MATH 104 – Practice Problems for Exam 2
1. Find the area between:
√
√
(a) x = 0, y = 1/ 1 + √
x2 , y = x/ 2
√
2
Answer: ln(1 + 2) −
4
2x
2x
(b) y = 3e , y = xe , x = 0
e6 7
Answer:
−
4
4
x
and the x axis, for 2 ≤ x ≤ 4.
(c) y = 2
x −1
ln 5
Answer:
2
2. Calculate the volume obtained by rotating:
(a) The region in problem 1a around the x-axis
π2 π
−
Answer:
4
6
(b) The region √
in problem
1a around the y-axis
!
5 2
Answer: 2π
−1
6
(c) The region in problem
! 1b around the x-axis
12
71
11e
−
Answer: π
32
32
(d) The region in problem 1b around the y-axis
Answer: π(e6 + 2)
(e) The region in problem 1c around the x-axis
Answer: π( 21 ln 3 − 41 ln 5 + 15 )
(f) The region in problem 1c around the y-axis
Answer: π(2 ln 3 − ln 5 + 4)
(g) The region in problem 1c around the line x = 1
Answer: 2π(ln 3 − ln 5 + 2)
(h) The region in problem 1c around the line y = −1
Answer: π( 21 ln 3 + 34 ln 5 + 15 )
3. Integrate: (straightforward)
(a)
Z
x4 e2x dx
Answer: 14 e2x (2x4 − 4x3 + 6x2 − 6x + 3) + C
(b)
Z
x2 tan−1 (x) dx
Answer: 13 x3 arctan x − 16 x2 + 61 ln(1 + x2 ) + C
Z
x
(c)
dx
2
x − 5x + 4
Answer: 34 ln(x − 4) − 13 ln(x − 1) + C
Z √
(d)
1 + 4x2 dx
√
√
Answer: 12 x 1 + 4x2 + 14 ln(2x + 1 + 4x2 ) + C
Z
1
√ dx
(e)
1 + √x
√
Answer: 2 x − 2 ln(1 + x) + C
√
Z
cos2 x
√
(f)
dx
√
√x
Answer: x + 12 sin(2 x) + C
sec(ln x) tan(ln x)
dx
x
Answer: sec(ln x) + C
(g)
Z
4. Integrate: (trickier)
(a)
Z
sin4 (2x) dx
3
Answer: 38 x − 16
cos(2x) sin(2x) − 18 cos(2x) sin3 (2x) + C
Z √ 2
x − 25
dx
(b)
√x
Answer: x2 − 25 + 5 arcsin(5/x) + C (Note that arcsin(5/x) = π − arcsec(x/5)
[Why?])
et
dt
e2t − 4
Answer: 14 (ln(et − 2) − ln(et + 2)) + C
Z √
(d)
1 + ex dx
√
√
√
Answer: 2 1 + ex + ln( 1 + ex − 1) − ln( 1 + ex + 1) + C
(c)
(e)
Z
Z
√
e
x
dx
√
√ √
Answer: 2 xe x − 2e x + C
5. Evaluate:
Z ∞
1
dx
x(ln x)3
e
Answer: 1/2
(a)
Z ∞
dx
x(x + 4)
0
Answer: π/2
(b)
(c)
√
Z 1s
1−y
y
Answer: π/2
dy
0
6. Find the general solution to each of the following differential equations:
dy
= y2
(a) x
dx
Answer: y = 1/(C − ln(x))
dy
=y
(b) (x2 + 1)
dx arctan x
Answer: y = Ce
7. Find the specific solution of each equation that satisfies the given condition:
dy
(a)
= xy,
y(1) = 3
dx
2
Answer: y = 3e(x −1)/2
dy
(b)
= xy + x,
y(0) = 10
dx
2
Answer y = 11ex /2 − 1
8. In a second-order chemical reaction, the reactant A is used up in such a way that
the amount of it present decreases at a rate proportional to the square of the
amount present. Suppose this reaction begins with 50 grams of A present, and
after 10 seconds there are only 25 grams left. How long after the beginning of the
reaction will there be only 10 grams left? Will all of the A disappear in a finite
time, or will there always be a little bit present?
Answer: 40 seconds, and there will always be a little bit present.
9. According to Newton’s law of heating and cooling, if the temperature of an object
is different from the temperature of its environment, then the temperature of the
object will change so that the difference between the object’s temperature and the
ambient temperature decreases at a rate proportional to this difference.
On a hot day, a thermometer was brought outdoors from an air-conditioned building. The temperature inside the building was 21◦ C, and so this is what the
thermometer read at the moment it was brought outside. One minute later the
thermometer read 27◦ C, and a minute after that it read 31◦ C. What was the temperature outside? (Impress us and express the answer without using logarithms or
the number e.)
Answer: 39◦ C
10. A super-fast-growing bacteria reproduces so quickly that the rate of production
of new bacteria is proportional to the square of the number already present. If
a sample starts with 100 bacteria, and after 3 hours there are 200 bacteria, how
long (after the starting time) will it take until there are (theoretically) an infinite
number of bacteria?
Answer: 6 hours
11.
12.
x2
dx =
0
1 − x2
π
2π
(b)
(a)
15
4
Answer: B
Z 1
√
Z ∞
(c)
2π
5
(d)
π
2
(e)
3π
5
(f) 2π
x2 e−2x dx =
0
(a) 1/4
Answer: A
13.
(b) 4/3
(c) 2
x3 − 6x − 4
dx =
4 x2 − x − 6
(a) 2 + ln(3)
(e) e1/2
(d) 3/8
(f) diverges
Z 6
(b) 4/3
(d) 47/24
Answer: C
(c) 12 + ln(3)
(e) 2 + ln(4/3)
(f) 10 + ln(3)
14. What is the volume of the solid obtained by rotating the region between the graph
1
and the x-axis for 0 ≤ x ≤ 1 around the y-axis?
of y = 2
x + 4x + 3
(a) π(ln 2 + 2 ln 3)
(b) 2π(4 ln 3 − 5 ln 2)
(c) 2π ln 12
(d) π(2 ln 3 + 3 ln 2)
Answer: F
15.
Z ∞
0
x2
(f) π(5 ln 2 − 3 ln 3)
(e) 2π ln 18
1
dx =
+ 2x + 2
(a) 1
(b) π
Answer: D
(c)
π
2
(d)
π
4
(e)
π
−1
2
(f) diverges
16. Find the surface area of the surface obtained by revolving the part of the graph of
y = x3 /9 where 0 ≤ x ≤ 2 around the x-axis.
38π
121π
76π
77π
98π
86π
(a)
(b)
(c)
(d)
(e)
(f)
27
72
9
48
81
27
Answer: E
17. Solve the initial-value problem:
(a) y = ln(sec x)
π
(d) ln(sec x)
4
Answer: A
dy
= ey sin x, y(0) = 0.
dx
1
(b) ln(cos x)
2
1
(e) ln(sec x)
2
18. The function
f (x) =


kx 0 ≤ x ≤ 1

0
(c)
π
+ ln(cos x)
4
(f) ln(cos x)
otherwise
is a probability density function for a certain value of k. Find the mean of that
probability density function.
3
(a)
4
Answer: D
(b)
1
2
(c)
1
4
(d)
2
3
(e)
1
3
(f) 2
19. If water leaks out of a small hole in a cylindrical bucket, then the height of the
water level above the bottom of the bucket decreases at a rate proportional to the
square root of the height. If the water level starts out at a height of 25 cm, and if
after 10 minutes it is down to 16 cm, how long after the start will the bucket be
empty?
(a) 30 min
(d) 60 min
Answer: C
(b) 40 min
(c) 50 min
(f) the bucket will never be completely empty
(e) 70 min
x3/2
between x = 0 and
20. What is the length of the part of the curve y = x1/2 −
3
x=2?
7√
7√
25 √
11 √
11 √
5√
2
(b)
2
(c)
2
(d)
2
(e)
2
(f)
2
(a)
3
3
6
6
3
6
Answer: A
21.
Z e
x4 ln x dx =
0
3e4
4e5
(a)
(b)
16
25
Answer: B
Z ∞√
1 + x2
22.
dx =
x6
1
2 √
(a)
( 2 + 1)
15
(c)
5e6
36
(b)
(d)
6e7
49
1 √
(4 2 + 1)
15
(e)
7e8
64
(c)
(f)
8e9
81
1 √
(2 2 − 1)
3
2 √
(d) ( 2 − 1)
5
Answer: A
23.
(e)
1 √
(4 2 − 1)
5
(f) diverges
Z 4
x
dx =
− 6x + 5
3
(a) 21 ln 3 − 3 ln 2
x2
(b)
1
4
(c) − 41 ln 3 − 3 ln 2
ln 3 − ln 2
(e) − 14 ln 3 − ln 2
(d) − 61 ln 3 − ln 2
Answer: E
(f) − 21 ln 3 − ln 2
24. What is the surface area of the surface obtained by rotating the part of the curve
2
y = x3 for 0 ≤ x ≤ 1 around the x-axis?
3
π √
π √
π √
(a)
(5 5 − 1)
(b)
(7 7 − 1)
(c)
(8 8 − 1)
18
21
24
√
√
√
π
π
π
(10 10 − 1)
(17 17 − 1)
(21 21 − 1)
(d)
(e)
(f)
27
36
45
Answer: A
25. Let y(x) be the solution of initial-value problem y 0 + 4xy = 0, y(0) = 1. Then
y(2) =
(a) e−2
Answer: D
(b) e−4
(c) e−6
(d) e−8
(e) e−10
(f) e−12
26. Some enterprising Penn scientists have created a sample of Unobtanium in their
lab. One of the remarkable properties of this material is that when it is heated,
contrary to Newton’s law of cooling, its temperature decreases to room temperature
at a rate proportional to the square root of the difference between its temperature
and the ambient temperature. In a laboratory kept at 20 degrees C, the sample
is heated to a temperature of 36 degrees C. After 2 minutes have passed, the
temperature of the sample is 29 degrees C. How long after the initial heating will
the sample’s temperature be equal to the room temperature?
27.
(a) 6 minutes
(b) 8 minutes
(c) 10 minutes
(d) 12 minutes
Answer B
(e) 14 minutes
(f) 16 minutes
Z π/8
tan 4t dt =
0
(a) 1
(b) ln 2
Answer: F
(c)
1
ln 2
2
(d) 1 −
1
ln 2
2
(e)
√
2 − ln 2
(f) diverges
28. The function
f (x) =


kxe−4x x ≥ 0

0
otherwise
is a probability density function for a certain value of k. Find the mean of that
probability density function.
1
2
1
1
2
(c)
(d)
(e)
(f)
(a) 1
(b)
3
2
5
3
8
Answer: C
29. The functions y1 (t) and
y (t) are both solutions of the autonomous differential
2
dy
y
but satisfy different initial conditions: y1 (0) = 1 and
equation
= 3 sin
dt
2
y2 (0) = −1. Either by solving the differential equation or, better, by thinking
about its geometry (slope field), calculate
lim (y1 (t) − y2 (t)).
t→∞
(a) 0
Answer: C
(b) 2π
(c) 4π
(d) 6π
(e) 8π
(f) ∞
MATH 104 – Second Midterm Exam - Fall 2014
1.
Z π
sin3 x cos2 x dx
0
4
(a)
63
Answer: D
2.
Z π/2
(b)
4
35
(c)
2
15
(d)
4
15
(e)
1
12
(f)
1
6
x sin(2x) dx
0
π
(a)
4
Answer: A
3.
Z 2
0
(b)
π
2
(c) π
(d) −
π
4
(e) −
π
2
(f) −π
x2
dx
9 − x2
(a) 2 ln 7 − 3 (b) 2 ln 9 − 5 (c)
Answer: C
3
2
ln 5 − 2 (d) 4 ln 3 − 5 (e) 5 ln 3 − 4 (f) ln 3 − 1
4.
Z √3/2
arcsin x dx
0
1
π
(a) √ −
2 3 2
4
1
π
(d) √ − ln
3
6 3 2
Answer A
5.
π
(b) √ − ln 2
3
√
3
π
+
−1
(e)
12
2
ex
dx
−∞ (1 + e2x )3/2
√
√
5 2
3 2
(b)
(a)
8
12
Answer: C
π
ln 2
(c) √ −
4
3
√
3
π
(f) −
6
2
Z 0
√
2
(c)
2
6. The function
√
5 2
(d)
24
√
5 2
(e)
6
√
2
(f)
4
k
x≥1
f (x) =  x6

0 otherwise



is a probability density function for a certain value of k. Find the mean of that
probability density function.
5
(a)
4
Answer: A
(b)
5
3
(c)
4
3
(d)
5
2
3
2
(f) 2
(e) 8
(f) 9
(e)
7. The solution of the initial-value problem
x
dy
+ 5y = 6x
dx
y(1) = 1
satisfies y(2) =
(a) 2
Answer: A
(b) 4
(c) 5
(d) 6
8. On a cold winter day, when the temperature outside is 10 degrees, Bart finds his
skateboard on the roof and brings it indoors, where the temperature is 70 degrees.
After being indoors for 20 minutes, the temperature of the skateboard rises to 34
degrees. What will the temperature of the skateboard be after another 20 minutes
(i.e., 40 minutes after being brought indoors)? Assume Newton’s law of cooling
(and heating) applies.
(a) 31.6 degrees
(d) 56.8 degrees
Answer: C
(b) 40.2 degrees
(e) 60.4 degrees
(c) 48.4 degrees
(f) 67.6 degrees
x
for 0 ≤ x ≤ 2π is rotated
2
around the x-axis to generate a solid. What is the volume of the solid?
9. The region between the x-axis and the graph of y = sin
π2
(a)
2
Answer: B
(b) π 2
(c) 2π 2
(d) 4π 2
(e) 8π 2
(f) 16π 2
10. Tank number 1 holds 100 liters of water in which 50 kg of salt is initially dissolved.
At time t = 0, pure water begins to flow into tank number 1 at a rate of 2 liters per
minute and the well-stirred mixture flows out at the same rate, into a second tank,
which initially contains 100 liters of pure water. The well-stirred mixture in the
second tank also flows out at the same rate. If S(t) is the amount of salt (in kg) in
the second tank at time t (minutes), what is the differential equation satisfied by S?
(a) S 0 = e−t/50 −
(d) S 0 = e−t/50 −
Answer: A
1
S
50
1
S
100
(b) S 0 = 12 e−t/50 −
(e) S 0 =
1
S
100
1 −t/50
1
e
− 50
S
2
(c) S 0 = e−t/100 −
(f) S 0 =
1
S
100
1 −t/100
1
e
− 50
S
2
MATH 104 – Second Midterm Exam - Fall 2015
1.
Z π/4
tan2 x sec4 x dx
0
4
(a)
63
Answer: D
2.
Z ln 2
(b)
7
24
(c)
5
12
(d)
8
15
(e)
12
35
(f)
1
6
x e−2x dx
0
3
1
− ln 2
16 8
7
1
(d)
−
ln 2
72 24
Answer: A
(a)
3 2
(b) − + ln 2
8 3
7 8
(e) − + ln 2
9 3
5
1
− ln 2
24 6
3
(f) − + 2 ln 2
4
(c)
3.
Z 1
0
16x2 + 2x + 3
dx
(x + 1)(16x2 + 1)
(b) ln 2 + 41 arctan 4
(a) 3 ln 2 + arctan 4
(d) 4 ln 2 − 21 arctan 2
Answer: E
4.
Z e2
(c) 2 ln 2 + 32 arctan 2
(e) ln 2 + 12 arctan 4
(f) ln 2 − arctan 2
9 x2 ln x dx
1
(a) 1 + 7e8
(d) 1 + 5e6
Answer: D
5.
Z ∞
1
(b) 1 + 9e10
(e) 4 + 8e3
(c) 3 + 5e8
(f) 2 + 7e6
12
dx
x(9 + (ln x)2 )3/2
(a) 1
Answer: D
(b) 2
(c)
6. The function
f (x) =
1
4





(d)
k
x2/3
0
4
3
(e)
5
4
(f)
1
3
0<x≤1
otherwise
is a probability density function for a certain value of k. Find the mean of that probability density function.
1
(a)
5
Answer: C
(b)
3
4
(c)
1
4
7. The solution of the initial-value problem
(d)
2
3
dy
√
= −4x y
dx
(e)
2
5
(f)
1
3
y(0) = 4
satisfies y(1) =
(a) 0
Answer: B
(b) 1
(c) 2
(d) 4
(e) 9
(f) 12
8. In a second-order chemical reaction, the reactant R is used up in such a way that
the amount of it present decreases at a rate proportional to the square of the amount
present. Suppose this reaction begins with 200 grams of R present, and after 12 seconds
there are only 100 grams left. How long after the beginning of the reaction will there be
only 40 grams left?
(a) 40 seconds
(d) 50 seconds
Answer: C
(b) 45 seconds
(e) 64 seconds
(c) 48 seconds
(f) 80 seconds
x
9. The region between the x-axis and the graph of y = cos for −4π ≤ x ≤ 4π is
8
rotated
around the x-axis to generate a solid. What is the volume of the solid?
π2
(a)
2
Answer: D
(b) π 2
(c) 2π 2
(d) 4π 2
(e) 8π 2
(f) 16π 2
10. Consider the initial-value problem: y 0 − 2y = 15 sin x
y(0) = A.
For which value of the constant A will the solution be periodic (with period 2π)?
(a) A = −4
(b) A = 4
(d) A = 1
Answer: E
(e) A = −3
(c) A = −1
(f) A = 3