Sem 2 Final Exam Review

Trigonometry H (2010-2011) Sem2 Final Exam Study Guide Name _________________________________________Period ______Date ___________

1. Determine whether the function has an inverse function. If it does, find the inverse function.

( )2

3 + 29, –9( )

+ 9 + 2, –9

x xf x

x x

<= ≥

2. Find the constant of proportionality for the following situation:"y is jointly proportional to x and z and inversely proportional to w."

7, 5, 4w x y= = = , and 3z =

3. Evaluate the indicated function for f (x) = x2 – 9 and g (x) = x – 5.

( fg )(1)

4. The cost of a widget has increased from $5.43 in 2002 to $6.47 in 2006. Estimate the cost of a widget in 2004 to the nearest cent.

5. Describe the sequence of transformations from the related common function ( )f x x= to g.

( ) 9g x x= − +

6. A security guard is paid $12 per hour for regular time and time-and-a-half for overtime. The weekly wage function is given by

15 , 0 40( )

22.5( 40) 600, 40h h

W hh h

< ≤= − + >

where h is the number of hours worked in a week.

The company increased its pay by 1 dollars per hour. What is the new weekly wage function?

7. If 2( ) –3 + 5 + 6f x x x= , use synthetic division to evaluate

18

f .

8. Find all real solutions of the polynomial equation 4 3 – 8 + 56 – 49 0x x x = .

9. Consider the function

3 2

2

7 – 3 – 2 + 3( ) – 3 – 2

x x xf xx x

=. Identify any slant asymptotes.

10. Solve:5 – 5 2

– 5xx

≥

11. Suppose the cost C (in millions of dollars) of cleaning p% of an oil spill from a particular lake is estimated by

31100

pCp

=− , 0 100.p≤ <

Find the cost of cleaning up 66% of the spill. Round to the nearest tenth of a million dollars.

12. A member of a collegiate track-and-field team recently adopted a new training regimen. His times, s, in the 400-meter event began to improve as described by the function

2.60 461 kts

e= +

+ , where s is his time and t is the number of months since beginning the new training regimen. Determine k if his times have improved 1.0 second after 6 months. (Substitute t = 0 to determine his 400 m time before the training change.) Round to the nearest thousandth.

13. An initial investment of $2000 grows at an annual interest rate of 7% compounded continuously. How long will it take to double the investment?

14. Solve for x: (8 ) 16x xe e− = . Round to 3 decimal places.

15. Solve for x: / 24 0.0052x− = . Round to 3 decimal places.

16. Condense the expression ( )3 log logx y− to the logarithm of a single term.

17. Condense the expression 3 3 3

1 [log log 5] [log ]3

x y+ − to the logarithm of a single term.

18. Identify the x-intercept of the function ( ) 3ln( 1)f x x= − .

19. Which investment option will pay the most interest?A) 10.6% compounded annuallyB) 10.4% compounded semiannuallyC) 10.2% compounded quarterlyD) 10.0% compounded continuouslyE) These investments all pay the same amount of interest.

20. Solve 2ln 13x = for x.

21. What is the half-life of a radioactive substance if 2.4 g decays to 0.80 g in 61 hours? Round to the nearest tenth of an hour.

22. Which of the following can be inserted to make the statement true?

( )

24arccos arcsin ________ , 0 22

x x−= ≤ ≤

23. Write an algebraic expression that is equivalent to ( )tan arccos 7x .

24. A communications company erects a 81-foot tall cellular telephone tower on level ground. Determine the angle of depression, θ (in degrees), from the top of the tower to a point 43 feet from the base of the tower. Round answer to two decimal places.

25. The angle of elevation of the sun is 27° . Find the length, l, of a shadow cast by a tree that is 44 feet tall. Round answer to two decimal places.

26. Use an inverse function to write θ as a function of x.

27. Evaluate 3arccos

2 without using a calculator.

28. Evaluate

1 2sin2

− −

without using a calculator.

29. Sketch the graph of the function below, being sure to include at least two full periods.

30. Use a graphing utility to graph the function below. Be sure to include at least two full periods.

31. Use a graphing utility to graph the function below. Be sure to include at least two full periods.

32. Describe the relationship between ( ) cos( )f x x= and ( ) cos3 – 11g x x= . Consider amplitude, period, and shifts.

33. Find the indicated trigonometric value in the specified quadrant.

13csc III tan4

Function Quadrant Trigonometric Value

θ θ= −

34. Evaluate the sine of the angle without using a calculator.5–4π

35. Evaluate the tangent of the angle without using a calculator.–120°

36. Determine two coterminal angles (one positive and one negative) for the given angle. Give your answer in degrees.

37. Use trigonometric identities to transform the left side of the equation into the right side. Assume all angles are positive acute angles, and show all of your work.

38. Verify the identity shown below.

39. Find all solutions of the following equation in the interval [ )0, 2π .

40. Find the exact value of ( )cos u v+ given that 7sin25

u = and

12cos13

v = −. (Both u and v

are in Quadrant II.)

41. Use the figure below to find the exact value of the given trigonometric expression.

θ

7

24

θ

7

24

42. Use a double-angle formula to find the exact value of cos 2u when 5sin , where

13 2u uπ π= < <

.

43. Given 14A = ° , 8b = , and 6a = , use the Law of Sines to solve the triangle (if possible) for the value of c. If two solutions exist, find both. Round answer to two decimal places.

44. Determine a value for b such that a triangle with 65A = ° and 13a = has only one solution.

45. Given 5a = , 10b = , and 6c = , use the Law of Cosines to solve the triangle for the value of B. Round answer to two decimal places.

46. A vertical pole 36 feet tall stands on a hillside that makes an angle of 19° with the horizontal. Determine the approximate length of cable that would be needed to reach from the top of the pole to a point 65 feet downhill from the base of the pole. Round answer to two decimal places.

47. Two automobiles leave from the same point in Chicago at the same time and travel along straight highways that differ by 78° . If their speeds are 63 mi/hr and 69 mi/hr, respectively, determine how far apart the cars are after 28 minutes. Round answer to two decimal places.

48. Find the angle between the vectors u and v if + 2=u i j and –4 + 3=v i j . Round answer to two decimal places.

49. A force of 50 pounds is exerted along a rope attached to a crate at an angle of 30° above the horizontal. The crate is moved 32 feet. How much work has been accomplished? Round answer to one decimal place.

50. A 650 -pound trailer is sitting on an exit ramp inclined at 40°on Highway 35. How much force is required to keep the trailer from rolling back down the exit ramp? Round answer to two decimal places.

51. Use DeMoivre's Theorem to find the indicated power of the folllowing complex number.

( )4–7 + 7 i

52. A residential building contractor borrowed $31,000 to complete a new home. Some of the money was borrowed at 5%, some at 6%, and some at 8%. How much was borrowed at each rate if the annual interest owed was $1850 and the amount borrowed at 6% is two times more than the amount borrowed at 8%?

53. Write the partial fraction decomposition of the improper rational expression.2

2 + 8

+ 8x x

x x+

54. Write the partial fraction decomposition of the rational expression.

( )

2

210 + 18 – 27

+ 3x xx x

55. Sketch the graph and label the vertices of the solution set of the system of inequalities. Shade the solution set.

56. Use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system.

– 6 + 8 –703 – 2 + 6 –58– + 6 – 8 70–2 + – 6 45

x y zx y zx y zx y z

= = = =

57. Use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system.

+ 7 – 8 57– + 4 –19

–2 – 14 + 16 57– + 6 + 3 –6

x y zx y z

x y zx y z

= = = =

58. The owner of the "Crazy 'Bout Nuts" shop wants to create his own blend of mixed nuts. To do so, he mixes peanuts ($5 per pound), pecans ($6 per pound), and cashews ($9 per pound) to obtain 100 pounds of mixed nuts costing $8 per pound. If he wants the amount of peanuts to be twice that of the pecans, how many pounds of each type of nut should he use? Use the matrix capabilities of a graphing utility to solve the resulting system of linear equations. Round answers to nearest hundredth of a pound.

1302 3 8 780

2 0

x y zx y z

x y

+ + = + + = − =

59. Use the matrix capabilities of a graphing utility to find the inverse of the matrix 8 1 24 13 16 3 32

8 1 24 13 16 3 32

− − − − (if it exists).

60. Write the system of linear equations as a matrix equation AX = B, and use Gauss-Jordan

elimination on the augmented matrix [ ]A BM to solve for the matrix X.+ 2 8

4 – 4 20x yx y

= =

61. Use the matrix capabilities of a graphing utility to find AB, if possible.–9 72 –2 –3 9 5 9

,–1 –1 2 –1 –5 21 4

A B

= =

62. Use Cramer's Rule to solve the following system of linear equations:16 – 24 + 16 8–24 + 16 + 8 732 + 8 – 24 2

x y zx y z

x y z

= = =

63. The following cryptogram was endoded using matrix A.

[ ] [ ] [ ] [ ]23 66 42 26 43 30 –4 44 11 0 25 25

0 1 1–1 1 –12 3 2

A =

Decode the cryptogram to find the secret message.

64. Determine a positive value for y such that a triangle with vertices (–2,4), (1,2) , and (0, )y has an area of 5 square units.

65. Given

0 –4 –812 8 –40 –16 –4

A = and

–12 8 0–4 4 –8–12 –4 –4

B = , find BA .

66. Use mathematical induction to prove the formula for every positive integer n. Show all your work.

( ) ( )–1 + 2 + 5 + 8 3 – 4 3 – 52nn n+ + =K

67. Find the sum using the formulas for the sums of powers of integers.

( )11

2

19 – 2

nn n

=∑

68. A deposit of $5000 is made in an account that earns 8% interest compounded monthly. The balance in the account after n months is given by

0.065000 1 , 1, 2,3,4

n

nA n = + =

K

Find the balance in the account after 9 years by finding the 108th term of the sequence. Round to the nearest penny.

69. Joey bought a new tractor for use in his landscaping business. He expects that the use of the tractor will generate $3500 of revenue in the first year with an increase in revenue of $500 per year for subsequent years. If the tractor's useful lifetime is 8 years, how much revenue will Joey receive from the use of the tractor?

70. Find the indicated nth term of the geometric sequence.

5th term: 5 9

3 3– , –16 256

a a= =

71. A college sent a survey to a sample of juniors. Of the 598 students surveyed, 286 live on campus, of whom 120 have a GPA of 2.5 or greater. The other 312 juniors live off-campus, of whom 135 have a GPA of 2.5 or greater. What is the probability that a survey participant lives on campus and has a GPA of 2.5 or greater?

72. In a sample of 23 hand-held calculators, 16 are known to be nonfunctional. If 6 of these calculators are selected at random, what is the probability that exactly 4 in the selection are nonfunctional? Round to the nearest thousandth.

73. Find the number of distinguishable permutations of the group of letters.

74. Eight weightlifters are competing in the dead-lift competition. In how many ways can the weightlifters finish first, second, and third (no ties)?

75. A combination lock will open when the right choice of three numbers (from 1 to 30) is selected. How many different lock combinations are possible?

76. Use the Binomial Theorem to expand the complex number. Simplify your result.

( )45 + 4i

77. Prove the inequality for the indicated integer values of n.8 , 247

nn n > ≥

78. Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form.

2 29 + 22 + 9 – 8 0x xy y =

79. Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

7x2 + 2y2 – 5x – 9y + 8 = 0

80. Find the center and vertices of the ellipse.

2 2 + 16 + 6 – 160 + 393x y x y = 0

81. Find the standard form of the parabola with the given characteristics.

focus: (6, –5) vertex: (6, –2)

82. Determine whether the planes are parallel, orthogonal, or neither.

3 + 5 + 4– – 2 + 13 2x y zx y z

==

83. Find the angle, in degrees, between two adjacent sides of the pyramid shown below. Round to the nearest tenth of a degree. [Note: The base of the pyramid is not considered a side.]

x

y

z

PQ

R

S

( ) ( ) ( ) ( )2,0,0 , 2, 2,0 , 0, 2,0 , 1,1,6P Q R S

84. Find a unit vector in the opposite direction of u.

–15, –2, –5=u

85. Find the standard form of the equation of the sphere with the given characteristics.

Endpoints of a diameter: ( ) ( )–6, –4, –4 , 0, –2, –6

86. Find a set of parametric equations for the line through the point and parallel to the specified vector. Show all your work.

( )5, –5, –1 , parallel to –8,7,1

87. Find the volume of the parallelpiped with the given vertices.

( ) ( ) ( ) ( )( ) ( ) ( ) ( )3, –4, –9 , 1, 4, –11 , 4,2, –17 , 2,10, –19 ,

–4,0, –2 , –6,8, –4 , –3,6, –10 , –5,14, –12

A B C D

E F G H

88. Find the distance between the point and the plane.

( )2, –2, –25 – 4 – 5 –5x y z =

89. Given that lim ( ) –4x c

f x→

= and

lim ( ) 7x c

g x→

=, find

[ ]2lim ( ) ( )x c

f x g x→

+.

90. Find

27

8lim11 24x

xx x→

−

+ +by direct substitution.

91. Find

0

3 3limy

yy→

+ −

92. Determine

215

15lim

225x

x

x→

−

−(if it exists) by evaluating the corresponding one-sided limits.

93. Use the limit process to find the slope of the graph of 3x + at ( )1, 2 .

94. Find a formula for the slope of the graph of 2( )

8f x

x=

+ .

95. Evaluate

( )10

3

13

kk

=+∑

using the summation formulas and properties.

96. Use the limit process to find the area of the region between 2( ) 2f x x= + and the x-axis

on the interval [ ]0,6 .

97. Use the limit process to find the area of the region between ( ) 10 6f x x= + and the x-axis

on the interval [ ]0,5 .

Answer Key

1.

1 – 29 , 2

3( ) – 2 – 9, 2

x xf x

x x

− <= ≥

Origin: Chapter 01- Functions and Their Graphs, 107

2.2815

k =

Origin: Chapter 01- Functions and Their Graphs, 1203. 32

Origin: Chapter 01- Functions and Their Graphs, 944. $5.95

Origin: Chapter 01- Functions and Their Graphs, 125. reflection in the x-axis; then vertical shift 4 units up


6.

16 , 0 40( )

24( 40) 640, 40h h

W hh h

< ≤= − + >


7.

1 4218 64

f =

Origin: Chapter 02- Polynomial and Rational Functions, 45

8. 1, 7, 7x = ±Origin: Chapter 02- Polynomial and Rational Functions, 74

9. 7 + 18y x=

Origin: Chapter 02- Polynomial and Rational Functions, 105

10.( )5, – 5,

3 −∞ ∪ ∞ Origin: Chapter 02- Polynomial and Rational Functions, 120

11. 144Origin: Chapter 02- Polynomial and Rational Functions, 109

12. 0.400 month–1

Origin: Chapter 03- Exponential and Logarithmic Functions, 8113. 9.90 years

Origin: Chapter 03- Exponential and Logarithmic Functions, 6814. 0.693

Origin: Chapter 03- Exponential and Logarithmic Functions, 5715. 14.361

Origin: Chapter 03- Exponential and Logarithmic Functions, 56

16.

3log x

y


17.

33

5log xy

Origin: Chapter 03- Exponential and Logarithmic Functions, 4718. x = 5

Origin: Chapter 03- Exponential and Logarithmic Functions, 3019. B


20.13/ 2 13/ 2,e e−

Origin: Chapter 03- Exponential and Logarithmic Functions, 5421. 38.5 hours


22. 2x

Origin: Chapter 04- Trigonometry, 146

23.

21 497

xx

−


24. 59.74°

Origin: Chapter 04- Trigonometry, 15725. 80.47l = feet

Origin: Chapter 04- Trigonometry, 15626.


27. 6π


28.–

4π


29.



Origin: Chapter 04- Trigonometry, 11932. The period of g(x) is five times the period of f(x).

Graph of g(x) is shifted downward 11 unit(s) relative to the graph of f(x).Origin: Chapter 04- Trigonometry, 113

33.

43 17Origin: Chapter 04- Trigonometry, 102

34.

22


35. 3Origin: Chapter 04- Trigonometry, 100

36. Answers may vary. One possible response is given below.



Origin: Chapter 05- Analytic Trigonometry, 2939.

Origin: Chapter 05- Analytic Trigonometry, 40

40.( ) 253cos

325u v+ =


41.


42.119cos 2169

u =

Origin: Chapter 05- Analytic Trigonometry, 7543. 2.08 and 15.39c =

Origin: Chapter 06- Additional Topics in Trigonometry, 744. 9b =

Origin: Chapter 06- Additional Topics in Trigonometry, 845. 52.41°

Origin: Chapter 06- Additional Topics in Trigonometry, 2346. 70.16 feet

Origin: Chapter 06- Additional Topics in Trigonometry, 3247. 32.73 miles

Origin: Chapter 06- Additional Topics in Trigonometry, 3348. 112.83°

Origin: Chapter 06- Additional Topics in Trigonometry, 6749. 1,472.2 foot-pounds

Origin: Chapter 06- Additional Topics in Trigonometry, 7750. 590.46 pounds

Origin: Chapter 06- Additional Topics in Trigonometry, 78

51. –16,384Origin: Chapter 06- Additional Topics in Trigonometry, 96

52. $13,000 at 5%; $12,000 at 6%; $6000 at 8%Origin: Chapter 07- Systems of Equations and Inequalities, 34

53.27 – 81

+ 8x

x x+

+Origin: Chapter 07- Systems of Equations and Inequalities, 44

54.2

1 9 9+ – + 3x x x

Origin: Chapter 07- Systems of Equations and Inequalities, 4255.

Origin: Chapter 07- Systems of Equations and Inequalities, 5156. x = 1, y = –9, z = –8

Origin: Chapter 08- Matrices and Determinants, 1657. no solution

Origin: Chapter 08- Matrices and Determinants, 17

58. peanuts: 35.00 lb; pecans: 17.50 lb; cashews: 87.50 lbOrigin: Chapter 08- Matrices and Determinants, 63

59. does not existOrigin: Chapter 08- Matrices and Determinants, 50

60.

8–4

X =


61.

10 24 –3 –29–53 –68 –10 76–21 –36 0 42–1 12 –6 –16


62.16 113 152, ,9 63 63

x y z= = =

Origin: Chapter 08- Matrices and Determinants, 8663. SEND MONEY

Origin: Chapter 08- Matrices and Determinants, 9764. 4

Origin: Chapter 08- Matrices and Determinants, 8865. 399


66. 1) When n = 1, ( )1

14 5 32

S = = +. The formula is valid for n = 1.

2) Assume that ( ) ( )4 + 9 + 14 + 19 5 – 1 5 + 3

2kk k+ + =K

is true.

( )( ) ( )

( )( )

( )( )

1

2

2

4 + 9 + 14 + 19 5 1 – 1 5 + 3 + 5 + 42

1 5 + 3 + 5 + 421 5 + 13 + 82

1 5 1 + 32

kkS k k k

k k k

k k

k k

+ = + + + =

=

=

+= +

K

By mathematical induction, the formula is true for all positive integers n.Origin: Chapter 09- Sequences, Series, and Probability, 53

67. –2821Origin: Chapter 09- Sequences, Series, and Probability, 59

68. $10,247.65Origin: Chapter 09- Sequences, Series, and Probability, 18

69. $42,000Origin: Chapter 09- Sequences, Series, and Probability, 35

70.2–

729Origin: Chapter 09- Sequences, Series, and Probability, 42

71.187615Origin: Chapter 09- Sequences, Series, and Probability, 96

72. 0.379Origin: Chapter 09- Sequences, Series, and Probability, 98

73.

Origin: Chapter 09- Sequences, Series, and Probability, 8174. 336

Origin: Chapter 09- Sequences, Series, and Probability, 8375. 27,000

Origin: Chapter 09- Sequences, Series, and Probability, 7676. –119 – 120i

Origin: Chapter 09- Sequences, Series, and Probability, 71

77. 1)

248 24.6 247

≈ > . The statement is true for n = 24.

2) Assuming that

87

kk >

for k > 24, show that

18 17

kk

+ > + .

8 , by assumption7

kk >

1

8 8 87 7 7

8 17 7

k

k

k

k k+

>

> +

Since 7 7 1 124 , then or 1 or 1.1 1 7 7

k k k k k k> > > > + > + Therefore,

1 18 1 81 or 1.7 7 7

k kk k k k

+ + > + > + > +

By mathematical induction, the relation is true for all 24n ≥ .Origin: Chapter 09- Sequences, Series, and Probability, 57

78.

( ) ( )2 2

12

14

x y′ ′− =

Origin: Chapter 10- Topics in Analytic Geometry, 4379. ellipse

Origin: Chapter 10- Topics in Analytic Geometry, 4080. center: (3, 8) vertices: (–1, 8), (7, 8)

Origin: Chapter 10- Topics in Analytic Geometry, 2181. (x – 6)2 = –12(y + 2)

Origin: Chapter 10- Topics in Analytic Geometry, 1582. orthogonal

Origin: Chapter 11- Analytic Geometry in Three Dimensions, 5583. 91.5°

Origin: Chapter 11- Analytic Geometry in Three Dimensions, 61

84.

1 –3,8, 277

Origin: Chapter 11- Analytic Geometry in Three Dimensions, 19

85. ( ) ( ) ( )2 2 2 + 3 + 3 + 5 11x x x+ + =

Origin: Chapter 11- Analytic Geometry in Three Dimensions, 886. Answers may vary. One possible answer is shown below.

–7 + 6 , 7 + 2 , –3 – 2x t y t z t= = =Origin: Chapter 11- Analytic Geometry in Three Dimensions, 43

87. 618Origin: Chapter 11- Analytic Geometry in Three Dimensions, 41

88.

3366

Origin: Chapter 11- Analytic Geometry in Three Dimensions, 6089. 121

Origin: Chapter 12- Limits and and Introduction to Calculus, 6

90.1–

150Origin: Chapter 12- Limits and and Introduction to Calculus, 9

91.3

6Origin: Chapter 12- Limits and and Introduction to Calculus, 14

92. limit does not existOrigin: Chapter 12- Limits and and Introduction to Calculus, 17

93.14Origin: Chapter 12- Limits and and Introduction to Calculus, 24

94. ( )22

8x−

+

Origin: Chapter 12- Limits and and Introduction to Calculus, 2695. 44,200

Origin: Chapter 12- Limits and and Introduction to Calculus, 4096. 84

Origin: Chapter 12- Limits and and Introduction to Calculus, 4797. 155

Origin: Chapter 12- Limits and and Introduction to Calculus, 46

Sem 2 Final Exam Review

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