math 2412 general review for precalculus last … 2412 general review for precalculus last updated...

Math 2412 General Review for Precalculus Last Updated 12/15/2015

Give the equation of the horizontal asymptote, if any, ofthe function.

1. h(x) = 9x2 - 9x - 86x2 - 7x + 8

2. h(x) = 2x3 - 2x - 99x + 5

3. g(x) = x + 9x2 - 49

Give the equation of the oblique asymptote, if any, of thefunction.

4. f(x) = x2 + 3x - 5x - 4

5. f(x) = 2x3 + 11x2 + 5x - 1x2 + 6x + 5

.

6. f(x) = -10x3 + 21x2 + 5x + 18-5x - 2

Graph the function.

7. f(x) = 3x(x - 1)(x + 2)

8. f(x) = x + 4x

9. f(x) = xx2 - 25

10. f(x) = x2 + x - 2x2 - x - 6

1

Solve the equation.11. log (4 + x) - log (x - 5) = log 4

12. log 2 (x + 4) + log 2 (x - 2) = 4

13. 32x + 3x - 6 = 0

Solve the equation. Express irrational answers in exactform and as a decimal rounded to 3 decimal places.

14. 23

x= 51 - x

Solve the exponential equation. Use a calculator to obtaina decimal approximation, correct to two decimal places, forthe solution.

15. 4(3x - 1) = 13

Solve the problem.16. The first recorded population of a particular

country was 25 million, and the populationwas recorded as 29 million 8 years later. Theexponential growth function A =25ektdescribes the population of this country t yearssince the first recording. Use the fact that 8years later the population increased by 4million to find k to three decimal places.

17. Conservationists tagged 70 black-nosedrabbits in a national forest in 2009. In 2010,they tagged 140 black-nosed rabbits in thesame range. If the rabbit population followsthe exponential law, how many rabbits will bein the range 9 years from 2009?

18. A fossilized leaf contains 13% of its normalamount of carbon 14. How old is the fossil (tothe nearest year)? Use 5600 years as thehalf-life of carbon 14.

19. The amount of a certain drug in thebloodstream is modeled by the functiony = y0 e- 0.40t, where y0 is the amount of thedrug injected (in milligrams) and t is theelapsed time (in hours). Suppose that 10milligrams are injected at 10:00 A.M. If asecond injection is to be administered whenthere is 1 milligram of the drug present in thebloodstream, approximately when should thenext dose be given? Express your answer to thenearest quarter hour.

20. A cup of coffee is heated to 194° and is thenallowed to cool in a room whose airtemperature is 72°. After 11 minutes, thetemperature of the cup of coffee is 140°. Findthe time needed for the coffee to cool to atemperature of 102°. Assume the coolingfollows Newton's Law of Cooling:

U = T + (Uo - T)ekt.(Round your answer to one decimal place.)

21. In a town whose population is 3000, a diseasecreates an epidemic. The number of people, N,infected t days after the disease has begun isgiven by the function

N(t) = 30001 + 21.2 e - 0.54t

. Find the number of

infected people after 10 days.

Use a calculator to solve the equation on the interval 0 < 2 . Round the answer to two decimal places.

22. 2 csc = 5

23. 5 tan - 4 = 0

Solve the equation on the interval 0 < 2 .24. sin2 + sin = 0

25. 2 cos2 - 3 cos + 1 = 0

26. tan + sec = 1

27. sin2 - cos2 = 0

28. cos2 - sin2 = 1 + sin

2

Use a graphing utility to solve the equation on the interval0° x < 360°. Express the solution(s) rounded to onedecimal place.

29. 3 cos2 x + 2 cos x = 1

Simplify the expression.

30. cos 1 + sin

+ tan

Establish the identity.

31. sec u + tan u = cos u1 - sin u

32. sin x1 - cos x

+sin x

1 + cos x= 2 csc x

Use the information given about the angle , 0 2 , tofind the exact value of the indicated trigonometricfunction.

33. tan =2021

, < <32

Find sin(2 ).

34. csc = -52

, tan > 0 Find cos(2 ).

35. tan =724

, < <32

Find tan(2 ).

Find the exact value of the expression.

36. cos sin-1 23

+ 2 sin-1 -13

37. sin =14

, 0 < <2

Find sin 2

.

38. sin =14

, tan > 0 Find cos 2

.

39. tan = 3, < <2

Find tan 2

.

Use the Half-angle Formulas to find the exact value of thetrigonometric function.

40. sin 75°

41. cos 75°

42. tan 75°

The polar coordinates of a point are given. Find therectangular coordinates of the point.

43. 3, 23

The rectangular coordinates of a point are given. Findpolar coordinates for the point.

44. (- 3, -1)

The letters x and y represent rectangular coordinates.Write the equation using polar coordinates (r, ).

45. x2 + y2 - 4x = 0

The letters r and represent polar coordinates. Write theequation using rectangular coordinates (x, y).

46. r = 51 + cos

Identify and graph the polar equation.47. r = 1 - cos

48. r = 4 sin(2 )

3

Graph the polar equation.

49. r = 21 - cos

Write the complex number in polar form. Express theargument in degrees, rounded to the nearest tenth, ifnecessary.

50. 1 + 3i

Write the complex number in rectangular form.

51. 4 cos 116

+ i sin 116

Find zw or zw

as specified. Leave your answer in polar

form.52. z = 2 + 2i

w = 3 - iFind zw.

53. z = 8 cos 2

+ i sin 2

w = 3 cos 6

+ i sin 6

Find zw

.

Write the expression in the standard form a + bi.54. (1 + i)20

Find all the complex roots. Leave your answers in polarform with the argument in degrees.

55. The complex fourth roots of -16

Use the vectors in the figure below to graph the followingvector.

56. u + z

The vector v has initial position P and terminal point Q.Write v in the form ai + bj; that is, find its position vector.

57. P = (-6, 1); Q = (4, -4)

Find the quantity if v = 5i - 7j and w = 3i + 2j.58. v + w

Find the unit vector having the same direction as v.59. v = -12i - 5j

Write the vector v in the form ai + bj, given its magnitudev and the angle it makes with the positive x-axis.

60. v = 5, = 45°

Solve the problem.61. Two forces, F1 of magnitude 60 newtons (N)

and F2 of magnitude 70 newtons, act on anobject at angles of 40° and 130° (respectively)with the positive x-axis. Find the direction andmagnitude of the resultant force; that is, findF1 + F2. Round the direction and magnitude totwo decimal places.

4

62. An audio speaker that weighs 50 poundshangs from the ceiling of a restaurant from twocables as shown in the figure. To two decimalplaces, what is the tension in the two cables?

Find the angle between v and w. Round your answer toone decimal place, if necessary.

63. v = 8i + 6j, w = 4i + 9j

Solve the problem.64. An airplane has an air speed of 550 miles per

hour bearing N30°W. The wind velocity is 50miles per hour in the direction N30°E. To thenearest tenth, what is the ground speed of theplane? What is its direction?

State whether the vectors are parallel, orthogonal, orneither.

65. v = 4i - 2j, w = 4i + 2j

Decompose v into two vectors v1 and v2, where v1 isparallel to w and v2 is orthogonal to w.

66. v = 3i - 5j, w = -3i + j

Solve the problem.67. An SUV weighing 4900 pounds is parked on a

street which has an incline of 10°. Find theforce required to keep the SUV from rollingdown the hill and the force of the SUVperpendicular to the hill. Round the forces tothe nearest hundredth.

Solve the problem. Round your answer to the nearesttenth.

68. Find the work done by a force of 4 poundsacting in the direction of 44° to the horizontalin moving an object 4 feet from (0, 0) to (4, 0).

Find the vertex, focus, and directrix of the parabola.Graph the equation.

69. y2 = -16x

Find the vertex, focus, and directrix of the parabola. Graphthe equation.

70. x2 - 12x = 12y - 96

Solve the problem.71. An experimental model for a suspension

bridge is built in the shape of a parabolic arch.In one section, cable runs from the top of onetower down to the roadway, just touching itthere, and up again to the top of a secondtower. The towers are both 4 inches tall andstand 40 inches apart. Find the verticaldistance from the roadway to the cable at apoint on the road 6 inches from the lowestpoint of the cable.

Find the center, foci, and vertices of the ellipse.72. 2x2 + 5y2 - 24x + 30y + 107 = 0

5

Solve the problem.73. A bridge is built in the shape of a semielliptical

arch. It has a span of 102 feet. The height of thearch 27 feet from the center is to be 12 feet.Find the height of the arch at its center.

Find the center, transverse axis, vertices, foci, andasymptotes of the hyperbola.

74. x2 - 4y2 + 8x + 16y - 4 = 0

Solve the problem.75. Two recording devices are set 4000 feet apart,

with the device at point A to the west of thedevice at point B. At a point on a line betweenthe devices, 200 feet from point B, a smallamount of explosive is detonated. Therecording devices record the time the soundreaches each one. How far directly north of siteB should a second explosion be done so thatthe measured time difference recorded by thedevices is the same as that for the firstdetonation?

Convert the polar equation to a rectangular equation.

76. r = 84 - 4 cos

77. r = 124 + cos

78. r = 123 + sin

Graph the curve whose parametric equations are given.79. x = 2t, y = t + 2; -2 t 3

Solve the problem.80. Ron throws a ball straight up with an initial

speed of 40 feet per second from a height of 7feet. Find parametric equations that describethe motion of the ball as a function of time.How long is the ball in the air? When is theball at its maximum height? What is themaximum height of the ball?

81. A baseball player hit a baseball with an initialspeed of 170 feet per second at an angle of 40°to the horizontal. The ball was hit at a height of3 feet off the ground. Find parametricequations that describe the motion of the ballas a function of time. How long is the ball inthe air? When is the ball at its maximumheight? What is the distance the ball traveled?

Find the parametric equations that define the curveshown.

82.

Solve the problem.83. Find parametric equations for an object that

moves along the ellipse x29

+y24

= 1 with the

motion described.

The motion begins at (0, 2), is clockwise, andrequires 2 seconds for a complete revolution.

Solve the system of equations by substitution.84.

x + 7y = -23x + y = 34

6

Solve the system of equations by elimination.85.

5x - 2y = -1 x + 4y = 35

Solve the problem.86. A retired couple has $200,000 to invest to

obtain annual income. They want some of itinvested in safe Certificates of Deposit yielding6%. The rest they want to invest in AA bondsyielding 12% per year. How much should theyinvest in each to realize exactly $20,400 peryear?

87. A movie theater charges $8.00 for adults and$5.00 for children. If there were 40 peoplealtogether and the theater collected $272.00 atthe end of the day, how many of them wereadults?

Solve the system of equations.88.

3x + 2y + z = -203x - 5y - z = 114x + y + 3z = -15

Solve the problem.89. The Family Arts Center charges $23 for adults,

$12 for senior citizens, and $9 for childrenunder 12 for their live performances onSunday afternoon. This past Sunday, the paidrevenue was $10,408 for 732 tickets sold. Therewere 40 more children than adults. How manychildren attended?

Solve the system of equations. If the system has nosolution, say that it is inconsistent.

90. x - y + 4z = 15x + z = 0-x + y - 4z = -3

Solve the system of equations.91.

x + 4y - z = 3 x + 5y - 2z = 53x + 12y - 3z = 9

Solve the problem using matrices.92. Find real numbers a, b, and c such that the

graph of the function y = ax2 + bx + c containsthe points (-2, -4), (1, -1), and (3, -19).

93. Jenny receives $1270 per year from threedifferent investments totaling $20,000. One ofthe investments pays 6%, the second one pays8%, and the third one pays 5%. If the moneyinvested at 8% is $1500 less than the amountinvested at 5%, how much money has Jennyinvested in the investment that pays 6%?

Solve the problem.94. Determinants are used to show that three

points lie on the same line (are collinear). Ifx1 y1 1x2 y2 1x3 y3 1

= 0,

then the points (x1, y1), (x2, y2), and (x3, y3)are collinear. If the determinant does not equal0, then the points are not collinear. Are thepoints (-7, 8), (0, -1), and (-14, 17) collinear?

Solve the system of equations using Cramer's Rule if it isapplicable. If Cramer's Rule is not applicable, say so.

95.5x - 8y - z = -75 x + 3y + 7z = 84

7x + y + z = 24

96.x - y + 2z = -4

2x + z =0-x + y - 2z = 16

Compute the product.97.

5 -1 5 2 -3 -3 8 -1 -5

2 4 5-7 -5 7 1 3 -9

Each matrix is nonsingular. Find the inverse of the matrix.Be sure to check your answer.

98.1 3 21 3 32 7 8

7

Show that the matrix has no inverse.99.

4 20 8-3 -1 1-1 7 4

Solve the system using the inverse matrix method.100.

2x + 4y - 5z = -8x + 5y + 2z = -1

3x + 3y + 3z = 15

Write the partial fraction decomposition of the rationalexpression.

101. x - 1(x - 4)(x - 3)

102. 8x2 + 17x + 6(x + 2)(x + 1)2

103. 12x + 3(x - 1)(x2 + x + 1)

104. 2x3 + 2x2

(x2 + 5)2

Solve the problem.105. The area of a rectangular piece of cardboard

shown is 736 square inches. The cardboard isused to make an open box by cutting a 4-inchsquare from each corner and turning up thesides. If the box is to have a volume of 1216cubic inches, find the dimensions of thecardboard that must be used.

106. A person at the top of a 600 foot tall buildingdrops a yellow ball. The height of the yellowball is given by the equation h = -16t2 + 600where h is measured in feet and t is thenumber of seconds since the yellow ball wasdropped. A second person, in the samebuilding but on a lower floor that is 408 feetfrom the ground, drops a white ball 3 secondsafter the yellow ball was dropped. The heightof the white ball is given by the equationh = -16(t - 3)2 + 408 where h is measured infeet and t is the number of seconds since theyellow ball was dropped. Find the time thatthe balls are the same distance above theground and find this distance.

Graph the solution set of the system of inequalities orindicate that the system has no solution.

107. 2x - y -8x + 2y 2

108. x + 2y 2 x - y 0

8

The sequence is defined recursively. Write the first fourterms.

109. a1 = 2, a2 = 5; an = an-2 - 3an-1

Express the sum using summation notation.

110. 14

+25

+12

+ ... +1417

Find the sum.111. 1 + 3 + 5 + ... + 1625

Solve.112. Suppose you just received a job offer with a

starting salary of $37,000 per year and aguaranteed raise of $1500 per year. How manyyears will it be before you've made a total (oraggregate) salary of $1,025,000?

Find the sum.

113.4

k = 1

25

k+1

Determine whether the infinite geometric series convergesor diverges. If it converges, find its sum.

114.

k=14 2

3k-1

Use the Principle of Mathematical Induction to show thatthe statement is true for all natural numbers n.

115. 12

+14

+18

+116

+ ... + 12n

= 1 - 12n

116. 1 · 2 + 2 · 3 + 3 · 4 + . . . + n(n + 1) =n(n + 1)(n + 2)

3

Expand the expression using the Binomial Theorem.117. (3x + 2)5

Use the Binomial Theorem to find the indicatedcoefficient or term.

118. The coefficient of x in the expansion of (3x+ 2)5

9

Answer KeyTestname: GENERAL PRECAL REVIEW

1. y = 32

2. no horizontal asymptotes3. y = 04. y = x + 75. y = 2x - 16. no oblique asymptote7.

8.

9.

10


10.

11. {8}12. {4}

13. ln 2ln 3

14. ln 5

ln 23

+ ln 51.337

15. {0.95}16. 0.01917. 35,840 rabbits18. 16,45319. 3:45 P.M20. 26.4 minutes21. 2737 people22. {0.41, 2.73}23. 0.67, 3.82

24. 0, , 32

25. 0, 3

, 53

26. {0}

27.4

, 34

, 54

, 74

28. 0, , 76

, 116

29. 70.5°, 180.0°, 289.5°30. sec

31. sec u + tan u = 1cos u

+sin ucos u

=1 + sin u

cos u=

1 + sin ucos u

·1 - sin u1 - sin u

=1 - sin2 u

cos u(1 - sin u)=

cos2 ucos u(1 - sin u)

=cos u

1 - sin u

32. sin x1 - cos x

+sin x

1 + cos x=

sin x[(1 + cos x)+(1 - cos x)](1 - cos x)(1 + cos x)

=2 sin x

1 - cos2x=

2 sin xsin2x

= 2 csc x.

11


33. 840841

34.1725

35. 336527

36. 7 5 + 8 227

37. 8 - 2 154

38. 8 + 2 154

39. 10 + 1-3

40. 12

2 + 3

41. 12

2 - 3

42. 2 + 3

43. -32

, 3 32

44. 2, -6

45. r = 4 cos 46. y2 = 25 - 10x47.

cardioid

12


48.

rose with four petals49.

50. 2(cos 60° + i sin 60°)51. 2 3 - 2i

52. 4 2 cos 12

+ i sin 12

53. 83

cos 3

+ i sin 3

54. -102455. 2(cos 45° + i sin 45°), 2(cos 135° + i sin 135°), 2(cos 225° + i sin 225°), 16(cos 315° + i sin 315°)56.

57. v = 10i - 5j58. 89

13


59. u = -1213

i -513

j

60. v =5 2

2i +

5 22

j

61. Direction: 89.40°; magnitude: 92.20 N62. Tension in right cable: 35.90 lb; tension in left cable: 41.59 lb63. 29.2°64. 576.6 mph; N25.7°W65. Neither

66. v1 =215

i -75

j, v2 = -65

i -185

j

67. 850.88 lb and 4825.56 lb68. 11.5 ft-lb69. vertex: (0, 0)

focus: (-4, 0)directrix: x = 4

70. vertex: (6, 5)focus: (6, 8)directrix: y = 2

71. 0.36 in.

72. (x - 6)25

+(y + 3)2

2= 1

center: (6, -3); foci: (7.7, -3), (4.3, -3); vertices: (8.2, -3), (3.8, -3)73. 14.14 ft

14


74. center at (-4, 2)transverse axis is parallel to x-axisvertices at (-6, 2) and (-2, 2)foci at (-4 - 5, 2) and (-4 + 5, 2)

asymptotes of y - 2 = -12

(x + 4) and y - 2 =12

(x + 4)

75. 422.22 ft76. y2 = 4x + 477. 15x2 + 16y2 + 24x - 144 = 078. 9x2 + 8y2 + 24y - 144 = 079.

80. x = 0, y = -16t2 + 40t + 72.664 sec, 1.25 sec,32 feet

81. x = 130.22t, y = -16t2 + 109.31t + 36.859 sec, 3.416 sec,893.179 feet

82. x = 2t + 2, y = -t + 5; 0 t 383. x = 3 sin ( t), y = 2 cos ( t), 0 t 284. x = 12, y = -2; (12, -2)85. x = 3, y = 8; (3, 8)86. $140,000 at 12% and $60,000 at 6%87. 24 adults88. x = -4, y = -5, z = 2; (-4, -5, 2)89. 258 children90. inconsistent91. x = -3z - 5, and y = z + 2, where z is any real number

or {(x, y, z) |x = -3z - 5, and y = z + 2, where z is any real number}92. a = -2 , b = -1, c = 293. $150094. Yes95. x = 1, y = 9, z = 8; (1, 9, 8)96. not applicable

15


97. 22 40 -27 22 14 16 18 22 78

98.-3 10 -3

2 -4 1-1 1 0

99.

4 20 8-3 -1 1-1 7 4

1 0 00 1 00 0 1

1 5 2-3 -1 1-1 7 4

14

0 0

0 1 00 0 1

1 5 2 0 14 7-1 7 4

14

0 0

34

1 0

0 0 1

1 5 20 14 70 12 6

14

0 0

34

1 0

14

0 1

1 5 2

0 1 12

0 12 6

14

0 0

356

114

0

14

0 1

1 5 2

0 1 12

0 0 0

14

0 0

356

114

0

-1128

-67

1

100. x = 5, y = -2, z = 2; (5, -2, 2)

101. 3x - 4

+-2

x - 3

102. 4x + 2

+4

x + 1+

-3(x + 1)2

103. 5x - 1

+-5x + 2

x2 + x + 1

104. 2x + 2x2 + 5

+-10x - 10

(x2 + 5)2

105. 16 in. by 46 in.106. 3.5 sec; 404 ft107.

16


108.

109. a1 = 2, a2 = 5, a3 = -13, a4 = 44110.

14

k = 1

kk + 3

111. 660,969112. 20 years

113. 8123125

114. Converges; 12

115. When n = 1, the left side of the statement is 12n

=121

=12

, and the right side of the

statement is 1 - 12n

= 1 - 121

= 1 - 12

=12

, so the statement is true when n = 1.

Assume the statement is true for some natural number k. Then,12

+14

+18

+116

+ ... + 12k

+1

2k+1= 1 - 1

2k+

12k+1

= 1 - 12k

1 - 12

= 1 - 12k+1

.

So the statement is true for k + 1. Conditions I and II are satisfied; by the Principle of Mathematical Induction, thestatement is true for all natural numbers.

17


116. First we show that the statement is true when n = 1.

For n = 1, we get 2 = 1(1 + 1)(1 + 2)3

= 2.

This is a true statement and Condition I is satisfied.

Next, we assume the statement holds for some k. That is,

1 · 2 + 2 · 3 + 3 · 4 + . . . + k(k + 1) = k(k + 1)(k + 2)3

is true for some positive integer k.

We need to show that the statement holds for k + 1. That is, we need to show that

1 · 2 + 2 · 3 + 3 · 4 + . . . + (k + 1)(k + 2) = (k + 1)(k + 2)(k + 3)3

.

So we assume that 1 · 2 + 2 · 3 + 3 · 4 + . . . + k(k + 1) = k(k + 1)(k + 2)3

is true and add the next term, (k + 1)(k + 2), to

both sides of the equation.

1 · 2 + 2 · 3 + 3 · 4 + . . . + k(k + 1) + (k + 1)(k + 2) = k(k + 1)(k + 2)3

+ (k + 1)(k + 2)

=k(k + 1)(k + 2)

3+

3(k + 1)(k + 2)3

=k(k + 1)(k + 2) + 3(k + 1)(k + 2)

3

=(k + 1)(k + 2)(k + 3)

3Condition II is satisfied. As a result, the statement is true for all natural numbers n.

117. 243x5 + 810x4 + 1080x3 + 720x2 + 240x + 32118. 240

18

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