Honors Pre-Calculus Name: Final Exam Review 2nd semester ...
Transcript of Honors Pre-Calculus Name: Final Exam Review 2nd semester ...
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 = __________
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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)_______________
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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.)_____________
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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.)_______________
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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.)____________________
_______________________ _______________________
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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.)____________________
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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 + …
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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.)____________________
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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.)____________________
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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.)____________________
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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
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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