Mathematics 2172-001
Calculus II
Summer Session II, 15
Final Examination
Instructor: Dr. Alexandra Shlapentokh
(1) One of the graphs in Figures
0, k < 0. Which one is it?
(a) Figure 1
(b) Figure 2
(c) Figure 3
(d) Figure 4
(e) Figure 5
(2) One of the graphs in Figures
0, k < 0. Which one is it?
(a) Figure 1
(b) Figure 2
(c) Figure 3
(d) Figure 4
(e) Figure 5
(3) One of the graphs in Figures
0, k > 0. Which one is it?
(a) Figure 1
(b) Figure 2
(c) Figure 3
(d) Figure 4
(e) Figure 5
(4) One of the graphs in Figures
0, k > 0. Which one is it?
(a) Figure 1
(b) Figure 2
(c) Figure 3
(d) Figure 4
(e) Figure 5
1–5 is the graph of y = aekx for real numbers a <
1–5 is the graph of y = aekx for real numbers a >
1–5 is the graph of y = aekx for real numbers a <
1–5 is the graph of y = aekx for real numbers a >
(5) One of the graphs in Figures 6–10 is the graph of y = a ln(|x|) for a real number
a > 0. Which one is it?
(a) Figure 6
(b) Figure 7
1
F IGURE 1.
F IGURE 2.
F IGURE 3.
F IGURE 4.
(c) Figure 8
(d) Figure 9
2
F IGURE 5.
(6)
(7)
(8)
(9)
(e) Figure 10
One of the graphs in Figures 6–10 is the graph of y = a ln(|x|) for a real number
a < 0. Which one is it?
(a) Figure 6
(b) Figure 7
(c) Figure 8
(d) Figure 9
(e) Figure 10
One of the graphs in Figures 6–10 is the graph of y = a ln(−x) for a real number
a > 0. Which one is it?
(a) Figure 6
(b) Figure 7
(c) Figure 8
(d) Figure 9
(e) Figure 10
One of the graphs in Figures 6–10 is the graph of y = a ln(x) for a real number
a > 0. Which one is it?
(a) Figure 6
(b) Figure 7
(c) Figure 8
(d) Figure 9
(e) Figure 10
1
Determine the limit: limn→∞ n+5
.
(a) The limit does not exist.
(b) ∞
(c) 1
(d) 0
(e) None of the above
3
F IGURE 6.
F IGURE 7.
F IGURE 8.
F IGURE 9.
4
F IGURE 10.
(10) Which of the following statements is true?
(a) 2log2 x = 2x
(b) ln(2x ) = x
(c) elog2 x = x
x
(d) log3 x = ln
ln 3
(e) None of the above
(11) Compute the derivative of y = log2 x.
(a) y 0 = x2 1ln 2
1
(b) y 0 =
x ln 2
1
0
(c) y =
x ln 3
(d) y 0 = x ln 2
(e) None of the above
(12) Compute the derivative of y = 2x .
(a) 2x ln 2
2x
(b)
ln 2
(c) 2x − ln 2
(d) 2x
(e) None of the above
(13) Compute the derivative of y = eπ
(a) 0
(b) 1
(c) 2
(d) eπ
(e) None of the above
(x − 1)(x + 1)2
.
(14) Compute the derivative of y = ln
(x − 3)(x − 4)3
1
2
1
3
(a)
−
−
−
x−1 x+1 x−3 x−4
1
1
1
3
(b)
+
−
−
x−1 x+1 x−3 x−4
1
2
1
3
(c)
+
−
−
x−1 x+1 x−3 x−4
5
F IGURE 11.
2
y
1
x
−20
−10
10
20
−1
−2
(d)
2
1
1
1
+
−
−
x−1 x+1 x−3 x−4
(e) None of the above
(x − 1)(x + 1)2
(15) Compute the derivative of y =
.
3
(x − 3)(x
− 4)
1
2
1
3
(a)
+
−
−
x−1 x+1 x−3 x−4
2
1
3
1
(x − 1)(x + 1)
+
−
−
(b)
x − 1 x + 1 x − 3 x − 4 (x − 3)(x − 4)3
1
2
1
3
(x − 1)(x + 1)2
(c)
+
−
−
x − 1 x + 1 x − 3 x − 4 (x − 3)(x − 4)3
1
2
1
3
(d)
+
−
−
(x − 1)(x + 1)2 (x − 3)(x − 4)3
x−1 x+1 x−3 x−4
(e) None of the above
(16) Compute the derivative of f (x) = x2x .
(a) The derivative does not exist.
(b) f 0 (x) = (2 ln x + 2)x2x
(c) f 0 (x) = (2 ln x − 2)x2x
(d) f 0 (x) = (2 ln x + 2)
(e) None of the above
(17) Which of the following statements are true.
(a) The range of y = arctan(x) is the same as the range of y = arcsin(x)
(b) The range of y = arctan(x) is the same as the range of y = arccos(x)
(c) The range of y = arccos(x) is the same as the range of y = arcsin(x)
(d) The domain of y = arctan(x) is the same as the domain of y = arcsin(x)
(e) None of the above
(18) One of the graphs in Figures 11 – 15 is the graph of y = arccos(ax) for a real
number a > 0. Which one is it?
(a) Figure 11
6
F IGURE 12.
2
y
1
x
−1
1
−1
−2
F IGURE 13.
y
3
2
1
x
−5 −4 −3 −2 −1
1
2
3
4
5
F IGURE 14.
5
y
3
1
x
1
2
3
−1
−3
(b) Figure 12
(c) Figure 13
(d) Figure 14
(e) Figure 15
(19) One of the graphs in Figures 11 – 15 is the graph of y = arcsin(ax) for a real
number a > 0. Which one is it?
7
2
y
F IGURE 15.
x
1
2
3
−2
−4
−6
−8
−10
−12
(20)
(21)
(22)
(23)
(24)
(a) Figure 11
(b) Figure 12
(c) Figure 13
(d) Figure 14
(e) Figure 15
One of the graphs in Figures 11 – 15 is the graph of y = arctan(ax) for a real
number a > 0. Which one is it?
(a) Figure 11
(b) Figure 12
(c) Figure 13
(d) Figure 14
(e) Figure 15
One of the graphs in Figures 11 – 15 is the graph of y = ax ln(x) for a real number
a > 0. Which one is it?
(a) Figure 11
(b) Figure 12
(c) Figure 13
(d) Figure 14
(e) Figure 15
One of the graphs in Figures 11 – 15 is the graph of y = a + ln(x)
for a real number
x
a > 0. Which one is it?
(a) Figure 11
(b) Figure 12
(c) Figure 13
(d) Figure 14
(e) Figure 15
Express cos2 (arcsin(x2 )) as a rational function of x.
(a) 1 + x
(b) 1 − x
(c) 1 − x2
(d) 1 − x4
(e) None of the above
Compute the derivative of f (x) = arctan(2x).
8
2
1 − 4x2
2
(b) f 0 (x) =
1 + x2
2
(c) f 0 (x) =
1 + 4x2
1
(d) f 0 (x) =
1 + 4x2
(e) NoneZof the above
1
√
dx
Compute
1 − 9x2
(a) arcsin(x) + C
1
(b) arcsin(3x) + C
3
1
(c) arccos(3x) + C
3
1
(d) arctan(x) + C
3
(e) NoneZof the above
1/6
3
√
Compute
dx
1 − 9x2
0
(a) π
(b) π/2
(c) π/3
(d) π/6
(e) NoneZof the above
1
Compute
dx
1 + 4x2
(a) arctan(x) + C
1
(b) arctan(2x) + C
2
1
(c) arccos(2x) + C
2
1
(d) arcsin(2x) + C
2
(e) NoneZof the above
1/2
2
Compute
dx
1 + 4x2
0
(a) π
(b) π/2
(c) π/3
(d) π/4
(e) NoneRof the above
2
Compute 1 ln(x)dx.
(a) ln 4
(b) ln 5 − 1
(c) ln 4 − 1
(d) ln 2 − 1
(a) f 0 (x) =
(25)
(26)
(27)
(28)
(29)
9
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(e) NoneRof the above
2
Compute 1 2x ln(x)dx.
(a) ln 16 − 3
(b) ln 16 − 7
(c) ln 16 − 32
(d) ln 2 − 27
(e) NoneRof the above
1
Compute 0 x21−4 dx
(a) − 17 ln 3
(b) − 14 ln 2
(c) − 15 ln 3
(d) − 14 ln 3
(e) NoneRof the above
π
Compute 0 2 cos2 (x)dx
(a) π
(b) 2π
(c) −π
(d) 12 π
(e) None of
the√above
R 1/2
1 − 4x2 dx
Compute 2 0
(a) π
(b) 2π
(c) −π
(d) 14 π
(e) NoneRof the above
∞
Compute 1 x12 dx
(a) 1
(b) 2
(c) 3
(d) ∞
(e) NoneRof the above
1 1
Compute 0 x1/2
dx
(a) 1
(b) 2
(c) 1/3
(d) ∞
(e) None of the above
P
3
Compute the following sum: ∞
n=1 n .
(a) The series diverges.
(b) 2
(c) 4
(d) 6
(e) None of the above
Compute the first 3 terms of the MacLaurin series for 2 cos x:
(a) 2 − x2 − 61 x4 + . . .
10
(38)
(39)
(40)
(41)
(42)
(43)
1 4
(b) 2 − x2 + 12
x + ...
1 4
2
(c) −1 − x + 6 x + . . .
(d) 1 − x2 + 16 x4 + . . .
(e) None of the above
2x
.
Compute the limit limx→∞ x+sin
ex
(a) ∞
(b) 0
(c) -1
(d) 2
(e) None of the above
Compute the area bounded by the curves y = x2 − 2 and y = x − 2.
(a) 65
(b) 76
(c) 11
6
(d) 16
(e) None of the above
Which of the formulas below computes the arc length for the curve y = 23 x3/2
between
R 1 √x = 0 and x = 1.
(a) 0 1 + x2 dx
R1
(b) 0 (1 + x)dx
R1
√
(c) 0 1 + xdx
R1√
(d) 0 1 + xdx
(e) None of the above
Compute the arc length for the curve y = 23 x3/2 between x = 0 and x = 1.
√
(a) 8 − 23
√
(b) 8√+ 1
(c) 23 ( 8 − 1)
√
(d) 8 − 2
(e) None of the above
Which of the following formulas computes the surface area of the solid obtained
by rotation of the region bounded by the graph of y = ex , the line x = 1, and the
line yR = 0 √
about x-axis?
1
(a) 0 2π 1 + e2x dx
R1 √
(b) 0 ex 1 + e2x dx
√
R1
(c) 0 2πex 1 + e2x dx
R1
√
(d) 0 2πex 1 + ex dx
(e) None of the above
Suppose the doubling time for a colony of a bacteria is 3 hours. How long before
the colony triples?
(a) log3 27
(b) log2 27
(c) ln 27
(d) log10 27
11
(e) None of the above
12
Key
1c, 2d, 3b, 4a, 5d, 6c, 7e, 8a, 9d, 10d, 11b, 12a, 13a, 14c, 15c, 16b, 17e, 18c, 19b, 20a
21d 22e, 23d, 24c, 25b, 26d, 27b, 28d, 29c, 30c 31d, 32a, 33d, 34a, 35b, 36a, 37b, 38b,
39d, 40d, 41c, 42e, 43b.
13