calculus i exam #3 questions reference list fall, 2003

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).

1

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.


x(1 + 2x4) dx.

2


(3eu + sec2 u) du.


(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.


x+2√−x2−4x

dx

7


1√−x2−4x

dx


tan2 θ sec2 θ dθ.


sin x1+cos2 x

dx.


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)).


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 .


56. Evaluate:∫ π

4

0cos3 x dx.

Ans. 5√

212 .



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


Ans. 435 .

Write sin3 (x) = sin (x)(1− cos2 (x)).Then let u = cos(x).


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


Ans. 160 .

Write cos5 (x) = cos (x)(1− sin2 (x))2.Then let u = sin (x).


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).


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.


0cos2 (y) tan3 (y) dy.

Ans. −14 + ln 2

2 .



Then let u = cos (y).


0cos2 (y) tan3 (y) dy.

11

Ans. −12 + ln ( 2√

3).



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.


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.


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.

12

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

13


y = x + 1, −∞ < x < ∞y = x2 − 2x + 1, −∞ < x < ∞.




ans: 92

sq. units

14


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


x = y2, 0 < y < ∞x = y3, 0 < y < ∞.




ans: 112

sq. units

16


y = x2, −∞ < x < ∞y = −2x4, −∞ < x < ∞|x| = 1, −∞ < y < ∞.




ans: 2215

sq. units

17


y = 2x2, −∞ < x < ∞y = x4 − 2x2, −∞ < x < ∞.




ans: 12815

sq. units

18


y = −x + 2, −∞ < x < ∞y = 4− x2, −∞ < x < ∞.




ans: 316

sq. units

19


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


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


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.

22

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.

23

2. A particle moves along the x-axis; its position as a function of time, t,is x(t) = t3 − 6t2 + 9t− 2, 0 ≤ t ≤ 5. Find the total distance traveledby the particle in 5 units of time. (Note that x(2) = 0.)

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calculus i exam #3 questions reference list fall, 2003

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