1. sin 2. Cos 8) Sin 10. sec. Honors Pre-Calculus Final Exam Review 2 nd semester June TRIGONOMETRY Solve for 0 2. without using a calculator:

Honors Pre-Calculus Name: _________________________________________ Final Exam Review – 2nd _{semester June 2014}

TRIGONOMETRY

Solve for

0







2



without using a calculator: 1.

2

1

sin







2.

csc





2

3.

tan







1

4.

cos







2

1)________________ 2)________________

3)________________

4)________________

Solve for



in degrees giving all solutions. 5.

sin







1

6.

2 3

cos



 7.

tan





undefined 5)________________ 6)________________

7)________________

Give the exact value of each expression.

8. Tan1 3 9.        2 1 cot Sin 1 10.

_

























5

3

sec

Cos

1 8)________________ 9)________________ 10)_______________ 11. Given

y

x

2

1

sin

3

1



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

12. Given:           2 3 cos 2 4 x



y , 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:



1



sin





1



csc





sin



13)_______________

14. Prove:









csc

tan

1

csc

sec

_





15. Solve:

2

cos

2





1



sin



for



if

0







2



. 15)_______________

16. Solve for x:

2

cos

x



1



0

,

0



x



2



. 16)_______________

17. Solve a right triangle ABC, if a = 6 and



A



30

0. (Give exact lengths.) 17) B __________ b = _________ c = __________

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

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,



R



137

0,t = 15, and s = 12. Find r to the nearest integer. 22)_______________

23. If ∆ABC has b = 30,



C = 400 and



A = 600, find a to the nearest tenth. 23)_______________

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.

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.

cos

75



b.

sin

105



33a.)_____________

33b.)_____________ 34. Simplify the following:

a.

cos

75



cos

15





sin

75



sin

15



b.

sin(

30





x

)



sin(

30





x

)

34a.)_____________ 34b.)_____________

35. Suppose angle A is acute and

13

5

cos

A



. Find:

a.

sin

A

b.

cos

2

A

c.

sin

2

A

35a.)_____________

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

a.

x

x

2

sin

2

cos

1



b.



1cot2





cos2 1



x x c.

1

sec

tan



x

x

36a.)_____________ 36b.)_____________ 36c.)_____________

37. Evaluate the given expression:

12

5

sin

2

1



2



37.)______________

38. Prove:



1tan2x





1cos2x



2 39. Solve the following for

0



x



2



.

a.

cos

2

x



sin

x



2

b.

2

sin

2

x



3

cos

x



3

a.)_______________ b.)_______________

c.

sin

x

tan

x



2

sin

x

c.)_______________

POLAR COORDINATES AND COMPLEX NUMBERS

40. Convert the following into rectangular form.

a.



6

,

90





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

z

₁



3



i

and

z

₂



4



4

i

, find

z

₁

,

z

₂

,

and

z

₁

z

₂ in polar form. 42.)

z

₁



__________

z

₂



_________

2 1

z

z

= ___________

43. If z = (4, 30°), find the following in

a



bi

form: 43a)

a. z3 b. z5 c. z -2

43b)

43c)

44. Find the cube roots of 3i . 44.)____________________ _______________________ _______________________

SEQUENCES AND SERIES

45. State whether each sequence is arithmetic, geometric or neither and find a formula for a_n. 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, a₃ 4 and

27

4

6



a

. Find a₁₀. 47.)____________________

48. In an arithmetic sequence, a₃ 23 and a₆ 50. Find

a

₂₄. 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.)____________________

50. Given the series











27

1

9

1

3

1

1

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

2

1

2







x

x

52.)____________________ _______________________

53. Express the following series in sigma notation:

8 + 5 + 2 – 1 – 4 – 7 – 10 – 13 53.)____________________ 54.

1

2

1

1 1











t

k

t

t

k k

a) Write the first 6 terms 54a)

b) Write an explicit formula 54b)

55. Evaluate the following:

a.







7 3

)

7

4

(

n

n

55a.)___________________ b.



  20 1 ) 1 ( 2 k k k 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)

58. Given the series











27

1

9

1

3

1

1

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 + …

61. Prove by Mathematical Induction.



4 1





2 1



1   



 n n i n i

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

3

lim

_

_n

_

n

n 62) 63.

4

2

2 2

lim

_

_x

x



_

x 63) 64. x x x 2 2 3

lim

7    64) 65. 1 2 2

lim

_ n_nn n 65) 66.

 

           1 2 3 1

lim

n _nn n 66) 67. 3 3

lim

3     x x x 67) 68. 5 1 3 5 2 2 1

lim

_ x_  x x x 68) 69. x x x 1 1

lim

0 69) 70.

_lim

  n

sin

n

70) 71.

_lim

0  x

x

x

1

1

1





₇₁₎

72.

_lim

1  x x x    1 2 3 72) 73.

_lim

  n

1

5

3

2 2





n

n

n

73) 74.

_lim

2  n

4

2

2 2





n

n

n

74) 75.

_lim

  n

5

1

3

5

2 2







n

n

n

75) 76.

_lim

  n 9 3 2 _  n n 76)____________________ 77.

_lim

2  x

2

8

3





x

x

77)____________________

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

78. f(x)4x5 2x3 9x1 5 78.)____________________ 79.





2 1 3 1 ) (   x x f 79.)____________________ 80. 2

6

)

(

x

x

f



80.)____________________ 81.

x

x

x

f

(

)





1

81.)____________________

82. f(x)



x32x

 

3x2

82.)____________________

83.

4

4

4

)

(

₂ 2









x

x

x

x

f

83.)____________________

84.

f

(

x

)





3

x

2



4

x



1



2

84.)____________________

85.





9

15

2

3

)

(

₂ 2









x

x

x

x

x

f

85.)____________________

86.

_f(x)3

 

_3x 7

_{86.)____________________}

Use the difference quotient,

h

x

f

h

x

f

h

)

(

)

(

lim

0







, t

o find the derivative.

87.

f

(

x

)



5



6

x

87.)____________________

88. f(x)x2 3x5 88.)____________________

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. f(x) x2 1; (2, 3) 90.)____________________ _______________________ 91. f(x)x3 x; (2, 6) 91.)____________________ _______________________ 92.

f

(

x

)



x



1

; (3, 2) 92.)____________________ _______________________ 93.

x

x

x

f

(

)



2



4

; (2, 6) 93.)____________________ _______________________

94. Find the equation of the tangent to the given curve: yx3 5x2 4x2; when

x



2

94.)____________________

95. If f'(x)3x2 4x3, find

f

(x

)

. 95.)____________________

****************************************************************************************************************************** 96. Use the first and second derivative to identify the local max and min, inflection point/s and

determine the intervals where the curve is concave up and concave down. Then graph the function. (DO NOT USE A GRAPHING CALCULATOR).

27 9 3 ) (x x3 x2  x f Local Max_______________ Local Min_______________ Pt. of Inflection_________________ Interval/s: Concave Up___________________ Concave Down_________________

97. A ball is thrown upward from the top of an 80 ft building so that its height in feet above the ground after

t

seconds is h(t)8064t16t2.

a. What is the instantaneous velocity at

t



1

second? 97a.)___________________

b. When is the velocity = 0? 97b.)___________________

c. What is the ball’s maximum height above the ground? 97c.)___________________

d. When does the ball hit the ground? 97d.)___________________

e. For what values of

t

is the ball falling? 97e.)___________________

Use derivatives to solve.

98. The number 120 is divided into two parts such that the product of one number times the square of the other is a maximum. Determine the two numbers.

98.)____________________

99. 800 yards of fencing is used to enclose a rectangular field with a fence down the middle parallel to one of the sides. What is the maximum area which can be enclosed?

99.)____________________

100. A cardboard poster is to have 50 square inches of printed material surrounded by a 2” border at the top, 2” at the bottom and 1” on each side. Find the minimum dimensions of the poster which has a minimum area.

100.)_____________________

101. An open square-base box is to be manufactured from the least amount of material. If the box is to have a volume of 32 cubic meters, what dimensions will minimize the amount of material used?

Honors Pre-Calculus

Final Exam Review Packet Answer Key 06/2014

1.

6

11

,

6

7





2.

4

3

,

4





3.

4

7

,

4

3





4. undefined 5.

270





360



n

6. n n       360 210 360 150 7.

90





180



n

8.

,

60



3



9. 3 10.

3

5



11a.

3

1

11b.

4



12a. 2 12b.

3

2



12c.

2



12d. -4

13.

cos

2



14. answers will vary 15.

2

3

,

6

5

,

6







16.

4

7

,

4





17.



B



60



;

b



6

3

;

c



12

18.

60



19.

342901

.

6

ft. 20.

104

.

5



21. 1.782 miles 22. 25 23. 26.4 24. 27.1 25.



C



39



;



B



109



;

b



27

.

0

or



C



141



;



B



7



;

b



3

.

5

26. 410.86 27.

36

.

9



or

143

.

1



28. 534.97 29.

69

.

5



,

69

.

5



,

41



30. 549.5 m 31. 28.7 32. 739.93 miles 33a. 4 2 6 33b. 4 2 6

34a.

cos

90





0

34b. cosx 35a.

13

12

35b.

169

119



35c.

169

120

36a. cotx 36b.

2

cot

2

x

36c.

x

x

cos

1

sin



37. 2 3

 38. answers will vary 39a.

2



39b.

3

4

,

3

2

,







39c.

0



,



,

1

.

11

,

4

.

25

40a.

6

i

40b. 3i 41a.       4 , 2 3



41b.       3 4 , 2



41c.       2 3 , 2



42.

12

2

8

;

4

2

4

;

6

11

2

_{2 1 2} 1







cis

z

z

cis

z

cis

z







43a.

64

i

43b. 512 3512i 43c. i 32 3 32 1  44. 3 2 10,3 2 130,3 2 250 cis cis

cis 45a. arith _{a 5n22}

45b. neither; an n 2n 2_  45c. geometric; 1 3 1 81          n n a 46a. 51 46b. 459 47.

2187

4

48. 212 49. 2340 50a. converge 50b.

4

3

51.

2187

1640

52. x x 2 3 3 ; 2 3 2 3     53.







   8 1 11 3 i n 54a. 1, 4, 9, 16, 25, 36 54b.

a

_n



n

2 55a. 65 55b. 5320 56.







  25 1 1 4 i n 57a.

2



x



4

57b.

3

1

3

1

_

_



x

58a.



         1 1 3 1 i n 58b.

2

3

59.

4

6

1 1







 n n

a

a

a

60.

3

1



61. omit!! 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.



20

x

6



6

x

4



9

x

2 79.





3 1 3 6   x 80. 3

12

x



81.

x

x

x

2

1



82.

15

x

4



18

x

2 83.





2 2 4  x 84.



12 8





3 24 1



x x x 85.









2 2 3 15 6 3    x x x _86.

 

3 4

3

7 x

87. -6 88. 2x – 3 89. 6x + 4 90.

4

;

y



4

x



5

91.

11

;

y



11

x



16

92.

4

5

4

1

;

4

1

_

_

x

y

93.

1

;

y



x



4

94.

y





4

x



6

95. f(x)x3 2x2 3xc

96.

Local Max=(-1,32) 97a. 32 97b. 2 sec. 97c. 144 ft

Local Min= (3,0)

Point of inflection=(1,16) 97d. 5 sec.

97e.

2



t



5

Concave Up: (1,∞)

Concave Down: (-∞,1) 98. 80 and 40 99.

3 2 666 , 26 yds2 100. 7 inches x 14 inches 101. 4 x 4 x 2

Honors Pre-Calculus Final Exam Review Name:

Additional Review June 2014

1. Find the area of



PQR

if

q



6

,

r



7

, and



P



50



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



0

,

2





.

x

x

)

cos

2

cos(

2







5.)_______________________ 6. Suppose that

7

5

sin

A



and

5

3

sin

B



where





A





2

and

0

2







B

. Find

sin(

A



B

)

. 6.)_______________________

8. Change into

a



bi

form:       3 13 , 4



8.)_______________________ 9. Find: 3 8i  9.)_______________________

10. Find



4



4

i



4 in trigonometric form. 10.)______________________

11.

1

lim

₂ 2 1







_x

x

x

x 11.)______________________ 12.

x

x

x 0

lim

 12.)______________________ 13.

_

_

_

2 3 5 4 lim 2      _{x x} x x x 13.)______________________ 14. x x x    4 2 lim 0 14.)______________________

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

x x x x f 5 3 1 2 7 ) (  2   at x1 15.)______________________ 16. 3 2

2

5

)

(

x

x

x

f







16.

f

'

(

2

)



_________

17. Determine two numbers whose sum is 36 such that the product of the first

and the cube of the second is a maximum. 17.)______________________

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 6 6 20 _7.       6 5 , 4  8. 22 3i 9. 2i, 3i, 3i 10. 1024cis 11. 2

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