Dual-Enrollment Final Exam Preparation

Dates:

May 7th and 8th: Part 1 (75 minutes) 20-25 questions covering 1st Semester Material

May 9th and 10th Part 2 (75 minutes) 35-40 Questions covering 2nd Semester Material

Exam:

Will be a mix of multiple choice and free response (Mostly Free Response)

Will be worth a 1/3 of your 2nd semester grade

Review: To help prepare you have

A List of Formulas you need to know (The Bolded ones will be given to you)

2nd Semester Cumulative Reviews

A List of Topics Possible Covered (Not all will be)

A set of 125 Practice Questions

Opportunity: If you submit neat and organized work for the 125 Practice Questions (All Completed

with Work Shown) by the start of the Final May 7th.

You will get the opportunity to OMIT 1 question from each Part of the Final (Your Pick)

Formulas You Must Know (To be Successful)

log(𝑎𝑥) = 𝑥𝑙𝑜𝑔(𝑎) log(𝑥𝑦) = 𝑙𝑜𝑔𝑥 + 𝑙𝑜𝑔𝑦 log (𝑥

𝑦) = 𝑙𝑜𝑔𝑥 − 𝑙𝑜𝑔𝑦

log𝑎 𝑏 =𝑙𝑜𝑔𝑏

𝑙𝑜𝑔𝑎=

𝑙𝑛𝑏

𝑙𝑛𝑎

𝐴 = 𝑃𝑒𝑟𝑡 𝑁𝑡 = 𝑁0𝑒𝑟𝑡

sin 𝜃 =𝑦

𝑟

sin 𝜃 = 𝑦

cos 𝜃 =𝑥

𝑟

cos 𝜃 = 𝑥

tan 𝜃 =𝑦

𝑥

tan 𝜃 =sin 𝜃

cos 𝜃

csc 𝜃 =1

sin 𝜃 sec 𝜃 =

1

cos 𝜃

cot 𝜃 =1

tan 𝜃

cot 𝜃 =cos 𝜃

sin 𝜃

𝑠𝑖𝑛2𝜃 + 𝑐𝑜𝑠2𝜃 = 1

𝑡𝑎𝑛2𝜃 + 1 = 𝑠𝑒𝑐2𝜃 1 + 𝑐𝑜𝑡2𝜃 = 𝑐𝑠𝑐2𝜃 cos 𝜃 = sin(90 − 𝜃) csc 𝜃 = sec(90 − 𝜃)

cot 𝜃 = tan(90 − 𝜃) sin(2𝑥) = 2 sin 𝑥 cos 𝑥

cos(2𝑥) = 𝑐𝑜𝑠2𝑥 − 𝑠𝑖𝑛2𝑥 = 1 − 2𝑠𝑖𝑛2𝑥

= 2𝑐𝑜𝑠2𝑥 − 1

𝐬𝐢𝐧 𝑨

𝒂=

𝐬𝐢𝐧 𝑩

𝒃=

𝐬𝐢𝐧 𝑪

𝒄

cos(𝑥 + 𝑦) = cos 𝑥 cos 𝑦 − sin 𝑥 sin 𝑦 𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 − 𝟐𝒂𝒃 𝐜𝐨𝐬 𝑪

cos(𝑥 − 𝑦) = cos 𝑥 cos 𝑦 + sin 𝑥 sin 𝑦 𝒃𝟐 = 𝒂𝟐 + 𝒄𝟐 − 𝟐𝒂𝒄 𝐜𝐨𝐬 𝑩

sin(𝑥 + 𝑦) = sin 𝑥 cos 𝑦 + cos 𝑥 sin 𝑦 𝑨𝒓𝒆𝒂 =𝟏

𝟐𝒂𝒃 𝐬𝐢𝐧 𝑪

𝒂𝟐 = 𝒃𝟐 + 𝒄𝟐 − 𝟐𝒃𝒄 𝐜𝐨𝐬 𝑨

sin(𝑥 − 𝑦) = sin 𝑥 cos 𝑦 − cos 𝑥 sin 𝑦 𝑨𝒓𝒆𝒂 = √𝒔(𝒔 − 𝒂)(𝒔 − 𝒃)(𝒔 − 𝒄), where 𝒔 =

𝒂+𝒃+𝒄

𝟐

Part 1: Possible Topics Covered

Chapter 1

o Identify a conic from 𝐴𝑥2 + 𝐵𝑦2 + 𝐶𝑥 + 𝐷𝑦 + 𝐸

o Put a conic section in (ℎ, 𝑘) form

o Solve Inequalities and put in Interval Notation

Chapter 2

o Sketch Polynomials and Rational Functions

o Solve Polynomial Equations, Polynomial Inequalities, and Rational Equations

o Apply the Rational Root Theorem (Find all possible rational roots)

o Divide Polynomials either Synthetic or Long

o Apply Factor Theorem to find solutions

Chapter 3

o Find Domain

o Perform/Solve Compositions

o Find Discontinuities (Holes, Vertical Asymptotes, and Jumps)

o Find Inverses

o Determine if a function is Even, Odd, or Neither

o Sketch and Evaluate Piece-wise Functions

o Express Absolute Value as Piece-wise Function

o Find the Difference Quotient

Chapter 4

o Expanding Logarithms

o Simplifying Logarithms

o Apply the Change of Base Formula (and Simplify)

o Solve Logarithmic and Exponential Equations

o Find/Apply Exponential Growth and Decay (Including Doubling and Half-Life)

Part 2- Possible Topics

Chapter 5

o Given a trig function and quadrant…Find another trig function

o Evaluate off Unit Circle

o Find Reference Angles

o Write as a Trig Function less than 90 or 45

o Convert between Radians and Degrees

o Find Arc Length

o Find Sector Area

Chapter 6

o Graph any of the six trig functions

o Find Domain of Trig Function

o Find Range of Trig Function

o Write a sine/cosine Equation (includes modeling word problems)

o Evaluate Arc Functions

Chapter 7

o Evaluate using Fundamental Identities

o Simplify and Verify using Fundamental Identities

o Evaluate using Sum, Difference and Double Angle Identities

o Simplify and Verify using Sum, Difference and Double Angle Identities

o Solve Basic Trig Equations on [0,2𝜋]

o Solve Trig Equations using Identities on [0,2𝜋]

o Solve a Trig Equations for all solutions

Chapter 8

o Find Angles and Sides of Triangle using Law of Sines and Law of Cosines

o Find Area of Triangle using Trig

o Find Area of a Triangle using Heron’s Formula

o Find Area of a Polygon

Practice Questions

Chapter 1

1. Identify the type of conic section 16𝑦2 − 25𝑥2 − 50𝑥 − 425 = 0

2. Identify the type of conic section 𝑥 + 2𝑦2 + 20𝑦 + 44 = 0

3. Identify the type of conic section 𝑥2 + 9𝑦2 + 2𝑥 − 72𝑦 + 109 = 0

4. Put in (h, k) form 9𝑥2 + 4𝑦2 + 36𝑥 − 24𝑦 + 36 = 0

5. Put in (h,k) form 9𝑥2 − 16𝑦2 − 54𝑥 + 64𝑦 − 127 = 0

6. Put in (h,k) form 𝑦2 + 4𝑥 = −25 + 6𝑦

7. Put in (h,k) form 3𝑥2 + 3𝑦2 − 6𝑥 + 3𝑦 = 4

8. Solve the following inequality (express in interval notation) 6𝑥2 − 2𝑥 − 20 < 0

Chapter 2

9. Sketch the polynomial 𝑓(𝑥) = 𝑥2(𝑥 + 2)(𝑥 − 1)(𝑥 + 1)3

10. Sketch 𝑃(𝑥) =2𝑥2+5𝑥

𝑥2+𝑥

11. Solve: 𝑥 + 7√𝑥 − 8 = 0

12. Solve: 2𝑥2(4𝑥 − 1) = 𝑥(1 − 4𝑥)

13. Solve: √2𝑥 + 5 − 1 = 𝑥

14. Solve: 3𝑥3 − 7𝑥2 − 2𝑥 + 8 = 0

15. Solve: 1

𝑥−2−

1

𝑥+2−

2

7= 0

16. Solve: 𝑥5 − 6𝑥3 = −5𝑥

17. Divide: 2𝑥3 − 3𝑥2 + 𝑥 + 1 by 𝑥 − 2

18. Divide: 2𝑥3 + 3𝑥2 − 2𝑥 − 3 by 2𝑥 + 3

19. Find the other solutions of 𝑃(𝑥) = 𝑥3 − 7𝑥2 + 21𝑥 − 27 given that 2 ± 𝑖√5 are solutions

20. Find k, such that 𝑥 − 3 is a factor of 3𝑥3 − 9𝑥2 + 𝑘𝑥 − 12

Chapter 3

21. Find the domain 𝑓(𝑥) =𝑥2−3𝑥+6

𝑥2−3𝑥−10

22. Find the domain 𝑓(𝑥) =2𝑥

√𝑥2−9

23. Even, Odd, or Neither 𝑓(𝑥) = (𝑥2 + 𝑥)2

24. Even, Odd, or Neither 𝑓(𝑥) =|𝑥|

𝑥+𝑥7

25. Describe any discontinuities in the following equation 3𝑥2+2𝑥

12𝑥2+5𝑥−2

26. Describe any discontinuities in 𝑓(𝑥) =𝑥2−4

𝑥2+3𝑥+2

27. Let 𝑓(𝑥) = 𝑥 + 2, 𝑔(𝑥) = 𝑥2 − 2𝑥. Find 𝑓[𝑓(4)]

28. Let 𝑓(𝑥) = 𝑥 + 2, 𝑔(𝑥) = 𝑥2 − 2𝑥. Find (𝑔 𝑓)(𝑥)

29. Find the inverse of 𝑓(𝑥) =𝑥−4

𝑥−2

30. Find k, that would make 𝑓(𝑥) continuous 𝑓(𝑥) = {3𝑘𝑥 + 4 𝑥 < 4−2𝑥 − 8 𝑥 ≥ 4

31. Find 𝑓(2) if 𝑓(𝑥) = {𝑥 + 3 𝑥 ≤ 0

3 0 ≤ 𝑥 < 22𝑥 − 1 𝑥 ≥ 2

32. Express 𝑓(𝑥) = |2𝑥 + 1| − 3 as a piecewise function

33. Let 𝑓(𝑥) = 𝑥2 + 4𝑥 − 5. Find the difference quotient of f(x)

34. Let 𝑓(𝑥) = 𝑥2 − 2. Find the difference quotient of f(x)

35. Sketch 𝑓(𝑥) = {𝑥 + 3 𝑥 ≤ 0

3 0 ≤ 𝑥 < 22𝑥 − 1 𝑥 ≥ 2

36. Sketch 𝑓(𝑥) = {

−𝑥 𝑥 ≤ −22 −2 < 𝑥 < 0

𝑥2

√𝑥0 ≤ 𝑥 ≤ 2

𝑥 > 2

Chapter 4

37. Evaluate 32 log3 6−2 log3 2

38. Evaluate log25 8 ∙ log8 5

39. Evaluate 𝑒𝑙𝑛𝑒3𝑙𝑛2

40. Evaluate log2 100 − 2 log2 5

41. Expand ln(𝑥3𝑦)2

42. Expand log (𝑥2𝑦

𝑧)

43. Simplify to a single log: log 𝑥 + 2 log 𝑦 − 3 log 𝑧

44. Simplify to a single log: ln 5 +1

2ln 36 − ln 3

45. Rewrite using the change of base formula log9 7, then get decimal apporximation

46. Solve: 32𝑥+1 = 15

47. Solve: log2(𝑥 + 2) + log2 5 = 4

48. Solve: 8 + 2𝑒𝑥 = 12

49. Solve: 𝑒2𝑥 − 4𝑒𝑥 + 3 = 0

50. Solve: 24𝑥+1 = 3𝑥

51. Solve: ln1

𝑥= −5

52. A population of bacteria doubles after 10 hours. What is the growth rate of the bacteria?

53. A house worth 200,000 in 1980 is now worth 325,000 in 2016. What is the relative growth rate?

Chapter 5

54. A sector has a radius of 15 cm and a central angle of 60°. What is the sector’s arc length?

55. A sector has an arc length of 6.4 cm and an area of 10.24 cm2. What is the central angle, 𝜃?

56. A sector has a central angle of 120° and an arc length of 9cm. What is the sector’s area?

57. A Point, P, moves on a circle, with radius 6 cm, at a speed of 𝜋

6 radians per second. How far does P

move after 8 seconds?

58. Convert 7𝜋

6 radians to Degrees

59. Convert −320°to Radians

60. Given P on a terminal ray at (5, −7). Find all 6 cos𝜃

61. Given: sec 𝜃 = 4 and sin 𝜃 > 0. Find tan 𝜃.

62. Given cot 𝜃 = √2 and cos 𝜃 > 0. Find all sec 𝜃

63. Evaluate each trig function

a. tan5𝜋

3

b. cos −2𝜋

3

c. csc5𝜋

4

d. cot3𝜋

2

e. sin −3𝜋

2

f. sec5𝜋

5

g. cos −𝜋

3

64. Evaluate sin2 3𝜋

4+ cos2 4𝜋

3

65. Evaluate 3 tan7𝜋

4− 2 cot

5𝜋

4

66. Find all 𝜃for 0 < 𝜗 ≤ 2𝜋

a. sec 𝜃 = −2

√3

b. tan 𝜃 = 1

c. cos 𝜃 = −√2

2

d. csc 𝜃 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

67. Rewrite as a trig function less than 90 degrees

a. cos 320

b. sec 250

c. cot 140

68. Rewrite as a trig function less than 45 degrees

a. csc 115

b. tan 120

Chapter 6

69. Sketch 𝑦 = 2 sec(𝑥 + 𝜋)

70. Sketch 𝑦 = csc (3𝑥 −𝜋

2) + 3

71. Sketch 𝑦 = −2 tan(2𝑥) + 3

72. Sketch 𝑦 =1

2cos

1

2(𝑥 −

𝜋

5) − 3

73. Write a sinusoidal equation that has a maximum at(−𝜋

6, 3) and a minimum at (

3𝜋

2, −1)

74. Find Domain and Range: 𝑓(𝑥) = 3 csc 4 (𝑥 −𝜋

3) − 1

75. Find Domain and Range: 𝑓(𝑥) = 2 tan (3𝑥 −3𝜋

4) + 2

76. Find Domain and Range: 𝑓(𝑥) = 3 sin(2𝑥) +1

3

77. Bilbo became the first hobbit to orbit the planet when one of Gandalf’s spells went horribly wrong.

Bilbo’s distance from the equator varied sinusoidally with time. After 45 minutes, Bilbo was the farthest

away at 3200 miles. Half a cycle later he was the closest at 1600 miles. It takes Bilbo 80 minutes to orbit

the planes. Write an equation that models how far Bilbo is away from the equator.

78. A tsunami is a fast-moving ocean wave caused by an underwater earthquake. The water first goes

down, from its normal level, then rises an equal distance above its normal level, then returns to its

normal level. Generally the period of a tsunami is around 15 minutes. Suppose a tsunami with a height

of 10 meters hits Honolulu, Hawaii which has a normal water depth of 9 meters. Write an equation that

models the water height.

79. Write an equation for the graph shown

80. Evaluate

a. cos-1 (−√3

2)

b. arctan √3

c. sin-1 (𝑐𝑜𝑠𝜋

2)

d. cos (𝑡𝑎𝑛−1√3)

e. sin(𝑠𝑖𝑛−1 √3

2)

f. cos (𝑎𝑟𝑐𝑠𝑖𝑛 (−1

2))

g. sin (arctan (−√3))

Chapter 7

81. Simplify 𝑐𝑠𝑐2𝑥(1 − 𝑐𝑜𝑠2𝑥)

82. Simplify 𝑐𝑜𝑡𝐴 × 𝑠𝑒𝑐𝐴 × 𝑠𝑖𝑛𝐴

83. Simplify (𝑐𝑠𝑐𝜃 − 𝑐𝑜𝑡𝜃)(𝑠𝑒𝑐𝜃 + 1)

84. Simplify sin𝐴 × tan𝐴 + sin (90 − 𝐴)

85. Simplify cos (𝜋

3+ 𝜃) + cos (

𝜋

3− 𝜃)

86. Simplify cos(𝑥 + 𝑦) cos(𝑥) + sin(𝑥 + 𝑦) sin (𝑥)

87. Simplify 𝑐𝑜𝑠2(4𝐴) − 𝑠𝑖𝑛2(4𝐴)

88. Simplify (1 + 𝑡𝑎𝑛2𝑥)(1 + cos 2𝑥)

89. Evaluate cot(20°) −cos(20°)

sin(20°)

90. Let cos(𝑥) =3

5 and tan(𝑥) < 0. Find sin (2𝑥)

91. Evaluate 2sin(67.5°) cos (67.5°)

92. Let sin(𝛼) =4

5, sin(𝛽) =

1

2. Let

𝜋

2< 𝛼 < 𝛽 < 𝜋. Find sin (𝛼 − 𝛽).

93. Evaluate sin(285°)

94. Evaluate cos (𝜋

4+ 𝜃) if cos(θ) =

1

2 with θ in Q4

95. Prove 𝑠𝑖𝑛𝜃

𝑠𝑖𝑛𝜃+𝑐𝑜𝑠𝜃=

𝑡𝑎𝑛𝜃

1+𝑡𝑎𝑛𝜃

96. Prove 2 csc(2𝑥) tan(𝑥) = 𝑠𝑒𝑐2(𝑥)

97. Prove sin(3𝑥) = 3 sin(𝑥) − 4sin3(𝑥)

98. Prove 1

1−𝑠𝑖𝑛𝑥+

1

1+𝑠𝑖𝑛𝑥= 2𝑠𝑒𝑐2𝑥

99. Prove sec(𝜃)+tan (𝜃)

sec(𝜃)−tan (𝜃)= (𝑠𝑒𝑐𝜃 + 𝑡𝑎𝑛𝜃)2

100. Solve 2 sin(2𝑥) + √3 = 0

101 √2 sin(2𝑥) = 1

102. Solve 2sin2𝑥 − 3 sin 𝑥 + 1 = 0

103. Solve cos2𝑥 − 3 cos 𝑥 − 4 = 0

104. Solve tan (𝑥

3) = −1

105. Solve (tan𝜃 − 1)(sec𝜃 − 1) = 0

106. Solve sin2(𝑥) − cos2(𝑥) = 1 + cos (𝑥)

107. Solve tan (𝜇) = 2sin (𝜇)

108. Solve 3(1 − cos(𝑥)) = sin2𝑥

Chapter 8

109. Given: a = 12, <A = 80, <B = 40. Find side length b

110.

111. A satellite dish can track the speed of a plane by recording the distance to the plane at two points

in time and the angle which the dish rotates. A satellite measured the distance to a plane at 35 miles.

After 15 minutes, the dish measured the distance to the plane at 58 miles. If the dish rotated 132

degrees, how far did the plane travel?

112. Find the altitude of an isosceles triangle with base 4.24 feet. The vertex angle of the triangle

measures 85 degrees.

113. Given: a = 3, b = 4, C = 40°. Find the area of triangle ABC

114. A triangular parcel of land has sides measuring 25 yards, 31 yards, and 50 yards. What is the area

of the land?

115. Find the area of a regular octagon inscribed in a circle with radius 6 cm.