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CALCULUS IEXAM #3 QUESTIONS REFERENCE LIST
Fall, 2003
The questions contained herein are provided as a reference for Exam #3,Calculus I, Fall, 2003. This document is composed of two parts:
• Part I. Symbol Manipulation Problems,
• Part II. Modeling Problems,
Approximately 80% or more of the questions on Exam #2 will be similar to(in some cases, copies of) those found in this document.
The thechnical terms used in this document are consistent with their defini-tions as found in The Terms of Calculus (which is an auxillary text referencedfor the course).
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PART ISYMBOL MANIPULATION PROBLEMS
EXAM #3 REFERENCECalculus I, Fall ’03
1. Evaluate∫ 2
1x−2 dx.
2. Evaluate∫ 1
0x
37 dx.
3. Evaluate∫ 2π
πcos (θ) dθ.
Ans. 0.
4. Evaluate∫ π
π4
sec2 (θ) dθ. (Be careful!)
5. Express the value of∫ √
3
16
1+x2 dx in the form (pq)π, where p and q de-
note integers with not common integer divisors other than ±1.Ans. 1/2. Maple verified.
6. Express as an integer, the value of∫ ln 8
ln 38ex dx.
Ans. 40.
7. Evaluate∫ 1
2
0dx√1−x2 dx.
8. Express, as an elementary function,∫
x−34 dx.
9. Express, as an elementary function,∫
x(1 + 2x4) dx.
2
10. Express, as an elementary function,∫
(3eu + sec2 u) du.
11. Express, as an elementary function,∫
(1−√
x)2 dx.
12. Express explicitly, as an integer, the value of γ in the following expres-sion:
∫ 9
82t dt = γ
ln 2.
13. Evaluate∫ 4
−2(3x− 5) dx.
14. Evaluate∫ 1
0(1− 2x− 3x2) dx.
15. Express as an integer the value of∫ 5
31
x+1dx− ln 3
2.
16. Evaluate∫ 0
−3(5y4 − 6y2 + 14) dy. Express your answer as an integer.
17. Evaluate∫ 3
1( 1
t2− 1
t4) dt. Express your answer in the form p/q where p
and q denote integers with no common integers divisors other than one.
18. Express, as a single fraction, the value of∫ 1
0
u(√
u + 3√
u) du.
3
Ans. 2935 .
19. Evaluate∫ 2
1
√x− 1 dx.
20. Express as a single fraction, the value of∫ π2
π8
sin 4t dt.
21. Express, as a single fraction, the value of∫ 4
0
√t dt.
Ans. 163 .
22. Evaluate∫ 1
02
(t+1)6dt. Express your answer in the form p/q where p and
q denote integers with no common integers divisors other than one.
23. Write an explicit expression for f(x), if f ′(x) = 4x4+3x2 and f(1) = 0.
24. Find a 2nd degree polynomial P (x) such that P (2) = 5, P ′(2) = 3 andP ′′(2) = 2. Express your answer in the form
P (x) = a0 + a1(x− 2) + a2(x− 2)2.
4
25. For the function f(x), 0 < x < ∞, it is known that f ′′(x) = 1x3 and
f(1) = 0 and f(2) = 0. The function f can be written in the formf(x) = 1
2x+ αx + β. Find values for both α and β. Express each of
their values in the form pq
where p and q denote integers with no com-
mon integer divisors other than one (as a reduced fraction).
α =
β =
26. For the real valued function f defined on the real numbers, it is knownthat f ′′(x) = x for all x and f(0) = −3 and f ′(0) = 2. Compute f(6).
27. For the real valued function f defined on the real line, it is known thatf ′′(x) = x2 + 3 cos x for all x and f(0) = 2 and f ′(0) = 3. Computef(π).
28. The value of∫ 2
1x2+1√
xdx can be expressed in the form 18
√2−γ5
. Expressγ as an iteger.
29. The value of∫ √
3
16
1+x2 dx can be written in the form βπ. Find β.
30. Write an explict expression for f(t), if f ′(t) = 11+t2
and f(0) = 0.
31. Express as an integer the value of∫ 2π
0| sin (x)| dx.
5
32. It is known that F (x) =∫
sin (πx) dx and F (0) = 1π
and F (1) = απ.
Express α as an integer.
33. The function f(x) is defined for all real numbers except x = 0. Itsderivative is 1
x, except for x = 0. Further f(−1) = 2. Find an explicit
form for f(x).
34. Evaluate∫ 2
0(x− 1)25 dx.
35. Evaluate∫ 3
01
2x+3dx.
36. Evaluate∫ 7
0
√4 + 3x dx.
37. Evaluate∫ 1
0x
1+x4 dx.
38. Evaluate∫ 1
0x e−x2
dx.
39. The value of∫ 3
2dx
x ln xcan be written in the form ln α. Find α.
40. The value of∫ π/4
0sec x tan x
√1 + sec x dx, can be represented in the
form (2/3)(1 +√
2)3/2 + α. Find α.
6
41. The value of∫ π
3
0cos4 x sin x dx can be expressed in the form β
160. Ex-
press β as an integer.Ans. β = 31. Maple checked.
42. Write an explicit expression for f(θ), if f ′(θ) = 3θ2 cos (θ3)and f(0) = π.
43. Evaluate∫ 1
0x2(1 + 2x3)3 dx. Express your answer in the form p
qwhere
p and q denote integers with no common divisorsother than 1 (as a reduced fraction).
44. Evaluate: β in the expression∫ π
4
0sin (x)√1−sin2 (x)
dx = ln β.
45. Evaluate:∫ 1
2
0sin−1 x√
1−x2 dx.
46. Express in terms of elementary functions:∫
x2√
1−xdx.
47. Express in terms of elementary functions:∫
x+2√−x2−4x
dx
7
48. Express in terms of elementary functions:∫
1√−x2−4x
dx
49. Express in terms of elementary functions:∫
tan2 θ sec2 θ dθ.
50. Express in terms of elementary functions:∫
sin x1+cos2 x
dx.
51. Express in terms of elementary functions:∫
sin3 x dx.
Ans.− cos (x) + cos3 (x)3 + C.
Write sin3 (x) = sin (x)(1− cos2 (x)).
52. Evaluate:∫ π
6
0sin3 x dx.
Ans. −3√
38 + 2
3 .Write sin3 (x) = sin (x)(1− cos2 (x)).
53. Evaluate:∫ π
4
0sin3 x dx.
Ans. −5√
212 + 2
3 .Write sin3 (x) = sin (x)(1− cos2 (x)).
54. Express in terms of elementary functions:∫
cos3 x dx.
8
Ans. − sin (x)− sin3 (x)3 + C.
Write cos3 (x) = cos (x)(1− sin2 (x)).
55. Evaluate:∫ π
6
0cos3 x dx.
Ans. 1124 .
Write cos3 (x) = cos (x)(1− sin2 (x)).
56. Evaluate:∫ π
4
0cos3 x dx.
Ans. 5√
212 .
Write cos3 (x) = cos (x)(1− sin2 (x)).
57. Express in terms of elementary functions:∫
sin3 (x) cos4 (x) dx.
Ans. − cos4 (x)5 + cos7(x)
7 + C.Write sin3 (x) = sin (x)(1− cos2 (x)).Then let u = cos(x).
58. Evaluate:∫ π
2
0sin3 (x) cos4 (x) dx.
Ans. 235 .
Write sin3 (x) = sin (x)(1− cos2 (x)).Then let u = cos(x).
9
59. Evaluate:∫ π
0sin3 (x) cos4 (x) dx.
Ans. 435 .
Write sin3 (x) = sin (x)(1− cos2 (x)).Then let u = cos(x).
60. Express in terms of elementary functions:∫
sin5 (x) cos5 (x) dx.
Ans. sin6 (x)6 − sin8 (x)
4 + sin10 (x)10 + C.
Write cos5 (x) = cos (x)(1− sin2 (x))2.Then let u = sin (x).Or write sin5 (x) cos5 (x) = ( sin (2x)
2 )5 =132 sin (2x)(1− cos2 (2x))2 andlet u = cos (2x). This leads to the answer−cos(2x)
64 + cos3 2x96 − cos5 (2x)
320 + C.
Evidently the two answers, without theconstants, differ by a non-zero constant.
61. Evaluate:∫ π
2
0sin5 (x) cos5 (x) dx.
Ans. 160 .
Write cos5 (x) = cos (x)(1− sin2 (x))2.Then let u = sin (x).
62. Express in terms of elementary functions:∫
sin3 (x)√
cos (x) dx.
10
Ans. − 23 cos3/2 (x) + 2
7 cos7/2 (x) + C.Write sin3 (x) = sin (x)(1− cos2 (x)).Then let u = cos (x).
63. Evaluate:∫ π/4
0sin3 (x)
√cos (x) dx.
Ans. 23 (1− (1/
√2)
32 ) + 2
7 ((1/√
2)72 − 1).
Write sin3 (x) = sin (x)(1− cos2 (x)).Then let u = cos (x).
64. Express in terms of elementary functions:∫
cos2 (y) tan3 (y) dy.
Ans. − ln | cos (y)|+ cos2 (y)2 + C.
Write cos2 (y) tan3 (y) = sin3 (y)cos (y) =
sin (y)( 1cos (y) − cos (y)).
Then let u = cos (y).Note that answer can also be written− ln | cos (y)| − sin2 (y)
2 + C.
65. Evaluate:∫ π/4
0cos2 (y) tan3 (y) dy.
Ans. −14 + ln 2
2 .
Write cos2 (y) tan3 (y) = sin3 (y)cos (y) =
sin (y)( 1cos (y) − cos (y)).
Then let u = cos (y).
66. Evaluate:∫ π/6
0cos2 (y) tan3 (y) dy.
11
Ans. −12 + ln ( 2√
3).
Write cos2 (y) tan3 (y) = sin3 (y)cos (y) =
sin (y)( 1cos (y) − cos (y)).
Then let u = cos (y).
67. Express in terms of elementary functions:∫ 1−sin (y)
cos (y)dy.
Ans.ln | sec (y) + tan (y)|+ ln | cos (y)|+ C.Let u = cos (y). Use∫
sec (y) dy = ln (| sec (y) + tan (y)|) + C.
68. Evaluate:∫ π/4
01−sin (y)cos (y)
dy.
Ans. ln (√
2 + 1)− ln (2)2 .
Let u = cos (y). Use∫sec (y) dy = ln (| sec (y) + tan (y)|) + C.
69. Evaluate:∫ π/3
π/61−sin (y)cos (y)
dy.
Ans. ln (√
3 + 2)− ln (3).Let u = cos (y). Use∫
sec (y) dy = ln | sec (y) + tan (y)|+ C.
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PART IIMODELING PROBLEMSEXAM #3 REFERENCE
Calculus I, Fall ’03
AREA PROBLEMS
1. (# pts.) Compute the area of the bounded region in the xy-plane,enclosed by the planar curves defined by the functions
x = y2, −∞ < y < ∞x = 0, −∞ < y < ∞
x = 2y + 3, −∞ < y < ∞.
Display neatly your computations in the box below. Express your an-swer in the form p
qwhere p and q denote integers with no common
integer divisors other than ±1.
ans: 323
sq. units
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2. (# pts.) Compute the area of the bounded region in the xy-plane,enclosed by the planar curves defined by the functions
y = x + 1, −∞ < x < ∞y = x2 − 2x + 1, −∞ < x < ∞.
Display neatly your computations in the box below. Express your an-swer in the form p
qwhere p and q denote integers with no common
integer divisors other than ±1.
ans: 92
sq. units
14
3. (# pts.) Compute the area of the bounded region in the xy-plane,enclosed by the planar curves defined by the functions
y =1
x, 0 < x < ∞
y =1
x2, 0 < x < ∞
x = 2, 0 < y < ∞.
Display neatly your computations in the box below. Your answer maycontain an expression of the form ln (a) where a denotes an integer.
ans: ln (2)− 1 sq. units
15
4. (# pts.) Compute the area of the bounded region in the xy-plane,enclosed by the planar curves defined by the functions
x = y2, 0 < y < ∞x = y3, 0 < y < ∞.
Display neatly your computations in the box below. Express your an-swer in the form p
qwhere p and q denote integers with no common
integer divisors other than ±1.
ans: 112
sq. units
16
5. (# pts.) Compute the area of the bounded region in the xy-plane,enclosed by the planar curves defined by the functions
y = x2, −∞ < x < ∞y = −2x4, −∞ < x < ∞|x| = 1, −∞ < y < ∞.
Display neatly your computations in the box below. Express your an-swer in the form p
qwhere p and q denote integers with no common
integer divisors other than ±1.
ans: 2215
sq. units
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6. (# pts.) Compute the area of the bounded region in the xy-plane,enclosed by the planar curves defined by the functions
y = 2x2, −∞ < x < ∞y = x4 − 2x2, −∞ < x < ∞.
Display neatly your computations in the box below. Express your an-swer in the form p
qwhere p and q denote integers with no common
integer divisors other than ±1.
ans: 12815
sq. units
18
7. (# pts.) Compute the area of the bounded region in the xy-plane,enclosed by the planar curves defined by the functions
y = −x + 2, −∞ < x < ∞y = 4− x2, −∞ < x < ∞.
Display neatly your computations in the box below. Express your an-swer in the form p
qwhere p and q denote integers with no common
integer divisors other than ±1.
ans: 316
sq. units
19
8. (# pts.) Compute the area of the bounded region in the xy-plane,enclosed by the planar curves defined by the functions
y = 2x, −∞ < x < ∞y = x2 − 4x, −∞ < x < ∞.
Display neatly your computations in the box below. Express your an-swer explictly as an integer.
ans: 36 sq. units
20
9. (# pts.) Compute the area of the bounded region in the xy-plane,enclosed by the planar curves defined by the functions
y = sin (x), −∞ < x < ∞y = 2 sin2 (x), −∞ < x < ∞
x = 0, −∞ < y < ∞x =
π
2, −∞ < y < ∞.
Display neatly your computations in the box below. Your answer maycontain the symobl π and a square root symbol, but should be free oftrigonometric functions.
ans: −√
32 + 1 + π
6 square units.
Use A =∫ π/6
0(sin (x)− sin2 (x)) dx +
∫ π/2
π/6(2 sin2 (x)− sin (x)) dx.
21
10. (# pts.) Compute the area of the bounded region in the xy-plane,enclosed by the planar curves defined by the functions
y = sin (x), 0 < x <π
2
y = sec2 (x), 0 < x <π
2x = 0, −∞ < y < ∞
x =π
4, −∞ < y < ∞.
Display neatly your computations in the box below. Your answer maycontain a square root symbol and should be free of trigonometric func-tions.
ans:√
22 square units.
Use A =∫ π/4
0(sec2 (x)− sin (x)) dx.
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DISPLACEMENT PROBLEMS
1. A ball is thrown vertically upward from a height of 6 feet with aninitial velocity of 60 ft./sec. How high will the ball go? Constructan appropriate mathematical model for this question and within themodel calculate the height of the ball. In your model use -32 ft/sec tobe the acceleration due to gravity. Please show your work in the boxprovide and box your answer.
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