Download Math 2412 General Review for Precalculus

General Review for Precalculus
Math 2412
Give the equation of the horizontal asymptote, if any, of
the function.
9x2 - 9x - 8
1. h(x) =
6x2 - 7x + 8
2. h(x) =
2x3 - 2x - 9
9x + 5
3. g(x) =
x+9
x2 - 49
8. f(x) = x +
Give the equation of the oblique asymptote, if any, of the
function.
x2 + 3x - 5
4. f(x) =
x-4
5. f(x) =
2x3 + 11x2 + 5x - 1
.
x2 + 6x + 5
6. f(x) =
-10x3 + 21x2 + 5x + 18
-5x - 2
9. f(x) =
Last Updated 12/15/2015
4
x
x
2
x - 25
Graph the function.
7. f(x) =
3x
(x - 1)(x + 2)
10. f(x) =
1
x2 + x - 2
x2 - x - 6
Solve the equation.
11. log (4 + x) - log (x - 5) = log 4
19. The amount of a certain drug in the
bloodstream is modeled by the function
y = y0 e- 0.40t, where y0 is the amount of the
drug injected (in milligrams) and t is the
elapsed time (in hours). Suppose that 10
milligrams are injected at 10:00 A.M. If a
second injection is to be administered when
there is 1 milligram of the drug present in the
bloodstream, approximately when should the
next dose be given? Express your answer to the
nearest quarter hour.
12. log 2 (x + 4) + log 2 (x - 2) = 4
13. 3 2x + 3 x - 6 = 0
Solve the equation. Express irrational answers in exact
form and as a decimal rounded to 3 decimal places.
2 x
= 51 - x
14.
3
20. A cup of coffee is heated to 194° and is then
allowed to cool in a room whose air
temperature is 72°. After 11 minutes, the
Solve the exponential equation. Use a calculator to obtain
a decimal approximation, correct to two decimal places, for
the solution.
15. 4 (3x - 1) = 13
temperature of the cup of coffee is 140°. Find
the time needed for the coffee to cool to a
temperature of 102°. Assume the cooling
follows Newton's Law of Cooling:
U = T + (Uo - T)ekt.
(Round your answer to one decimal place.)
Solve the problem.
16. The first recorded population of a particular
country was 25 million, and the population
was recorded as 29 million 8 years later. The
exponential growth function A =25ekt
21. In a town whose population is 3000, a disease
creates an epidemic. The number of people, N,
infected t days after the disease has begun is
given by the function
3000
N(t) =
. Find the number of
1 + 21.2 e - 0.54t
describes the population of this country t years
since the first recording. Use the fact that 8
years later the population increased by 4
million to find k to three decimal places.
infected people after 10 days.
17. Conservationists tagged 70 black-nosed
rabbits in a national forest in 2009. In 2010,
they tagged 140 black-nosed rabbits in the
same range. If the rabbit population follows
the exponential law, how many rabbits will be
in the range 9 years from 2009?
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
18. A fossilized leaf contains 13% of its normal
amount of carbon 14. How old is the fossil (to
the nearest year)? Use 5600 years as the
half-life of carbon 14.
-4=0
Solve the equation on the interval 0
24. sin2 + sin = 0
25. 2 cos2
26. tan
2
- 3 cos
+ sec
+1=0
=1
27. sin2
- cos2
=0
28. cos2
- sin2
= 1 + sin
<2 .
42. tan 75°
Use a graphing utility to solve the equation on the interval
0° x < 360°. Express the solution(s) rounded to one
decimal place.
29. 3 cos2 x + 2 cos x = 1
The polar coordinates of a point are given. Find the
rectangular coordinates of the point.
2
43. 3,
3
Simplify the expression.
cos
+ tan
30.
1 + sin
Establish the identity.
31. sec u + tan u =
32.
The rectangular coordinates of a point are given. Find
polar coordinates for the point.
44. (- 3, -1)
cos u
1 - sin u
The letters x and y represent rectangular coordinates.
Write the equation using polar coordinates (r, ).
45. x2 + y2 - 4x = 0
sin x
sin x
+
= 2 csc x
1 - cos x 1 + cos x
The letters r and represent polar coordinates. Write the
equation using rectangular coordinates (x, y).
5
46. r =
1 + cos
Use the information given about the angle , 0
2 , to
find the exact value of the indicated trigonometric
function.
20
3
,
Find sin(2 ).
< <
33. tan =
21
2
34. csc
=-
35. tan
=
5
, tan
2
7
,
24
>0
<
<
3
2
Identify and graph the polar equation.
47. r = 1 - cos
Find cos(2 ).
Find tan(2 ).
Find the exact value of the expression.
2
1
36. cos sin-1 + 2 sin-1 3
3
37. sin
=
1
, 0<
4
<
1
, tan
4
>0
Find cos
<
Find tan
38. sin
=
39. tan
= 3,
<
2
2
Find sin
2
2
2
.
48. r = 4 sin(2 )
.
.
Use the Half-angle Formulas to find the exact value of the
trigonometric function.
40. sin 75°
41. cos 75°
3
Graph the polar equation.
2
49. r =
1 - cos
Use the vectors in the figure below to graph the following
vector.
56. u + z
Write the complex number in polar form. Express the
argument in degrees, rounded to the nearest tenth, if
necessary.
50. 1 + 3i
Write the complex number in rectangular form.
11
11
+ i sin
51. 4 cos
6
6
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)
z
Find zw or
as specified. Leave your answer in polar
w
form.
52. z = 2 + 2i
w= 3-i
Find zw.
53. z = 8 cos
w = 3 cos
Find
Find the quantity if v = 5i - 7j and w = 3i + 2j.
58. v + w
2
6
+ i sin
+ i sin
Find the unit vector having the same direction as v.
59. v = -12i - 5j
2
Write the vector v in the form ai + bj, given its magnitude
v and the angle it makes with the positive x-axis.
60. v = 5, = 45°
6
z
.
w
Solve the problem.
61. Two forces, F1 of magnitude 60 newtons (N)
Write the expression in the standard form a + bi.
54. (1 + i)20
and F2 of magnitude 70 newtons, act on an
object at angles of 40° and 130° (respectively)
with the positive x-axis. Find the direction and
magnitude of the resultant force; that is, find
F1 + F2 . Round the direction and magnitude to
Find all the complex roots. Leave your answers in polar
form with the argument in degrees.
55. The complex fourth roots of -16
two decimal places.
4
62. An audio speaker that weighs 50 pounds
hangs from the ceiling of a restaurant from two
cables as shown in the figure. To two decimal
places, what is the tension in the two cables?
Find the vertex, focus, and directrix of the parabola.
Graph the equation.
69. y2 = -16x
Find the angle between v and w. Round your answer to
one decimal place, if necessary.
63. v = 8i + 6j,
w = 4i + 9j
Find the vertex, focus, and directrix of the parabola. Graph
the equation.
70. x2 - 12x = 12y - 96
Solve the problem.
64. An airplane has an air speed of 550 miles per
hour bearing N30°W. The wind velocity is 50
miles per hour in the direction N30°E. To the
nearest tenth, what is the ground speed of the
plane? What is its direction?
State whether the vectors are parallel, orthogonal, or
neither.
65. v = 4i - 2j,
w = 4i + 2j
Decompose v into two vectors v1 and v2 , where v1 is
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 one
tower down to the roadway, just touching it
there, and up again to the top of a second
tower. The towers are both 4 inches tall and
stand 40 inches apart. Find the vertical
distance from the roadway to the cable at a
point on the road 6 inches from the lowest
point of the cable.
parallel 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 the
force required to keep the SUV from rolling
down the hill and the force of the SUV
perpendicular to the hill. Round the forces to
the nearest hundredth.
Find the center, foci, and vertices of the ellipse.
72. 2x2 + 5y2 - 24x + 30y + 107 = 0
Solve the problem. Round your answer to the nearest
tenth.
68. Find the work done by a force of 4 pounds
acting in the direction of 44° to the horizontal
in moving an object 4 feet from (0, 0) to (4, 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 the
arch 27 feet from the center is to be 12 feet.
Find the height of the arch at its center.
Solve the problem.
80. Ron throws a ball straight up with an initial
speed of 40 feet per second from a height of 7
feet. Find parametric equations that describe
the motion of the ball as a function of time.
How long is the ball in the air? When is the
ball at its maximum height? What is the
maximum height of the ball?
Find the center, transverse axis, vertices, foci, and
asymptotes of the hyperbola.
74. x2 - 4y2 + 8x + 16y - 4 = 0
81. A baseball player hit a baseball with an initial
speed of 170 feet per second at an angle of 40°
to the horizontal. The ball was hit at a height of
3 feet off the ground. Find parametric
equations that describe the motion of the ball
as a function of time. How long is the ball in
the air? When is the ball at its maximum
height? What is the distance the ball traveled?
Solve the problem.
75. Two recording devices are set 4000 feet apart,
with the device at point A to the west of the
device at point B. At a point on a line between
the devices, 200 feet from point B, a small
amount of explosive is detonated. The
recording devices record the time the sound
reaches each one. How far directly north of site
B should a second explosion be done so that
the measured time difference recorded by the
devices is the same as that for the first
detonation?
Find the parametric equations that define the curve
shown.
82.
Convert the polar equation to a rectangular equation.
8
76. r =
4 - 4 cos
77. r =
12
4 + cos
78. r =
12
3 + sin
Solve the problem.
83. Find parametric equations for an object that
x2 y2
moves along the ellipse
+
= 1 with the
9
4
Graph the curve whose parametric equations are given.
79. x = 2t, y = t + 2; -2 t 3
motion described.
The motion begins at (0, 2), is clockwise, and
requires 2 seconds for a complete revolution.
Solve the system of equations by substitution.
84.
x + 7y = -2
3x + y = 34
6
Solve the system of equations by elimination.
85.
5x - 2y = -1
x + 4y = 35
Solve the problem using matrices.
92. Find real numbers a, b, and c such that the
graph of the function y = ax2 + bx + c contains
the points (-2, -4), (1, -1), and (3, -19).
Solve the problem.
86. A retired couple has $200,000 to invest to
obtain annual income. They want some of it
invested in safe Certificates of Deposit yielding
6%. The rest they want to invest in AA bonds
yielding 12% per year. How much should they
invest in each to realize exactly $20,400 per
year?
93. Jenny receives $1270 per year from three
different investments totaling $20,000. One of
the investments pays 6%, the second one pays
8%, and the third one pays 5%. If the money
invested at 8% is $1500 less than the amount
invested at 5%, how much money has Jenny
invested 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). If
x1 y1 1
87. A movie theater charges $8.00 for adults and
$5.00 for children. If there were 40 people
altogether and the theater collected $272.00 at
the end of the day, how many of them were
adults?
x2 y2 1 = 0,
x3 y3 1
then the points (x1 , y1), (x2 , y2 ), and (x3 , y3 )
Solve the system of equations.
88.
3x + 2y + z = -20
3x - 5y - z = 11
4x + y + 3z = -15
are collinear. If the determinant does not equal
0, then the points are not collinear. Are the
points (-7, 8), (0, -1), and (-14, 17) collinear?
Solve the system of equations using Cramer's Rule if it is
applicable. If Cramer's Rule is not applicable, say so.
95.
5x - 8y - z = -75
x + 3y + 7z = 84
7x + y + z = 24
Solve the problem.
89. The Family Arts Center charges $23 for adults,
$12 for senior citizens, and $9 for children
under 12 for their live performances on
Sunday afternoon. This past Sunday, the paid
revenue was $10,408 for 732 tickets sold. There
were 40 more children than adults. How many
children attended?
96.
Solve the system of equations. If the system has no
solution, say that it is inconsistent.
90.
x - y + 4z = 1
+ z= 0
5x
-x + y - 4z = -3
x - y + 2z = -4
+ z =0
2x
-x + y - 2z = 16
Compute the product.
97.
5 -1 5
2 4 5
-7 -5 7
2 -3 -3
8 -1 -5
1 3 -9
Solve the system of equations.
91.
x + 4y - z = 3
x + 5y - 2z = 5
3x + 12y - 3z = 9
Each matrix is nonsingular. Find the inverse of the matrix.
Be sure to check your answer.
1 3 2
98. 1 3 3
2 7 8
7
Show that the matrix has no inverse.
99.
4 20 8
-3 -1 1
-1 7 4
106. A person at the top of a 600 foot tall building
drops a yellow ball. The height of the yellow
ball is given by the equation h = -16t2 + 600
where h is measured in feet and t is the
number of seconds since the yellow ball was
dropped. A second person, in the same
building but on a lower floor that is 408 feet
from the ground, drops a white ball 3 seconds
after the yellow ball was dropped. The height
of the white ball is given by the equation
h = -16(t - 3)2 + 408 where h is measured in
Solve the system using the inverse matrix method.
100.
2x + 4y - 5z = -8
x + 5y + 2z = -1
3x + 3y + 3z = 15
feet and t is the number of seconds since the
yellow ball was dropped. Find the time that
the balls are the same distance above the
ground and find this distance.
Write the partial fraction decomposition of the rational
expression.
x-1
101.
(x - 4)(x - 3)
102.
8x2 + 17x + 6
(x + 2)(x + 1)2
103.
12x + 3
(x - 1)(x2 + x + 1)
104.
2x3 + 2x2
2
(x2 + 5)
Graph the solution set of the system of inequalities or
indicate that the system has no solution.
107. 2x - y -8
x + 2y 2
Solve the problem.
105. The area of a rectangular piece of cardboard
shown is 736 square inches. The cardboard is
used to make an open box by cutting a 4-inch
square from each corner and turning up the
sides. If the box is to have a volume of 1216
cubic inches, find the dimensions of the
cardboard that must be used.
108. x + 2y 2
x-y 0
8
The sequence is defined recursively. Write the first four
terms.
109. a 1 = 2, a 2 = 5; a n = a n-2 - 3a n-1
Express the sum using summation notation.
1 2 1
14
110. + + + ... +
4 5 2
17
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 a
guaranteed raise of $1500 per year. How many
years will it be before you've made a total (or
aggregate) salary of $1,025,000?
Find the sum.
4
2 k+1
113.
5
k=1
Determine whether the infinite geometric series converges
or diverges. If it converges, find its sum.
114.
4
k=1
2 k-1
3
Use the Principle of Mathematical Induction to show that
the statement is true for all natural numbers n.
1 1 1
1
1
1
+ ... +
=1115. + + +
2 4 8 16
n
2
2n
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 indicated
coefficient or term.
118. The coefficient of x in the expansion of (3x
+ 2)5
9
Answer Key
Testname: GENERAL PRECAL REVIEW
1. y =
2.
3.
4.
5.
6.
7.
3
2
no horizontal asymptotes
y= 0
y=x+ 7
y = 2x - 1
no oblique asymptote
8.
9.
10
Answer Key
Testname: GENERAL PRECAL REVIEW
10.
11. {8}
12. {4}
ln 2
13.
ln 3
14.
ln 5
2
ln
+ ln 5
3
1.337
15. {0.95}
16. 0.019
17. 35,840 rabbits
18. 16,453
19. 3:45 P.M
20. 26.4 minutes
21. 2737 people
22. {0.41, 2.73}
23. 0.67, 3.82
3
24. 0, ,
2
25. 0,
3
26. {0}
27.
4
,
,
5
3
3 5
7
,
,
4 4
4
28. 0, ,
7 11
,
6
6
29. 70.5°, 180.0°, 289.5°
30. sec
31. sec u + tan u =
32.
1
sin u 1 + sin u 1 + sin u 1 - sin u
1 - sin2 u
cos2 u
cos u
+
=
=
·
=
=
=
cos u cos u
cos u
cos u
1 - sin u cos u(1 - sin u) cos u(1 - sin u)
1 - sin u
sin x
sin x
sin x[(1 + cos x)+(1 - cos x)]
2 sin x
2 sin x
+
=
=
=
= 2 csc x.
1 - cos x 1 + cos x
(1 - cos x)(1 + cos x)
1 - cos2 x
sin2 x
11
Answer Key
Testname: GENERAL PRECAL REVIEW
33.
840
841
34.
17
25
35.
336
527
36.
7 5+8 2
27
37.
8 - 2 15
4
38.
8 + 2 15
4
39.
10 + 1
-3
40.
1
2
2+
3
41.
1
2
2-
3
42. 2 + 3
3 3 3
43. - ,
2 2
44. 2, -
6
45. r = 4 cos
46. y2 = 25 - 10x
47.
cardioid
12
Answer Key
Testname: GENERAL PRECAL REVIEW
48.
49.
rose with four petals
50. 2(cos 60° + i sin 60°)
51. 2 3 - 2i
52. 4 2 cos
53.
12
+ i sin
12
8
cos + i sin
3
3
3
54. -1024
55. 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 - 5j
58. 89
13
Answer Key
Testname: GENERAL PRECAL REVIEW
59. u = 60. v =
12
5
ij
13
13
5 2
5 2
i+
j
2
2
61. Direction: 89.40°; magnitude: 92.20 N
62. Tension in right cable: 35.90 lb; tension in left cable: 41.59 lb
63. 29.2°
64. 576.6 mph; N25.7°W
65. Neither
21
7
6
18
i - j, v2 = - i j
66. v1 =
5
5
5
5
67. 850.88 lb and 4825.56 lb
68. 11.5 ft-lb
69. vertex: (0, 0)
focus: (-4, 0)
directrix: x = 4
70. vertex: (6, 5)
focus: (6, 8)
directrix: y = 2
71. 0.36 in.
(x - 6)2 (y + 3)2
+
=1
72.
5
2
center: (6, -3); foci: (7.7, -3), (4.3, -3); vertices: (8.2, -3), (3.8, -3)
73. 14.14 ft
14
Answer Key
Testname: GENERAL PRECAL REVIEW
74. center at (-4, 2)
transverse axis is parallel to x-axis
vertices at (-6, 2) and (-2, 2)
foci at (-4 - 5, 2) and (-4 + 5, 2)
1
1
asymptotes of y - 2 = - (x + 4) and y - 2 = (x + 4)
2
2
75. 422.22 ft
76. y2 = 4x + 4
77. 15x2 + 16y2 + 24x - 144 = 0
78. 9x2 + 8y2 + 24y - 144 = 0
79.
80. x = 0, y = -16t2 + 40t + 7
2.664 sec, 1.25 sec,
32 feet
81. x = 130.22t, y = -16t2 + 109.31t + 3
6.859 sec, 3.416 sec,
893.179 feet
82. x = 2t + 2, y = -t + 5; 0 t 3
83. x = 3 sin ( t), y = 2 cos ( t), 0 t 2
84. 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 adults
88. x = -4, y = -5, z = 2; (-4, -5, 2)
89. 258 children
90. inconsistent
91. 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 = 2
93. $1500
94. Yes
95. x = 1, y = 9, z = 8; (1, 9, 8)
96. not applicable
15
Answer Key
Testname: GENERAL PRECAL REVIEW
97.
22 40 -27
22 14 16
18 22 78
-3 10 -3
98. 2 -4 1
-1 1 0
4 20 8 1 0 0
-3 -1 1 0 1 0
99.
-1 7 4 0 0 1
1
4
1
00
1 52
4
-3 -1 1
-1 7 4 0 1 0
0 01
0 0
1 5 2
1 3 1 0
0 1
2 56 14
0 12 6 1 0 1
4
1
4
1
00
1 5 2 4
0 14 7 3
10
-1 7 4 4
0 01
0 0
1 5 2
1
1 3
0
0 1
56
14
2
0 0 0 - 11 - 6 1
28 7
100. x = 5, y = -2, z = 2; (5, -2, 2)
3
-2
+
101.
x-4 x-3
102.
4
4
-3
+
+
x + 2 x + 1 (x + 1)2
103.
5
-5x + 2
+
x - 1 x2 + x + 1
104.
2x + 2 -10x - 10
+
x2 + 5 (x2 + 5)2
105. 16 in. by 46 in.
106. 3.5 sec; 404 ft
107.
16
1
00
4
1 5 2 3
10
0 14 7 4
0 12 6 1
01
4
Answer Key
Testname: GENERAL PRECAL REVIEW
108.
109. a 1 = 2, a 2 = 5, a 3 = -13, a 4 = 44
110.
14
k
k+ 3
k=1
111. 660,969
112. 20 years
812
113.
3125
114. Converges; 12
115. When n = 1, the left side of the statement is
statement is 1 -
1
2n
=1-
1
21
=1-
1
1
1
=
= , and the right side of the
n
1
2
2
2
1 1
= , so the statement is true when n = 1.
2 2
Assume the statement is true for some natural number k. Then,
1 1 1
1
1
1
1
1
1
1
1
1.
+ + +
+ ... +
+
= 1+
=1=12 4 8 16
k
k+1
k
k+1
k
2
k+1
2
2
2
2
2
2
So the statement is true for k + 1. Conditions I and II are satisfied; by the Principle of Mathematical Induction, the
statement is true for all natural numbers.
17
Answer Key
Testname: GENERAL PRECAL REVIEW
116. First we show that the statement is true when n = 1.
1(1 + 1)(1 + 2)
For n = 1, we get 2 =
= 2.
3
This is a true statement and Condition I is satisfied.
Next, we assume the statement holds for some k. That is,
k(k + 1)(k + 2)
1 · 2 + 2 · 3 + 3 · 4 + . . . + k(k + 1) =
is true for some positive integer k.
3
We need to show that the statement holds for k + 1. That is, we need to show that
(k + 1)(k + 2)(k + 3)
1 · 2 + 2 · 3 + 3 · 4 + . . . + (k + 1)(k + 2) =
.
3
So we assume that 1 · 2 + 2 · 3 + 3 · 4 + . . . + k(k + 1) =
both sides of the equation.
1 · 2 + 2 · 3 + 3 · 4 + . . . + k(k + 1) + (k + 1)(k + 2) =
k(k + 1)(k + 2)
is true and add the next term, (k + 1)(k + 2), to
3
k(k + 1)(k + 2)
+ (k + 1)(k + 2)
3
=
k(k + 1)(k + 2) 3(k + 1)(k + 2)
+
3
3
=
k(k + 1)(k + 2) + 3(k + 1)(k + 2)
3
=
(k + 1)(k + 2)(k + 3)
3
Condition II is satisfied. As a result, the statement is true for all natural numbers n.
117. 243x5 + 810x4 + 1080x3 + 720x2 + 240x + 32
118. 240
18