Math Final Exam Review Problems (6181)

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Math 2412 - Final Exam Review Problems (6181)

1. Verify the identity: tan

(x) sin(x) + cos(x) = sec(x)

2. Verify the identity: sin

(x) − sin(x) cos

2

(x) = sin

3

_(x)

3. Verify the identity:

tan

2

_(x)

tan

2

_{(x) + 1}

= sin

2

_(x)

4. Verify the identity:

tan

2

(x)

tan

2

_{(x) + 1}

= sin

2

_(x)

5. Verify the identity: cos

(x) +

sin

2

_(x)

cos

(x) + 1

= 1

6. Without the aid of a calculator, find the exact value of cos

(

11π₁₂

) cos(

3π₄

) + sin(

11π₁₂

) sin(

3π₄

).

7. Let x

=

tan

(10

○

) + tan(20

○

)

1 − tan(10

○

) tan(20

○

)

.

a) Write x as a single function of sine, cosine, or tangent.

b) Without the aid of a calculator, find the exact value of x.

8. Given the following information

i) sin

(A) =

15₁₇

and the terminal side of A lies in Quadrant I and

ii) sin

(B) =

24₂₅

and the terminal side of B lies in Quadrant II,

find the exact value of each of the following trigonometric functions.

a) cos

(A + B)

b) sin

(A + B)

c) cot

(A + B)

d) sec

(−A − B)

9. Given that cot

(θ) = 9 and the terminal side of θ lies in Quadrant III, find the exact value of each of the

following trigonometric functions.

a) cos

(2θ)

b) sin

(2θ)

c) tan

(2θ)

10. Use the appropriate half-angle formula to find the exact value of sin

(75

○

).

11. Express cos

(5θ) + cos(7θ) as a product of sines and/or cosines.

12. Verify the identity:

cos

(x) − cos(y)

sin

(x) + sin(y)

= − tan[

x

− y

2 ]

13. Find all solutions to 7 sin

(x) + 1 = 5 sin(x).

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15. Find all solutions to tan

(3x) = 1 on [0, 2π).

16. Solve the equation 2 sin

2

(x) + sin(x) = 1 on [0, 2π). Select the best answer choice below.

A

O {

π 6

,

5π 6

}

O {

B π 6

,

5π 6

,

π 2

,

3π 2

}

O {

C π 6

,

5π 6

,

3π 2

}

D

O {

π 3

,

2π 3

,

π 2

,

3π 2

}

O {

E π 3

,

2π 3

,

3π 2

}

O No solution

F G

O None of

O through

A

O.

F

17. Solve the equation 2 sin

2

(2x) − sin(2x) = 1 on [0, 2π).

18. Solve the equation cot

(θ)(tan(θ) +

√

3 ) = 0 on [0, 2π). Select the best answer choice below.

A

O {

π 2

,

3π 2

,

2π 3

,

5π 3

}

O {

B π 2

,

3π 2

,

π 6

,

5π 6

}

O {

C 2π 3

,

5π 3

}

D

O {

π 2

,

3π 2

,

π 3

,

4π 3

}

O {

E π 3

,

4π 3

}

O No solution

F G

O None of

O through

A

O.

F

19. Find all solutions to sin

2

(x) − 9 cos(x) + 9 = 0 on [0, 2π).

0.0.1 WEWEWEWEWE

Solving Triangles

Use the given information to solve the triangle. Round lengths to the nearest hundredth and angles to the nearest tenth of a degree. Place your answers in the appropriate answer blanks. Sketch the triangle(s) showing the given information. If no triangle is formed from the given information, then write “No Solution”.

If two triangles are formed from the given information, then fill in the information for△2; otherwise, leave the information blank.

20. Given A= 30○, a= 7, and c = 16, solve the triangle. In the case that there is only one triangle which satsifies the given information leave the information for the second trangle (△2) blank.

△1: A= 30○, B= , C= , and a= 7, b = , c= 16. △2: A= 30○, B= , C= , and a= 7, b = , c= 16.

21. Given a= 4, b = 14, and c = 12, solve the triangle. In the case that there is only one triangle which satsifies the given information leave the information for the second trangle (△2) blank.

△1: A= , B= , C= , and a= 4, b = 14, c = 12. △2: A= , B= , C= , and a= 4, b = 14, c = 12.

22. Given B = 45○, a= 5, and c = 8, solve the triangle. In the case that there is only one triangle which satsifies the given information leave the information for the second trangle (△2) blank.

△1: A= , B= 45○, C= , and a= 5, b = , c= 8. △2: A= , B= 45○, C= , and a= 5, b = , c= 8.

23. Given C = 60○, a= 2, and b = 5, solve the triangle. In the case that there is only one triangle which satsifies the given information leave the information for the second trangle (△2) blank.

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△1: A= , B= , C= 60○, and a= 2, b = 5, c = . △2: A= , B= , C= 60○, and a= 2, b = 5, c = .

24. Find the area of the triangle △ABC having the following properties: B = 50○, a= 5 ft, and c = 2 ft.

25. Given a= 400 ft and b = 300 ft and the measure of angle A is twice the measure of angle B, solve the triangle. In the case that there is only one triangle which satsifies the given information leave the information for the second trangle (△2) blank.

You may round the side to the nearest tenth of a foot and the angle to the nearest degree. △1: A= , B= , C= , and a= 400, b = 300, c = .

△2: A= , B= , C= , and a= 400, b = 300, c = . 26. Find the polar coordinates of (−2√3, 2).

27. Find the rectangular coordinates of(−3,11π₄ ).

28. Convert x= 14 to a polar equation that expresses r in terms of θ.

A

O r= 14

sin(θ) O rB = cos(θ)14 O rC 2=sin(θ)14

D

O r2₌ 14

sin(θ) O rE =cos(2θ)14 O None ofF O throughA O.E 29. Convert r= 16 sin(θ) to a rectangular equation.

A O(x + 8)2_{+ y}2_{= 64} _O_B _{(x − 8)}2_{+ y}2_{= 64} _{O x}_C 2_{+ (y + 8)}2_{= 64} D O x2_{+ (y − 8)}2_{= 64} E O(x + 8)2_{+ y}2_{= 16} F O(x − 8)2_{+ y}2_{= 16} G O x2_{+ (y + 8)}2_{= 16} _{O x}_H 2_{+ (y − 8)}2_{= 16} _{O None of}_I _{O through}_A _O._H 30. Convert r sin(θ) = 10 to a rectangular equation.

31. Convert r2cos(2θ) = 1 to a rectangular equation.

32. Convert x2= 7y to a polar equation that expresses r in terms of θ.

33. Convert r cos(θ +π₆) = 5 to a rectangular equation.

34. Determine whether r= 4 cos(θ) is symmetric about the

a) polar axis, b) the line θ= π₂, and/or c) about the pole. Place a check in the appropriate box.

Yes No Symmtery Test Inconclusive a) polar axis

b) θ= π₂ c) pole

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36. Write 7[cos (2π₃ ) + i sin (2π₃)] in rectangular form.

37. Given z1= 3 [cos (π₅) + i sin (π₅)] and z2= 8 [cos (π₈) + i sin (π₈)], find z1⋅ z2. Write your answer in polar form.

38. Given z1= 72 [cos (15○) + i sin (15○)] and z2= 8 [cos (5○) + i sin (5○)], find z1 z2 . Write your answer in polar form.

A

O 9[cos(3○_{) + i sin(3}○_)] _{O 9}_B _[cos(15○_{) sin(5}○_{) + i sin(15}○_{) cos(5}○_)]

C

O 9[cos(10○_{) + i sin(10}○_)] _{O 9}_D _[cos(3○_{) − i sin(3}○_)]

E

O 9[cos(20○_{) + i sin(20}○_)] _{O None of}_F _{O through}_A _O._E 39. Use DeMoivre’s Theorem to simplify (−1 − i)4_.

40. Find all the complex roots of w= 16 [cos (4π₃ ) + i sin (4π₃)]; write your answers in rectangular form.

41. Solve x3− 4 + 4i√3= 0. Write your answers in polar form.

A

O 8[cos(100○_{) + i sin(100}○_)] _{O 2}_B _[cos(300○_{) + i sin(300}○_)] 8[cos(220○) + i sin(220○)]

8[cos(340○) + i sin(340○)]

C

O 8[cos(300○_{) + i sin(300}○_)] _{O 2}_D _[cos(100○_{) + i sin(100}○_)] 2[cos(220○) + i sin(220○)] 2[cos(340○) + i sin(340○)]

42. Solve x3= 8i.

43. Use Polar Graph paper to graph r= 1 + 2 sin(θ).

44. Let u= 2i +√2j and v= (1 +√2)i + (1 −√2)j. a) Find the exact value of∣∣u∣∣.

b) Find the exact value of ∣∣v∣∣. c) Find uv.

d) Find the angle between u and v.

e) Find the unit vector in the direction of u. f) Find the projection of u onto v. g) Find 2u− 3v.

45. Which of the following expressions would be the correct form for the partial fraction decomposition of 7x

2_{+ 15x − 11} (x − 6)(x + 4)2? A O A x− 6+ B (x + 4)2 OB A x− 6+ B (x + 4)+ C (x + 4)2 C O A x− 6+ Bx+ C (x + 4)2 OD A x− 6+ B (x + 4)+ Dx+ E (x + 4)2 E O A x− 6+ Bx+ C (x + 4) + Dx+ E (x + 4)2 O None ofF O throughA O.E 46. Complete the partial fraction decomposition of 4x− 5

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47. Write the partial fraction decomposition of 8x− 12 (x + 4)2? 48. Write the partial fraction decomposition of 6x

2_{− x + 1} (x + 1)(x2+ 3)? 49. Determine whether each statement is TRUE (T) or FALSE (F).

T F The foci of an ellipse are located along the major axis.

T F The foci of a hyperbola must satisfy the equation of the hyperbola. T F The vertices of a hyperbola are located along the transverse axis. T F The center of the circle x2_{+ y}2_{− 6y + 4 = 0 is on the y-axis.}

T F The major axis is the smaller of the two axes of symmetry of an ellipse.

T F A curve in the xy-plane can have more than one set of parametric equations that describe it. T F The parametric equations{x= 2 tan(t)

y= 3 sec(t) will describe a parabola. T F The parametric equations{x= t

2

y= t6 have the same graph as { x= t3 y= t9 . T F If the parametric curve{x= f(t)

y= g(t) satisfies g(2) = 8, and f(2) = −3, then (−3, 8) is on its graph.

50. Consider the equation of the parabola: 2y= −y2_{− 8x + 11.}

(a) Complete the square to place the equation in standard form. (b) State the coordinates of the vertex.

(c) State the coordinates of the focus.

(d) State the equations of the directrix and axis of symmetry.

51. Find the asymptotes of y 2

25− x2

9 = 1.

52. Consider the ellipse with center at(3, −2), focus at (3, 1), and vertex at (3, 3). a) Plot the given points and find the coordinates of the remaining vertices and focus. b) Write the equation of the ellipse in standard form.

53. Identify the conic 25(x − 3)2_{+ 9(y − 2)}2_{= 225.} 54. Identify the center of the parabola (y + 5)2_{= 8x − 4.}

55. Consider the equation of the hyperbola 4x2− 8x − 68 = 9y2+ 36y. (a) Complete the square to place the equation in standard form. (b) Find the vertices.

(c) Find the foci.

(d) State the equation of the transverse axis (i.e., major axis) of the hyperbola.

56. Consider the conic: 5x2− 4xy + 8y2= 36. Determine a positive angle of rotation (in radians) needed to eliminate the xy term. Do not perform the rotation.

57. Given {x= 1 − t 3

y= 4 − t6 , find the rectangular equation. Sketch the curve and indicate the orientation.

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59. Identify 3x2− 2√3xy+ y2+ 2x + 2√3y− 8 = 0 without applying a rotation of axes.

60. State the center of (x + 3)2+ (y − 12)2= 16.

61. Find a set of parametric equations for the rectangular equation (x − 3)2_{= 5y.}

62. After rotation of π₆ radians, the equation of a conic

C

is (x ′₎2 9 + (

y′)2

25 = 1. Sketch the original conic

C

. Include the x′y′-coordinate axes.

Find the coordinates of the foci (

in the xy-plane).

63. Consider the polar graph r

=

6

4 + 2 sin(θ)

.

a) State the eccentricity p

b). Identify the conic.

64. Rewrite xy

= 4 in a rotated x

′

_y

′

_{-system without an x}

′

_y

′

_term.

65. Given

{

x

= sin(2t)

y

= 3 cos(t)

, eliminate the parameter to obtain the rectangular equation.

66. Determine whether each statement is TRUE (T) or FALSE (F).

T F The sequence 2, 6, 24, 120, . . . is a geometric sequence.

T F The sequence b

n

= 5n + 12 is an arithmetic sequence.

T F A sequence is a function from the natural numbers to the reals.

T F For the geomtric sequence a

n

= 3(2)

−n+1

, r is

1

2 .

T F 2

+ 4 + 8 + 16 + . . . =

2

1 − 2

.

T F 10

+ 5 +

5₂

+

5₄

+ . . . =

10

1 −

1₂

.

T F If a

1

= 6 and a

5

= −2 in an arithmetic sequence, then a

3

= 2.

67. Find an expression for the n

th

term (i.e., general term) of the sequence

−2, 1, 4, 7, 10, . . . .

68. Evaluate and simplify:

15!

2! 13!

.

69. Let n

∈ N. Evaluate and simplify: (

n

+ 2)!

n!

.

70. Find the sum:

5

∑

i=1

(−1)

i

_{(i − 1)!}

2

n−1

.

71. For the arithmetic sequence (a

n

) whose second and fourth terms are

−2 and −12 respectively, find

a) a

8

b) S

8

.

72. Use concepts of sequences to find the sum of the first 100 even integers; i.e., 2

+ 4 + 6 + ⋯ + 200.

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74. Evaluate and simplify the following binomial coefficeient:

(

13

4 ).

75. Use the Binomial Theorem to expand

(x − 2)

4

.

76. Find the exact value of the sum of the first 20 terms of the sequence:

1

6 ,

1

3 ,

2

3 ,

4

3 ,

8

3 , . . ..

77. Find the fourth term of

(2x − y)

7

.

78. Use the Principle of Mathematical Induction to prove

4 + 8 + ⋯ + 4n = 2n(n + 1)

for each n

∈ N.

(a) Identify S

(n):

(b) Base Case: Show that S

(1) is true.

(c) State the Inductive Hypothesis:

Assume that S

(k) is true:

(d) Identify S

(k + 1):

(e) Inductive Step: Use the Inductive Hypothesis to show that S

(k + 1) is true.

79. Use the Principle of Mathematical Induction to prove

3 + 11 + ⋯ + (8n − 5) = n(4n − 1)

for each n

∈ N.

(a) Identify S

(n):

(b) Base Case: Show that S

(1) is true.

(c) State the Inductive Hypothesis:

Assume that S

(k) is true:

(d) Identify S

(k + 1):

(e) Inductive Step: Use the Inductive Hypothesis to show that S

(k + 1) is true.

80. Find the exact value of the following sum:

∞

∑

i=1

15 (

2

3 )

n−1

.

81. Use the concepts of geometric sequences to write 1.23456 as a rational number in simplest form.

82. Let a

n

be the sequence defined by a

n

= {

3 (2)

n−1

, if n is odd

10 + 3(n − 1) , if n is even

.

Find the sum: a

1

+ a

2

+ ⋯ + a

10

.

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83. Use properties of limits to find the indicated limit.

lim

x→5

(4x − 8)

Select the correct choice below and fill in any answer blanks in your choice.

◯ A. lim

_x→5

(4x − 8) =

◯ B. The limit does not exist.

84. a. Find the slope of the tangent line to the graph of f

(x) = 6x + 4 at the point (2,16).

b. Find the slope-intercept equation of the tangent line to the graph of f

(x) = 6x + 4 at

the point (2,16).

85. Use properties of limits to find the indicated limit.

lim

x→6

x

2

− 36

x

− 6

Select the correct choice below and fill in any answer blanks in your choice.

◯ A. lim

_x→6

x

_x

2

− 36

_{− 6}

=

◯ B. The limit does not exist.

86. Determine for what numbers, if any, the given function is discontinuous.

T

(x) =

⎧⎪⎪⎪

⎪⎨

⎪⎪⎪⎪

⎩

7x

if x

< 6

43 if x

= 6

x

2

+ 6 if x > 6

Select the correct choice below and fill in any answer blanks in your choice.

◯ A. The function T is discontinuous at

.

◯ B. The function is continuous everywhere.

87. a. What is the derivative of f

(x)?

b. What is the slope of the tangent line to the graph of f at the given x-values?

f

(x) = x

2

− 4x + 6; x =

5₂

, x

= 3

88. Use properties of limits to find the indicated limit.

lim

x→36

√

x

− 6

x

− 36

Select the correct choice below and fill in any answer blanks in your choice.

◯ A. lim

_x→36

√

x

− 6

x

− 36

=

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89. A piecewise function is given. Use properties of limits to find the indicated limits, or state that a limit does

not exist.

f

(x) =

⎧⎪⎪⎪

⎪⎨

⎪⎪⎪⎪

⎩

7 − x

if x

< 7

3 if x

= 7

x

2

− 49 if x > 7

a. Select the correct choice below and fill in any answer blanks in your choice.

◯ A. lim

_x→7−

f

(x) =

◯ B. The limit does not exist.

b. Select the correct choice below and fill in any answer blanks in your choice.

◯ A. lim

_x→7+

f

(x) =

◯ B. The limit does not exist.

c. Select the correct choice below and fill in any answer blanks in your choice.

◯ A. lim

_x→7

f

(x) =

◯ B. The limit does not exist.

90. An explosion causes debris to rise vertically with an initial velocity of 112 feet per second. The function

s

(t) = −16t

2

+112t describes the height of the debris above the ground, s(t), in feet, t seconds after the explosion.

a. What is the instantaneous velocity of the debris 1 second after the explosion? 2 seconds after the explosion?