miles.
B) No. The closest they ever got to each other was 5.4 nautical miles.
C) Yes. They were within 3 nautical miles of each other.
D) Yes. They were within 4 nautical miles of each other.
5
19)
Provide an appropriate response.
20) Find the standard equation for the position s of a body moving with a constant acceleration a along
a coordinate line. The following properties are known:
d2 s
i.
= a,
dt2
ii.
20)
ds
= v0 when t = 0, and
dt
iii. s = s0 when t = 0,
where t is time, s0 is the initial position, and v0 is the initial velocity.
at2
A) s = at2 + v0 t + s0
B) s =
- v0 t - s0
2
C) s =
at2
+ s0
2
D) s =
at2
+ v0 t + s0
2
Find the extreme values of the function and where they occur.
21) f(x) = (x - 4)2/3
A) Absolute minimum value is 0 at x = 4.
C) There are no definable extrema.
21)
B) Absolute maximum value is 0 at x = -4.
D) Absolute minimum value is 0 at x = -4.
Find the most general antiderivative.
22)
23)
∫(
∫
t-
4
t) dt
22)
A)
3 3/2 5 5/4
t
- t
+C
2
4
B)
C)
2 3/2 4 5/4
t
- t
+C
3
5
D)
t-
3
t+ C
-1 1/2 1 -3/4
t
- t
+C
2
4
y
8
dy
+
7
y
23)
A)
2 3/2
y
+ 16 y + C
21
B)
2 3/2
y
- 16 y + C
21
C)
1
14
D)
3 3/2 1
y
+
14
16
y-
1
+C
16 y
y+ C
Find the area of the shaded region.
24)
A)
23
3
24)
B) 5
C) 3
6
D)
5
3
Evaluate the integral by using substitution.
x cos 24x2 sin 24x2
25)
dx
8
∫
A)
sin3/2 x
+C
576
B)
25)
sin3/2 24x2
+C
576
C)
sin3/2 x2
+C
24
sin 24x2
+C
384
D)
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed.
26) f(x) = x2 between x = 2 and x = 6 using a left sum with four rectangles of equal width.
26)
A) 86
B) 54
Express the sum in sigma notation.
27) 2 + 3 + 4 + 5 + 6 + 7
5
A) ∑ k + 2
k=2
C) 69
D) 62
27)
5
B)
∑
4
k
∑
C)
k=0
(-1)2k k
5
∑
D)
k = -1
k+2
k=0
Evaluate the integral.
28)
∫ sin (6x - 2) dx
A)
28)
1
cos (6x - 2) + C
6
C) -
B) -cos (6x - 2) + C
1
cos (6x - 2) + C
6
D) 6 cos (6x - 2) + C
Find the average value of the function over the given interval.
29) f(x) = 2x + 14 on [-7, 7]
A) 28
B) 196
C) 7
Evaluate the integral.
4 2
t +1
30)
dt
t
1
92
A)
5
29)
D) 14
∫
30)
B)
72
5
C) 32
D)
77
5
Solve the problem.
31) Suppose that f and g are continuous and that
∫
11
f(x) dx = -4 and
7
Find
∫
∫
11
g(x) dx = 9.
7
11
7
A) -26
f(x) - 2g(x) dx .
B) -13
C) 14
7
D) -22
31)
Estimate the value of the quantity.
32) The table shows the velocity of a remote controlled race car moving along a dirt path for 8 seconds.
Estimate the distance traveled by the car using 8 subintervals of length 1 with left-end point values.
Time Velocity
(sec) (in./sec)
0
0
1
10
2
21
3
17
4
27
5
30
6
32
7
12
8
5
A) 149 in.
Express the sum in sigma notation.
33) 4 + 8 + 12 + 16 + 20
5
A) ∑ 4(k + 1)
k=1
B) 139 in.
C) 298 in.
32)
D) 154 in.
33)
6
B)
∑
5
∑
C)
4k
k=1
4
4(k - 1)
∑
D)
k=2
4(k + 1)
k=0
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
34) y = 7x, y = 7, x = 0
34)
343
343
343
A) 98π
B)
π
C)
π
D)
π
2
3
4
Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method.
35) The region in the first quadrant bounded by x = 4y - y2 and the y-axis about the y-axis
35)
A)
512
π
5
B)
512
π
15
C)
1024
π
15
D)
128
π
5
Find the area enclosed by the given curves.
π
π
36) y = sin x, y = csc 2 x, ≤ x ≤
3
2
A) 1 -
37) y =
3
2
B)
36)
3 1
2
2
C)
3 1
2
3
D)
1 2
x , y = -x2 + 6
2
A) 32
3 1
+
2
3
37)
B) 16
C) 4
8
D) 8
Solve the problem.
38) A water tank is formed by revolving the curve y = 3x4 about the y-axis. Find the volume of water
in the tank as a function of the water depth, y.
2π 3/2
3π 3/2
A) V(y) =
y
B) V(y) =
y
3 3
2 3
C) V(y) =
π
2 3
y1/2
D) V(y) =
9
π 9
y
9
38)
Answer Key
Testname: 1710 FINAL REVIEW
1) C
2) A
3) C
4) D
5) A
6) B
7) D
8) B
9) A
10) D
11) C
12) B
13) B
14) D
15) D
16) C
17) A
18) B
19) B
20) D
21) A
22) C
23) A
24) A
25) B
26) B
27) D
28) C
29) D
30) B
31) D
32) A
33) D
34) B
35) B
36) C
37) B
38) A
10