Honors Pre-Calculus Name: _________________________________________ Final Exam Review – 2nd semester June 2016

TRIGONOMETRY

Solve for 20 without using a calculator:

1. 2

1sin 2. 2csc 3. 1tan 4. 2cos 1)________________

2)________________ 3)________________ 4)________________

Solve for in degrees giving all solutions.

5. 1sin 6. 2

3cos 7. tan undefined 5)________________

6)________________ 7)________________ Give the exact value of each expression.

8. 3tan 1 9.

2

1sincot 1

10.

5

3cossec 1

8)________________

9)________________ 10)_______________

11. Given xy2

1sin

3

1 , find

a. amplitude 11a.)_____________ b. period 11b.)_____________

c. graph at least two periods of the function d.

2

12. Given:

23cos24

xy , find 12a)______________

a. Amplitude b. Period 12b)______________ c. Phase shift

d. Vertical shift e. graph at least two periods of the function 12c)______________ 12d)______________

13. Simplify: sincsc1sin1 13)_______________

14. Prove:

csc

tan1

cscsec

15. Solve: sin1cos2 2 for if 20 . 15)_______________

16. Solve for x: 01cos2 x , 20 x . 16)_______________

17. Solve a right triangle ABC, if a = 6 and 030A . (Give exact lengths.) 17) B __________

b = _________

c = __________

3

18. A boy flying a kite is standing 20 ft from a point directly under the kite. 18)_______________ If the string to the kite is 40 ft long, find the angle of elevation of the kite. 19. An airplane is at an elevation of 30,000 ft and approaches the airport 19)_______________ with an angle of descent of 5°. What is the distance between the airport and the point on the ground directly below the airplane?

20. Given ∆ABC with a = 30, b = 20, c = 40, find the largest angle. 20)_______________ 21. The captain of a clipper ship spots two other ships on the ocean. 21)_______________ One ship is 5 miles away while the other is 5.2 miles away. The angle between the two sightings is 20º. How far apart are the two observed ships? (to three decimals)

22. In ∆RST, 0137R ,t = 15, and s = 12. Find r to the nearest integer. 22)_______________

23. If ∆ABC has b = 30, C = 400 and A = 60

0, find a to the nearest tenth. 23)_______________

4

24. Two angles of a triangle measure 29º and 51º. The longest side is 55cm. 24) ______________ Find the length of the shortest side to the nearest tenth.

25. Solve for ABC if a = 15, c = 18, and A = 32º. 25) C = __________ B = __________ b = __________ OR (if two triangles)

C = __________ B = __________ b = __________ 26. Find the area of ∆ABC if b = 32, c = 27, and A = 108º. 26)___ ____

27. The area of ∆PQR is 15. If p = 5 and q= 10, find all possible measures of 27)___ ____ R. 28. Find the area of a regular pentagon inscribed in a circle with 28) ____ radius 15.

5

29. The sides of an isosceles triangle have lengths 7, 10 and 10. What are the measures of its angles? 29.)______________ 30. At a distance of 200 meters, the angle of elevation to the top of a building is

70. Approximately how tall is the building? 30)_______________

31. Two ships leave a port on courses that differ by 70 and each travels at 25 knots. In terms of nautical miles, how far apart are the ships after 1 hour? 31)_______________ 32. After leaving an airport, a plane flies for 1.75 hours at a speed of 200 k/h on a

course of 100. The plane then flies for 2 hours at a speed of 250 k/h on a

course of 40. At this time, how far from the airport is the plane? 32.)______________ 33. Find the exact value of the following:

a. 75cos b. 105sin 33a.)_____________

33b.)_____________ 34. Simplify the following:

a. 15sin75sin15cos75cos b. )30sin()30sin( xx 34a.)_____________

34b.)_____________

6

35. Suppose angle A is acute and 13

5cos A . Find:

a. Asin b. A2cos c. A2sin 35a.)_____________

35b.______________ 35c.)_____________ 36. Simplify the following:

a. x

x

2sin

2cos1 b. 12coscot1 2 xx c.

1sec

tan

x

x 36a.)_____________

36b.)_____________ 36c.)_____________

37. Evaluate the given expression: 12

5sin21 2

37.)______________

38. Prove: 22cos1tan1 2 xx

39. Solve the following for 20 x .

a. 2sin2cos xx b. 3cos3sin2 2 xx a.)_______________

b.)_______________

c. xxx sin2tansin c.)_______________

7

POLAR COORDINATES AND COMPLEX NUMBERS

40. Convert the following into rectangular form.

a. 90,6 b.

6,2

40a.)_____________

40b.)_____________ 41. Convert the following into polar form:

a. (3,3) b. (-1, 3 ) c. (0, -2) 41a.)_____________

41b.)_____________ 41c.)_____________

42. If iz 31 and iz 442 , find 2121 ,, zzandzz in polar form. 42.) 1z __________

2z _________

21zz = ___________

43. If z = (4, 30°), find the following in bia form: 43a)

a. z3 b. z5 c. z -2 43b) 43c)

44. Find the cube roots of i3 . 44.)____________________

_______________________ _______________________

8

SEQUENCES AND SERIES

45. State whether each sequence is arithmetic, geometric or neither and find a

formula for na .

a. 17, 12, 7, 2, … b. 3, 8, 15, 24, 35,… c. -81, 27, -9, 3,… 45a.)___________________ _______________________ 45b.)___________________ _______________________ 45c.)___________________ _______________________ 46 a. Find the 17th term of 3, 6, 9, … 46a.)___________________

b. Find the sum of the first 17 terms of 3, 6, 9, … 46b.)___________________

47. In a geometric sequence, 43 a and 27

46 a . Find 10a . 47.)____________________

48. In an arithmetic sequence, 233 a and 506 a . Find 24a . 48.)____________________

49. An auditorium has 30 rows of seats. There are 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on. Determine the seating capacity in the auditorium. 49.)____________________

9

50. Given the series 27

1

9

1

3

11

a) Does it converge or diverge? 50a) b) Find the sum, if possible. 50b)

51. Find the sum of the first 8 terms of ,27

1,

9

1,

3

1,1 51)

52. Find the interval of convergence and the sum in terms of x of:

...9

4

3

21

2

xx

52.)____________________

_______________________ 53. Express the following series in sigma notation: 8 + 5 + 2 – 1 – 4 – 7 – 10 – 13 53.)____________________

54. 12

1

1

1

ktt

t

kk

a) Write the first 6 terms 54a) b) Write an explicit formula 54b)

10

55. Evaluate the following:

a.

7

3

)74(n

n 55a.)___________________

b.

20

1

)1(2k

kk 55b.)___________________

56. Express in sigma notation: 5 + 9 + 13 + …+ 101 56.) 57. For what values of x do the following converge? a) 1 + (x-3) + (x-3)2 + … 57a) b) 1 + 3x + 9x2 + … 57b)

58. Given the series 27

1

9

1

3

11

a) Express the series in sigma notation. 58a) b) Find the sum. 58b) 59. Write a recursive definition for the following sequence: 6, 10, 14, 18, 22, … 59) 60. For what value of x does the following sequence converge 60)

to 5

3? 1 + 2x + 4x2 + …

11

LIMITS, DERIVATIVES AND APPLICATIONS OF DERIVATIVES (MAX/MIN PROBLEMS)

Find the following limits, if they exist. If they do not exist, write “does not exist.”

62. 2

3lim

n

n

n

62)

63. 4

22

2lim

x

x

x

63)

64. x

x

x 2

23lim

7

64)

65. 1

2 2

lim

n

nn

n

65)

66.

1

231lim n

nn

n

66)

67. 3

3lim

3

x

x

x

67)

68. 5

1352

2

1lim

x

xx

x

68)

12

69. x

x

x 11lim

0

69)

70. limn

nsin 70)

71. lim0x

x

x1

1

1

71)

72. lim1x

x

x

1

23 72)

73. limn

15

32

2

n

nn 73)

74. lim2n

4

22

2

n

nn 74)

75. limn

5

1352

2

n

nn 75)

13

76. limn

9

3

2

n

n 76)____________________

77. lim2x

2

83

x

x 77)____________________

Find the derivative of the following (using the special rules/techniques).

78. 5924)( 135 xxxxf 78.)____________________

79. 2

13

1)(

xxf 79.)____________________

80. 2

6)(

xxf 80.)____________________

81. x

xxf

1)(

81.)____________________

82. 23 32)( xxxxf 82.)____________________

14

Find the derivative of the following (using the special rules/techniques).

83. 44

4)(

2

2

xx

xxf 83.)____________________

84. 22 143)( xxxf 84.)____________________

85.

9

1523)(

2

2

x

xxxxf 85.)____________________

86. 3 73)( xxf 86.)____________________

Use the difference quotient, h

xfhxf

h

)()(lim

0

, to find the derivative.

87. xxf 65)( 87.)____________________

88. 53)( 2 xxxf 88.)____________________

89. 643)( 2 xxxf 89.)____________________

15

Find the slope of the graph at the given point. Use the result to find an equation of the tangent line to the graph at the point.

90. 1)( 2 xxf ; (2, 3) 90.)____________________

_______________________

91. xxxf 3)( ; (2, 6) 91.)____________________

_______________________

92. 1)( xxf ; (3, 2) 92.)____________________

_______________________

93. x

xxf4

2)( ; (2, 6) 93.)____________________

_______________________

94. Find the equation of the tangent to the given curve: 245 23 xxxy ; when 2x

94.)____________________

95. If 343)(' 2 xxxf , find )(xf . 95.)____________________

16

Honors Pre-Calculus Final Exam Review Packet Answer Key 2015

1. 6

11,

6

7 2.

4

3,

4

3.

4

7,

4

3 4. undefined

5. n 360270 6. n

n

360210

360150 7. n 18090 8. 60,

3

9. 3 10. 3

5 11a.

3

1 11b. 4

12a. 2 12b. 3

2 12c.

2

12d. -4

13. 2cos 14. answers will vary 15. 2

3,

6

5,

6

16.

4

7,

4

17. 12;36;60 cbB 18. 60 19. 6.342901 ft.

20. 5.104 21. 1.782 miles 22. 25 23. 26.4

24. 27.1 25. 0.27;109;39 bBC or 5.3;7;141 bBC

26. 410.86 27. 9.36 or 1.143 28. 534.97 29. 41,5.69,5.69

30. 549.5 m 31. 28.7 32. 739.93 miles 33a. 4

26

33b. 4

26 34a. 090cos 34b. xcos 35a.

13

12

35b. 169

119 35c.

169

120 36a. xcot 36b. x2cot2

36c. x

x

cos1

sin

37.

2

3 38. answers will vary 39a.

2

39b. 3

4,

3

2,

39c. 25.4,11.1,,0 40a. i6 40b. i3

41a.

4,23

41b.

3

4,2

41c.

2

3,2

42. 12

28;4

24;6

112 2121

ciszzciszcisz 43a. i64 43b. i5123512

17

43c. i32

3

32

1 44. 2502,1302,102 333 cisciscis 45a. arith 225 nan

45b. neither; nnan 22 45c. geometric; 1

3

181

n

na 46a. 51

46b. 459 47. 2187

4 48. 212 49. 2340

50a. converge 50b. 4

3 51.

2187

1640

52. x

x23

3;

2

3

2

3

53.

8

1

113i

n 54a. 1, 4, 9, 16, 25, 36 54b. 2nan

55a. 65 55b. 5320 56.

25

1

14i

n 57a. 42 x

57b. 3

1

3

1 x 58a.

1

1

3

1

i

n

58b. 2

3 59.

4

6

1

1

nn aa

a

60. 3

1 62. 3 63.

4

1

64. 0 65. 66. does not exist (oscillates) 67.

68. 4

3 69. 2 70. does not exist (oscillates) 71. does not exist

72. 6

3 73.

5

3 74.

2

1 75. 5

76. 1 77. 12 78. 246 9620 xxx

79. 313

6

x 80.

3

12

x 81.

xx

x

2

1 82.

24 1815 xx

83. 2

2

4

x 84. 143812 2 xxx 85.

2

2

3

1563

x

xx 86. 3

4

37 x

87. -6 88. 2x – 3 89. 6x + 4 90. 54;4 xy

91. 1611;11 xy 92. 4

5

4

1;

4

1 xy 93. 4;1 xy 94. 64 xy

95. cxxxxf 32)( 23

18

Honors Pre-Calculus Final Exam Review: Additional Review 2016

1. Find the area of PQR if 7,6 rq , and 50P and find p . 1. Area PQR = _______

p = _______________

2. The perimeter of a regular decagon is 240. Find its area. 2.)_______________________

3. Find all other angles and sides of ΔABC if A = 59, a = 22, and b = 19. 3.)_______________________ 4. Two planes leave an airport at the same time. One flies 425 mph at N 5º W while the other flies 530 mph at N 67º E. How far apart are the planes after 3½ hours? 4.)_______________________

5. Solve the following trig equation for 2,0 .

xx cos)2cos(2 5.)_______________________

6. Suppose that 7

5sin A and

5

3sin B where

A

2 and

20

B .

Find )sin( BA . 6.)_______________________

19

7. Change into polar form: i232 7.)_______________________

8. Change into bia form:

3

13,4

8.)_______________________

9. Find: 3 8i 9.)_______________________

10. Find 444 i in trigonometric form. 10.)______________________

11. 1

lim2

2

1

x

xx

x 11.)______________________

12. x

x

x 0lim

12.)______________________

20

13. 23

54lim

2

xx

xx

x 13.)______________________

14. x

x

x

42lim

0 14.)______________________

15. Find the slope of the curve at the given point: x

xxxf1

352

7)( 2 at 1x 15.)______________________

16. 23 25)( xxxf 16. )2('f _________

1. 16.1 units2 ; p = 5.6 2. 4424.4 3. c = 24.6, C = 73.2º, B = 46.8º 4. 1987 miles 5. 6.

35

6620 7.

6

5,4

8. i322 9. iii 3,3,2 10. cis1024 11.

2

1 12. does not exist 13. 5 14. 4

1 15. 3 16. -34