Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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35. Vertical asymptotes at x 1 and x 2. The graph moves
upward as x approaches 1 from the left and downward as
x approaches 1 from the right. The graph moves down-
ward as x approaches 2 from the left and upward as x ap-
proaches 2 from the right.
37. The limit does not exist.
39. 0 41. 1/2 43. 10/3 45. y 1/2
Chapter 13 Test, page 925
1. (a) 2 (b) Does not exist (c) 1 (d) 3
2.
1
9
3. 9 4.
3
5
2
5. (a)
(b) 1 (c) 1 (d) 5 (e) 2
6. 2 7. Does not exist 8. 6
9. (a) 0 (b) Not defined
10. f' (x) 2x 5 11. 22
12. Given e 0. Let d e/3. Then x 2 d c
x 2 e/3 c 3x 2 e c 3x 2 e c
3(x 2) e c 3x 6 e c (3x 1) 5 e.
13. k(2)
4
8
(
2
(
)
2
2)
2
7
9
4
2
9
By the properties of limits:
lim
x2
k(x) lim
x2
4x
8
2
x
7
2
9
4
2
9
Therefore, lim
x2
k(x) k(2) and hence, k is continuous at
x 2.
14. (a) 0 (b) 0 (c) (d)
15. Not continuous at x 3 because the graph jumps there
16. (a) x 10 (b) 1 (c) 1
17. Continuous at every real number except 0
18. 2
19. (a) Yes (b) No
20. 0 21. b 2/3 and b 1
22. 4
lim
x2
(8
x
2
)
lim
x2
(4x
2
7)
lim
x2
8 x
2
lim
x2
(4x
2
7)
x
y
2 1323144
2
2
4
6
1056 ANSWERS
23.
24. y 5
Appendix 1
Section 1.A, page 934
1. 36 3. 73 5. 5
7. 125/64 9. 1/3 11. 112
13.
8
1
1
6
15. 211/216 17.
12
8
9
19. x
10
21. .03y
9
23. 24x
7
25. 9x
4
y
2
27. 21a
6
29. 384w
6
31. ab
3
33. 8x
1
y
3
35. a
8
x
3
37. 3xy 39. 2
12
41. 2
12
43. x
7
45. e
9
c 47. b
2
c
2
d
6
49. a
12
b
8
51.
c
10
1
d
6
53.
10
1
8x
55.
a
b
7
6
c
57. c
3
d
6
59. a
1
a
61. negative 63. negative 65. negative
67. 3
s
69. a
6t
b
4t
71.
b
c
rs
2
rt
st
73. Many possible examples, including this one. If a 3, b 4,
and r 2, then a
2
b
2
3
2
4
2
9 16 25,
but (a b)
2
(3 4)
2
7
2
49
75. If a 2, b 3, r 1, s 1, then a
r
b
s
(2
1
)(3
1
) 6 but
(ab)
rs
[(2)(3)]
11
6
2
36.
77. If c 2, r 3, s 1, then
c
c
r
s
2
2
3
1
2
2
4
but c
r/s
2
3/1
2
3
8.
79. False for all nonzero a. For instance, (3)
2
(3)(3) 9,
but 3
2
9.
Section 1.B, page 938
1. 8x 3. 2a
2
b
5. x
3
4x
2
2x 3 7. 5u
3
u 4
9. 6z
3
w
2
zw
3
12z
2
w 4z 8
Interval
Function [0, 1] (0, 1] [0, 1) (0, 1)
f(x)
1
x
No Yes No Yes
g(x)
x
2
x
2
1
No No Yes Yes
H(x) x
Yes Yes Ye s Ye s
K(x)
2x
2
x
2
1
No No No No
11. 3x
3
15x 8 13. 5xy x
15. 15y
3
5y 17. 12a
2
x
2
6a
3
xy 6a
2
xy
19. 12z
4
30z
3
21. 12a
2
b 18ab
2
6a
3
b
2
23. x
2
x 2 25. 2x
2
2x 12
27. y
2
7y 12 29. 6x
2
x 35
31. 3y
3
9y
2
4y 12 33. x
2
16
35. 16a
2
25b
2
37. y
2
22y 121
39. 25x
2
10bx b
2
41. 16x
6
8x
3
y
4
y
8
43. 9x
4
12x
2
y
4
4y
8
45. 2y
3
9y
2
7y 3
47. 15w
3
2w
2
9w 18
49. 24x
3
4x
2
4x
51. x
3
6x
2
11x 6
53. 3x
3
5x
2
y 26xy
2
8y
3
55. 3 57. 6
59. 6 61. 1
63. 5 65. x 25
67. 9 6 y
y 69. 3
x
2
4x 3
71. 3ax
2
(2a 3b)x 2b
73. abx
2
(a
2
b
2
)x ab
75. x
3
(a b c)x
2
(ab ac bc)x abc
77. 16x
nk
79. y
rs
4y
r
y
s
4
81. 4x
2
32x
83. The statement 3(y 2) 3y 2 is false when y 1, since
3(1 2) 9 but 3(1) 2 5. The mistake is the value 2.
The correct statement is 3( y 2) 3y 6.
85. The statement (x y)
2
x y
2
is false when x 1 and
y 2, since (1 2)
2
9 but 1 2
2
5. The mistake is the
first part of the expression x y
2
. The correct statement is
(x y)
2
x
2
2xy y
2
.
87. The statement (7x)(7y) 7xy is false when x 1 and
y 1, since (7
1)(7
1) 7
2
49, but 7
1
1 7. The
mistake is dropping a 7 on the left side. The correct state-
ment is (7x)(7y) 49xy.
89. The statement y y y y
3
is false when y 1, since
1 1 1 3 but 1
3
1. The mistake is the use of 3 as an
exponent. The correct statement is y y y 3y.
91. The statement (x 3)(x 2) x
2
5x 6 is false when
x 3, since (3 3)(3 2) 0 but 3
2
5(3) 6 12.
The mistake is the sign on the 6. The correct statement is
(x 3)(x 2) x
2
5x 6.
93. If x represents the number chosen, the values at each step
are x, x 1, (x 1)
2
x
2
2x 1, x 1, (x 1)
2
x
2
2x 1, (x
2
2x 1) (x
2
2x 1) 4x;
4
x
x
4.
95. Answers will vary.
ANSWERS 1057
Section 1.C, page 944
1. (x 2)(x 2) 3. (3y 5)(3y 5)
5. (9x 2)
2
7. (5
x)(5
x)
9. (7 2z)
2
11. (x
2
y
2
)(x y)(x y)
13. (x 3)(x 2) 15. (z 3)(z 1)
17. ( y 9)( y 4) 19. (x 3)
2
21. (x 5)(x 2) 23. (x 9)(x 2)
25. (3x 1)(x 1) 27. (2z 3)(z 4)
29. 9x(x 8) 31. 2(5x 1)(x 1)
33. (4u 3)(2u 3) 35. (2x 5y)
2
37. (x 5)(x
2
5x 25) 39. (x 2)
3
41. (2 x)(4 2x x
2
) 43. (x 5)
3
45. (x 1)(x
2
x 1)
47. (2x y)(4x
2
2xy y
2
)
49. (x 2)(x
2
2x 4)(x 2)(x
2
2x 4)
51. ( y
2
5)( y
2
2) 53. (9 y
2
)(3 y)(3 y)
55. (z 1)(z
2
z 1)(z 1)(z
2
z 1)
57. (x
2
3y)(x
2
y) 59. (x z)(x y)
61. (a 2b)(a
2
b)
63.
x 8
x 8
(x 4)
65. If x
2
1 (x c)(x d ) x
2
(c d )x cd, then
c d 0 and cd 1. But c d 0 implies that c d
and hence that 1 cd (d )d d
2
, or equivalently,
that d
2
1. Since there is no real number with this prop-
erty, x
2
1 cannot possibly factor in this way.
Section 1.D, page 951
1.
9
7
3.
19
8
5
5.
x
x
2
1
7.
a
2
a
a
b
b
b
2
9.
1
x
11.
2
3
9
5
13.
1
4
2
2
1
15.
ce
de
3cd
17.
b
2
b
c
c
2
19.
x(x
1
1)
21.
(x
x
4
3
)
2
23.
x(
2
3
x
x
4
4)
25.
27.
29.
2 31.
3
2
c
33.
3
x
y
2
35.
x
1
2x
3
37.
3( y
5y
2
5)
39.
u
u
1
41.
(
(
2
u
u
v
v
)
)
(
(
4
2
u
u
3
3
v
v
)
)
43.
3
2
5
4
6x
5
38x
4
84x
3
71x
2
14x 1
4x(x 1)
3
(x 2)
3
x
2
xy y
2
x y
x
3
y
3
45.
v
u
w
2
47.
x
2
x
3
49.
(x y
x
)
2
(
y
x
2
2y)
51.
cd
c
(c
d
d )
53.
y
x
y
x
55.
3( y
y
1)
57.
x(2x
1
2h)
59.
x
x
y
y
61. p/4
63. 1/25
65. The statement
1
a
1
b
a
1
b
is false when a 1 and
b 1, since 1/1 1/1 2 but
1
1
1
1
2
. The correct
statement is
1
a
1
b
a
a
b
b
.
67. The statement
a
1
b
2
a
1
b
is false when a 1
and b 1, since
1
1
1
2
1
4
but
1
1
1
1
2
.
The correct statement is
a
1
b
2
a 2
1
ab b.
69. The statement
u
v
u
v
1 is false when u 1 and v 2,
since
u
v
u
v
1
2
2
1
2
1
2
, not 1. The correct statement
is
u
v
u
v
u
2
v
u
v
2
.
71. The statement x
y
x
1
y
x y is false
when x 1 and y 1, since (1 1)
(1
1
1)
1 but
1 1 2. The correct statement is
x
y
x
1
y
1.
Appendix 2, page 961
1. b 20 25
3. c 63 37
5. b 50 52
, c 5 7. b 75 53
, c 5
9. Angle A 53.13°
Angle R 36.87°
r 3, s 4, t 5
1058 ANSWERS
11. Angle A 65° and Angle R 25°.
Thus, ⬔A ⬔T, ⬔C ⬔R, AC TR.
By ASA, the triangles are congruent.
13. ⬔u ⬔v, that is, ⬔POD ⬔QOD. ⬔PDO ⬔QDO,
since all right angles are congruent. OD OD. By ASA,
PDO and QDO are congruent. Hence, corresponding parts
are congruent and PD QD.
15. ⬔u ⬔v, that is, ⬔BAC ⬔DCA. AB DC and AC
AC. By SAS, the triangles ABC and CDA are congruent.
17. ⬔x ⬔y, that is, ⬔BAC ⬔DAC. ⬔u ⬔v, that is,
⬔BCA ⬔DCA. AC AC. By ASA, the triangles ABC
and ADC are congruent.
19. c 7.5, r 28, t 35 21. r 3, t 3.5
23. Since
A
R
B
S
1
1
5
0
A
R
C
T
and ⬔A ⬔R, the triangles are
similar by S/A/S. a 15, ⬔S ⬔T, and both have measure
60°, since the triangles are equilateral.
25. Since
A
R
C
S
6
4
.8
.5
8
9
5
5
3
.2
.5
5
B
T
C
S
and ⬔C ⬔S, the
triangles are similar by S/A/S. (S/S/S can also be used.)
⬔T ⬔B, and both have measure 62.46°. ⬔A ⬔R, and
both have measure 42.54°.
27. Since
a
r
17
5
.5
b
s
and ⬔C ⬔T, the triangles are
similar by S/A/S. Since the triangles are isosceles right
triangles, ⬔A, ⬔B, ⬔R, ⬔S all have measure 45°.
c 52
and t 17.52
.
29. Since ⬔A ⬔R and ⬔B ⬔T, the triangles are similar by
AA. ⬔C ⬔S and both have measure 41.5°; s 2.4.
31. 55 ft 33. 80 ft
35. Distance from Marietta to Columbus: 92.4 mi., distance
from Marietta to Cincinnati: 171.2 mi.
37. In the figure, AD CR, CD AR, AC CA. Hence, by
SSS, triangle ADC triangle CRA, and so ⬔x ⬔u. Sim-
ilarly, ⬔v ⬔y. Since ⬔x ⬔w ⬔y 180°, it follows
that ⬔u ⬔w ⬔y 180°.
Astronomy
black holes, 927–929
brightness of star, 487
comets, 685, 762
earth’s orbit around sun, 682–683
periodic orbits of sun and moon,
571–572, 685
planetary orbits, 349, 412–413
radio telescope, 709–710, 712
satellite orbit, 604, 605, 686
spacecraft, 686
speed of light, 14
sun and moon cycles, 465
Athletics
baseball diamond geometry, 42, 604
baseball thrown, dropped, or hit, 444–445,
454, 498, 503, 542, 736, 815–816
batting averages, 286–287
car race speed, 72, 114
football field geometry, 50
football thrown, dropped, or kicked, 740
golf ball trajectory, 734–735, 740
gunshot, 740
jogging distances, 175
rowing and running, 110–111,
114–115, 354
running, 113
stadium seating, 841
swimming distances and time, 652, 665
swimming pool depth, 52
swimming water pressure, 34
Biology and Life Sciences
AIDS cases, number of, 30, 50–51, 100,
217, 851
Alzheimer’s disease, 131
blood alcohol content, 149
blood pressure, 215, 465, 482–483,
487, 509
computer-aided tomography (CAT)
scanners, 805
death rates, comparable, 782
disease modeling, 425
drug concentration in bloodstream, 849, 851,
855–856, 875
FDA drug approval, 129
fungus, growth of, 203
healthy weight range, 15, 66–67
heart disease, death from, 74, 126
illness, temperature and, 216
infant mortality rates, 417
infection, spread of, 384, 392, 408
life expectancy, 130–131, 367, 418, 843
medication, concentration of, 91
multiple births, 416
radioactive decay, 400–401
sizes in scientific notation, 14
trees, size of, 133–134
Business and Manufacturing
advertising spending, 128, 194, 214, 215,
287, 384
airline passengers, 131
automobile markets, 784
book sales, 836
break-even point, 783–784
car leases, 316
computer software “learning curve,” 421
cost functions, 188, 315, 316, 317
costs by unit of production, 120, 132,
336, 337
hourly earnings, 134, 216–217
input-output analysis, 823–824
ladder building, 843
minimum wage, 214
mixtures, 111, 132, 783, 804–805
net revenue, 875
output by labor and materials, 350, 784,
806, 821
pensions, 874
prices, 132, 249
production costs, 63–64, 67, 91
production demand, 66
profit margin by unit of production,
157–158, 160
profits, 121–124, 128, 216, 233, 248,
282, 843
radio programming, 161
research and development spending, 130
retail expenditures, 875
revenue growth, 44
revenue maximization, 119
salary and wages, 408, 421
sales averages per call, 249
sales growth, 844
sales totals, 791
subdivisions, 316
supply and demand, 137–139
ticket prices, 778–779, 782, 783, 804
time spent discussing budgetary items, 34–35
unemployment statistics, 489
van loads, 805
wireless telephone service, growth in,
174–175, 207, 408
yield, maximizing, 282
Chemistry and Physics
air quality standards, 820
angle of elevation, sound waves and, 564
balloon filling, 206–207, 213, 221
boiling point of water, 66
cable-hung weight, 652, 653
camera focal length, 303
concentration in solution, 66, 407
daylight hours, 509, 566
density of light, 35–36
exponential decay, 362, 407, 408
flywheels, 512
gas in parts per million, 366, 818
GPS device position, 789, 791
gravitation, 302, 303
gunshot, 566, 569
light rays, 564
mixtures of metals, 780–781
object on a ramp, 310–311, 313, 321–322,
649–650, 651–652, 659, 662, 664
parabolic satellite dish, 712, 765
particles path, 700
pendulum swings, 39, 488, 490
pistons, 488, 512
pressure of an enclosed gas, 38
puddle, shrinking, 200
pulling an object along ground, 660, 662,
664, 665
radioactive waste, 303
resistance and resistors, 31, 36–37, 38, 302
rocket travel, 91, 317, 335–336, 554, 570, 595
salt solution, 111, 317
solution, removing items from, 368
sound wave, speed of, 450
spring-hung weight, 38, 484–485, 566, 569
temperature change, 269
temperature of filled container, 209–210
volume of a container, 215
weight supported by beam, 38
whispering room, 685
Construction
arch, 340, 364–365, 367, 765
banner, 619
box, 109–110, 114, 117, 119, 132, 133, 161,
166–167, 172, 269, 277, 282, 299–300,
302–303
Index of Applications
I-1
bridge, 248, 685, 712, 765
bus shelter, 615
cable angle, 554
circular saw blade, 441
concrete walks, 105–106, 112
crane angle, 616
cut pieces, 132
cylinder or can, 117–118, 119, 172–173,
282, 791
design-making machine, 768–769
drawbridge, 594, 596
fencing dimensions, 115–116, 119–120, 132,
160–161, 245, 249, 335, 336, 686
flagpole, 591, 593, 597
gate angle, 604
gutter dimensions, 249
lamp post height, 592
lights on a pond, 615
lot size, 619
movie screen, 552–553, 554
paint coverage, 619
pole angle, 615
pool depth, 594
posters, 132
ramps, 1, 595
ravine, surveying, 590, 593
road grade, 544, 593, 622
room dimensions, 791
rope angle, 356, 441
sculpture height, 601, 604
sinkhole size, 605
spotlight illumination, 114, 593
spotlight on a sign, 555
stained glass window, 685–686
statue, 615
surveying, 614, 622, 666–667
tower, 815
track size, 250
tunnels, 605
window, 133
wire or cable angle, 590–591, 593, 601–602
wiring, 113, 356–357, 790
Consumer Affairs
air conditioner size, 129
budget, items in, 303
cable television subscribers, 367
car depreciation, 62–63, 68
charitable giving, 134
chicken consumption, 818
clothing sales, 818–819
comparable product prices, 782
consumer confidence levels, 51,
168–169
consumer price index, 128, 174
credit card debt, 802, 851
debt, 128, 804
digital device growth, 368
electric bills, 90
I-2 Index of Applications
electronics sales, 818
federal debt, growth in, 14, 30, 74,
172, 417
fish consumption, 406
fuel economy, 119
health care expenditures, 30, 59
heating system choices, 783
house value, appreciation in, 68, 71
income per item sold, 160
income taxes, 142–143, 152, 160
Internet phone calls, 864
IRA contributions, 49
life insurance policy, 133, 851
median price of home, 233
mixtures, 783, 819
mortgage payments, 836
nursing home room costs, 819
pay increases, 367
pension accrual, 51
pension guarantee resources, 91–92
personal income, growth in, 844
postage, 913
postage rates, 163, 171, 230
poverty level, 68, 125–126
prescription drug spending, 131
price and demand, 778
prices, calculating, 284–285
public debt, 287
radiator mixture, 112
rent increases, 246
salary and wages, 51, 842, 843, 851
sales of product, 49
satellite radio listening, 836
shared construction project, 113
shipping containers, 160
single-parent families, 149
smokers, 175
spending, 836
taxi prices, 913
telephone bills, 317
television spending, 851
ticket prices, 778–779, 782, 783,
804, 821
unemployment, 119, 127
water usage, 66
Education
computer ownership, 851
degrees in population, 67, 836
earnings by educational level, 130
enrollment, 129–130, 214, 286
exam scores, 111
federal student assistance, spending on,
40–41
gender of college students, 418, 851
high school enrollment, 101
home schooling rates, 417
lecture hall seating, 843
private school tuition, 161, 874
public college and university costs, 49,
100, 337
reading, 175, 235
revenues for public schools, 128
SAT scores, 90, 134
scholarships, 149–150
students per computer, 417
Financial
airline passengers, 50
allocating investments, 784, 804
compound interest, 873
consumer price index for urban consumers
(CPI-U), increase in, 13–14
continuously compounded interest, 236,
360, 370–375, 407
cost-benefit equation, 31
credit card interest, 368
department expenses, variance in, 15
doubling investment, 221–222
effective annual interest rate, 112
equilibrium between national output and
national expenditures, 30
exponential growth, 408
GDP, 128–129
inflation, 368–369
interest rates, 174
mortgage principal and interest
payments, 45–46
present value of future payments, 374, 407
prime rate, 149
stock price growth and decay, 216
stock purchases, 110–111
stock volume, 214, 235
total return, 104
Miscellaneous
adults on probation, 418
alcohol-related car crashes, 51, 408
alternating current circuit, 554
antifreeze combination, 104–105
camera angle, 554–555
cement walk construction, 105–106
cigarette smokers, 287, 423
death rate in motor vehicle
accidents, 418
driver fatality rates, 31, 71–72
earthquakes, 390, 391, 423
emptying tank, 68
forest fire, 614
games of chance, 877–878
identity theft complaints, 25
jogging board, 249
memory skills, 421
merry-go-round revolutions, 438–439, 488
name popularity, 214–215
paper folding, 368
radio transmission, 487
rings on a peg game, 873
rumor, spreading, 391, 422–423
Sumerian place-value system, 385
sun rays and earth’s circumference, 441
theft rate, 89
touch-tone telephone, 543
typing speed, 408
wheel rotations, 441, 483–484
Population
ancestors, calculating, 851
equivalency, estimating, 95
exponential decay, 423
exponential growth, 363–364, 384, 402, 408,
423, 837
growth, 31, 90, 203, 233, 238, 250, 269,
341, 361–362, 366, 367, 368, 410–411,
414, 416
growth and decline, 91, 782
inhibited growth, 216, 278, 364, 402–403
linear growth, 73
marriage rate in U.S., 67
at past date, calculating from present, 30,
283–284, 349, 820, 851
property crime per 100,000, 286
seedling survival rate, 338
television watching, proportion of, 367
of world, 14
Index of Applications I-3
Time and Distance
airplane travel, 33, 107–108, 112, 173, 554,
588–589, 593, 595, 596, 604, 605,
612–614, 615–616, 650, 652, 664,
779–780, 783
balloon travel, 72
bicycle travel, 112, 439
boat or ship travel, 594, 595, 600, 621, 783
buoys, 594
cannon fire, 248
car travel, 66, 146–147, 160, 175, 201, 206,
214, 215–216, 231–232, 248–249
dropped objects, 30, 38, 106–107, 143–144,
149, 152–153, 204, 205, 210–211,
212–213, 233, 249, 317, 848, 851
Ferris wheel, 52, 487–488, 742
focal length of a digital camera
lens, 28–29
general travel, 149
height of an object, 26, 356
height of a person, 15
hiking, 913
horizon, objects beyond, 624
illumination, 357
kite height, 594
ladder angle, 112, 589–590
lamp post height, 595
launched objects, 30
listening stations, 699
pedestrian overpass, 595
pendulum, period of, 14–15
pole height, 622
ravine depth, 590
river, canal or pond width, 588, 594, 616
Seattle Space Needle height, 594
shadow, 203–204
ship travel, 605, 695–697, 699
speed limits, 338
stretched rope, 113, 357
taxicab distances, 75–76
thrown objects, 249, 317
traffic flow, 806
train travel, 112, 599–600, 621
two travelers at different speeds, 113
view from given height, 101
well depth, 112
wheelchair access ramp, 64
Weather
air temperature, 143, 174, 208, 286, 384,
450, 465, 485, 489, 490, 509, 913
atmospheric pressure, 30, 74, 366–367, 408
Fahrenheit and Celsius scales, 67–68, 227
lightning and thunder, 38
precipitation, 489
rainfall, 90
temperature and cricket chirps, 130
volume of weather balloon, 203
wind-chill factor, 15, 423
A
Absolute value, 9–11
of complex number, 626–627
equations, 32
function, 165, 170
inequalities, 317–320
properties of, 10
Acute angles, 584–585
Addition and subtraction identities, 524, 526
Algebra
identities, 457–465
nonlinear equations, 784–785
with trigonometric functions, 458
Algebraic equations
absolute value equations, 32–33
basic principles for solving, 19
fractional equations, 27–29
higher-degree equations, 26–27
quadratic equations, 20–26
variation, 33–37
Algebraic methods, 318–320
Algebraic notation, functional notation
versus, 152
Algebraic solutions
advantages and disadvantages, 99
exponential and logarithmic functions,
399–408
of trigonometric equations, 560–562
Ambiguous case (SA), 608–611
Amplitude, 480
Analytic geometry
circles, 671–672
parametric equations, 713–714
vertical and horizontal shifts, 676–678
conic sections, 753–762
design, parametric equations to make,
768–769
ellipses, 672–676
applications, 682–686
graphing, 679–682
parametric equations, 714–715
standard equations, 679
hyperbolas, 686–690
applications, 695–700
graphing techniques, 693–694
parametric equations, 715–716
vertical and horizontal shifts, 690–693
parabolas
applications, 709–712
definition, 700
equations, 700–704
graphing techniques, 707–708
latus rectum, 703
parametric equations, 716–717
vertical and horizontal shifts, 705–707
plane curves, 727–742
polar coordinates, 743–753
rotation of axes, 722–727
Analytic geometry (continued)
rotations and second-degree equations,
718–720
second-degree equations, 720–721
Angles, 428, 574
acute, 584–585
angular speed, 438
area of a circular sector, 437–438
coterminal, 429, 432
defined, 428
degree, 429
degrees and radians, 575
depression, 590
elevation, 590
of inclination, 532
initial side, 586
linear speed, 438
negative angle identities, 576
periodicity identities, 576
point-in-the-plane decription, 586
point-in-the-plane description, 575
Pythagorean identities, 576
radian measure, 430–433
right angle, 584
right triangle, 584
description of trigonometric functions,
576–579
solving, 579–582
standard position, 429, 586
terminal side, 586
trigonometric functions, 574
between two lines, 533
vectors in the plane, 654–656
Angle Theorem, 655
Angular speed, 438
Applied problems
annual simple interest, simple rule of,
104–106
distance and rate of speed, 106–111
English language, 103
mathematical language, 103
optimization applications, 114–118
setting up, 101–103
Archimedean spiral, 750
Architectural arches, 340
Arc length and angular speed, 435–441
Arcsine function, 545–546
Area formulas for triangles, 617–620
Area of a circular sector, 437–438
Argument, 627
Arithmetic, 3–4
Arithmetic progression. See arithmetic
sequence
Arithmetic sequences, 837–840
Arrcosine function. See inverse cosine
function
Asymptotes, 687
nonvertical, 304–307
Augmented matrix, 795
Average rate of change, 206, 228
secant lines and, 208–209
Axis
major and minor, 673
of parabola, 241
rotation, 722–727
B
Back substitution, 794
Base b, definition of logarithms to, 393
Behavior when x is large, 272, 295–297
Bernett, Laura, 595
The Big-Little Principle, 288
Binomial coefficient, 858
Binomial Theorem, 857–862
games of chance, 877–878
Black holes, 927–929
Bounds Test, 266
C
Calculator use. See also Graphing
Explorations
computations, 81
absolute value of a complex number
(ABS key), 627
angle, converting from degrees to
radians, 434
argument u, 627
binomial expansions, 861
complex arithmetic, 323, 626, 629
complex format, 344
composite functions, evaluating, 198
dot products, 654
identity, solving equation that
might be, 521
implied multiplication, 81
inverse A
1
of matrix A, 812
least squares regression line, 127
limits, 883
linear equations, 795
magnitude of a vector (NORM), 642
polynomial division, 253
polynomial equation solvers, 26
polynomial solvers, 267, 326, 635
principal square root, 8
quadratic, cubic, and quartic regression
commands, 283
rectangular to polar form, 628
repeating decimals, converting to
fractions, 17
roots of polynomial, 329
square root of negative number, 325
synthetic division, 261
unit vector in same director as v, 646
vector arithmetic, 643
Subject Index
I-4
Calculator use (continued)
entries, 182
Abs key, 10
correlation coefficient, 125
COS
1
, 497
COS key, 458
DEGREE mode, triangle trigonometry
and, 578
equation solver, 96
evaluating functions, 154
functional notation, 153
GRAPH error message (TI-83/86), 80
graphical intersection finder, 94
graphical maximum finder, 85
graphical root finder, 93
graphing equations (“variable” key), 79
graphing sum of function in
memory, 195
horizontal line test, 218
inequality symbols, 164
INT or FLOOR, 145
keystrokes, saving, 681
long decimals, reducing in
matrices, 798
matrix, 795
matrix in reduced row echelon
form, 799
matrix in row echelon form, 797
natural exponential function, 363
natural logarithms of numbers in list
L
2
, 413
negative numbers, 6
“No Solutions Found” message, 800
“on” equations, 81
quadratic, cubic, and quantic regression
commands, 283
r(cos u), 628
scatter plot or line graph, 88
scientific notation, 7
SIN
1
, 497
sin
3
4, 458
square window, 83
statistics editor, 88
TAN
1
, 497
TRACE, 87
TRACE key, 81
two functions, graphing, 182
vectors in component form, 641
ZOOM, 82
graphings
circles, 675
inverse function, 224
points of inflection, 275
rational function, 298
views
auto scaling feature, 86
conic sections, 681
decimal window, 83
hidden behavior, 167
n n identity matrix, 810
parametric graphing mode, 177, 729
polynomial graph, 271
RADIAN mode for trigonometric
functions, 444, 454
rectangular to polar coordinates,
converting, 745
table of t-x-y values, 178
Subject Index I-5
Calculator use (continued)
table setup screen, 147
trigonometric functions, 467
vertical asymptote in center, 293
Cardioid, 750
Catalog of functions, 162–163
Center (c)
of circle, 46, 673
of hyperbola, 687
Central angle, 435
Change in x, change in y, 54
Change of base formula, 397
Circle, 670, 671–672, 751
center (c), 46
equation, 46, 69
at the origin, 48
parametric equations, 713–714
vertical and horizontal shifts, 676–678
Closed interval, 5, 908
Coefficient, 251
Cofunction identities, 527–528
Columns, 807
augmented matrix, 795
Common difference of arithmetic
sequence, 838–839
Common logarithmic function, 376
Common logarithms, 393–394
properties, 380
Common ratio of geometric
sequence, 844
Complete graph, 85
Completing the square, 22
Complex numbers, 321–324
conjugate solutions, 326
nth root, 633–635
nth roots of unity, 636–638
powers, computing with DeMoivre’s
Theorem, 632–633
square roots of negative numbers,
324–326
Complex plane, 626
Component of projection of u along v,
658–659
Components, 641
Composite functions, properties of, 909
Composition of functions, 196–197
Compound interest formula, 369–372
Computer-aided tomography (CAT)
scanners, 805
Conic section, 670
polar equations, 755
Conjugate of complex number, 323, 330
Conjugate Roots Theorem, 331
Conjugate solutions of complex
numbers, 326
Constant
on an interval, function, 167
limit of, 918–919
of proportionality, 34
of variation, 33, 34
Constant limits, 891–892
Constant polynomial, 251
Constant sequence, 828–829
Constant term, 251
Continuity, 271, 903–907
on an interval, 907–908
continuous functions, 907
Continuity (continued)
Intermediate Value Theorem, 910–911
properties of continuous functions,
908–910
rational graph, 293
Continuous compounding, number e and,
372–373
Contractions, 182
Converges, 853
Coordinate plane
circles, 46–48
coordinates, 39–40
distance formula, 41–44
graphs, 44–46
line graph, 41
origin, 39
point, coordinates of a, 40
quadrants, 39
rectangular (Cartesian) coordinate
system, 40
scatter plot, 41
x-axis, 39
x-coordinate, 40
y-axis, 39
y-coordinate, 40
Correlation coefficient, 124
Cosecant function, 497
graphs of, 501–502
period of, 500
Cosine functions, 442–457
graphs, 466–472
Cotangent function, 497
graph of, 501–502
period of, 500
reciprocal identities of, 499
Coterrminal angles, 429
Cube functions, 162
catalog, 170
Cube root
of 1, 326
of unity, 326
Cubic regression, 283–285
Cycloid, 737
D
Damped and compressed trigonometric
graphs, 493–495
Decimal representation, 17
Decimals, repeating and nonrepeating, 17
Decimal window, calculator, 83
Decreasing functions, 167
Decreasing on an interval, function, 167
Degree, 575
of angle, 429
of polynomial, 251
Degree measure, 429
Demand curve, 137
DeMoivre’s Theorem, 632–633
Dependent system, 777, 800
Depression, angle of, 590
Derivative of function, 896
Design, parametric equations to make,
768–769
Difference quotient, 154–155, 210, 228
limits of, 895
Differences, 195–196
Direction angles, vectors in the plane,
647–649
Directrix, 700, 757
Direct variation, 34
Discovery projects
architectural arches, 340
black holes, 927–929
designing machines to make designs,
768–769
exponential and logistic modeling of
diseases, 425–426
feedback, 236–238
games of chance, 877–878
input-output analysis, 823–824
pistons and flywheels, 512
sphere, life on a, 624
sun and the moon, 571–572
supply and demand, 137–139
surveying, 666–667
taxicab geometry, 75–76
Discrete algebra, 825–878
arithmetic sequences, 837–840
binomial theorem, 857–862
geometric sequences, 844–850
infinite series, 852–856
mathematical induction, 864–873
partial sums, 834–835, 840–842
sequences, 826–831
summation notation, 831–834
Discriminant, 24, 720
Distance Formula, 41, 69, 753
Distance on the number line, 11
Distance to zero, 12
Distance traveled, 205
Distributive law, 4
Division
in polar form, 629
of polynomials, 334
Division Algorithm, 253
Domain, 144, 288
Domain convention, 156
Dot product, vectors in the plane, 653–654
Double-angle identities, 535–537
E
Eccentricity, 686, 753
Elementary operations, 793
Elevation, angle of, 590
Elimination method
linear equations in two variables, 775–777
xy term, 725
Ellipse, 670, 672–676
applications, 682–686
graphing, 679–682
parametric equations, 714–715
standard equations, 674, 679
Endpoints, 5
Equal matrices, 807
Equation memory, 79
Equations
absolute value, 32
of circle, 46, 69, 671
of ellipses, 674, 679
fractional, 27, 99
higher-degree, 69
of parabolas, 701, 707
parametric graphing, 175–178
I-6 Subject Index
Equations (continued)
solving, 96
applied problems, 104–111
basic principles, 19
graphical, 92–95
indirect equation solving, 97–99
method, choosing, 99
numerical equation solving, 96–97
Equation Theorem, 911
Equilibrium point, 138, 778
Equilibrium price, 138, 778
Equilibrium quantity, 778
Equivalent equations, 19, 792
Equivalent systems, 792
Equivalent vectors in the plane, 640–641
Escape velocity, 927
Estimating instantaneous rates of
change, 228
Even and odd functions, 192–193
Event horizon, 928
Expansions, 182
Exponential and logistic modeling of
diseases, 425–426
Exponential decay, 362–363
Exponential equations, 399
Exponential functions, 357–369,
358–359, 381
algebraic solutions, 399–408
with base a, 357–360
compound interest and the number e,
369–375
models, 409–418
radical equations, 350–357
radicals and rational exponents, 342–350
Exponential growth, 360–362
Exponential models, 409
Exponent Laws, 345
Extraneous solution, 27
F
Factoring method, 20
need for one side to equal zero, 21
reducing equations, 560–562
Factoring polynomials, 265
Factorization
over the complex numbers, 329
over the real numbers, 331
Factors, polynomial division, 254
Factor Theorem, 256
Fibonacci numbers, 830
Fibonacci sequence, 830, 837
First-degree equations, 19–20
Fixed costs, 69
Foci, 672, 687, 700, 757
Fractional equations, 27, 99
Fraction as zero, 98
Functional notation, 152–153
applications, 157–158
domains, 156–157
Functions, 142–144. See also specific
functions listed by name
defined by equations and graphs, 145
equations used to define, 145–147
even and odd, 192–193
graphing, 147–148
graphs and transformations, 179–180
combining transformations, 185–186
Functions (continued)
expansions and contractions, 182–183
horizontal shifts, 181–182
reflections, 184–185
vertical shifts, 180–181
graphs of, 161–163
catalog of basic functions, 170
increasing and decreasing functions, 167
local maxima and minima, 165–167
piecewise-defined functions, 164–165
reading, 168–169
step functions, 163
vertical line test, 169
greatest integer function, 145
inverse, 217, 219–222
decreasing function, 219
graphs of, 224–225
horizontal line test, 218
increasing function, 219
round-trip properties, 222–223
limits of, 880–883
one-to-one, 217
operations on, 195–196
applications, 200–201
composition, 196–200
rates of change, 204–206
average, 206–208
average, geometric interpretation of,
208–210
difference quotient, 210–212
instantaneous, 212–213
Fundamental Theorem of Algebra, 328
G
Games of chance, 877–878
Gaussian elimination, 792, 794
Gauss-Jordan method, 798–800
Geometric interpretation of scalar
multiplication, 643
Geometric interpretation of vector
addition, 644
Geometric sequences, 844–850
common ratio of, 844
nth term of, 845
partial sums of, 847–850
Geometry. See analytic geometry
Graph. See also graphing tools
of an equation, 44
complete graphs, 85–87
of equation “by hand,” 78–79
of equations with technology, 79–81
of functions, 161
and identities, 473
of inverse functions, 224–225
multiplicity and, 273
parabolas, 707–708
of piecewise-defined functions, 164
polar, 748–751
quadratic function (parabola), 240
rational functions, 293–296, 296–299
reading, 168
of rotated conics, 718–720
scatter plots and line graphs, 87–88
second-degree equations, 720–721
of sine, cosine, and tangent functions,
466–472
symmetries, 189–193
Graphical equation solving, 92–93
intercept method, 93–94
intersection method, 94–95
systems of nonlinear equations, 786–788
of trigonometric equations, 562–563
Graphical intersection finder, 94
Graphical root finder, 93, 99
Graphing Explorations
absolute value inequalities, 318, 320
basic graphs, 467
circles and ellipses, 681
complete graphs, 87
compound interest, 371, 373
conic sections, 713
coordinates, approximating, 85
damped and compressed trigonometric
graphs, 493–495
discriminant, 721
eliminating the parameter, 731, 734
expansions and contractions, 182, 183
exponential decay, 362
exponential functions, 357–359, 365
functions, limits of, 883, 884
graphical root finder, 94
horizontal line test, 218
horizontal shift, 181
hyperbolas, 692, 716, 756
identities, 473, 515, 518, 523
infinite series, 856
intersection method, 95
inverse functions, 225
limits involving infinity, 915, 918, 921
limits of functions, 883
local extrema, 274
local maxima and minima, 166
logarithmic functions, 382, 387, 388, 414
minimum finder, 115
natural exponential function, 363
negative angle identities, 463
nonexistence of limits, 884
one-sided limits, 886
parametric equation, 178
partial sum, 854
periodic graphs, 477
perpendicular lines, 84
phase shift, 482, 483
piecewise-defined functions, 164
plane curves, 730
polynomial functions, 270, 277, 278, 284
polynomial or one-step solver, 98
projectile motion, 736
Pythagorean Identity, 460
quadratic functions, 241
radical equations, 353, 355
rational functions, 293, 295, 296, 298,
299, 300, 304, 305, 306, 311
reflections, 184, 185
root finder or polynomial solver, 110, 111
roots of unity, 638
sine and cosine functions, 443, 455
sinusoidal functions, 490–492
statistics editor, 127
step functions, 163
symmetry, 190
systems of equations, 776, 786
TRACE key, 83
transformations, 179
vertical shift, 181
Subject Index I-7
Graphing Explorations (continued)
when x is large, 271
x-axis symmetry, 191
zooming, 82
Graphing tools
maximum/minimum findeer, 84–85
special viewing windows, 82–84
trace feature, 81–82
zoom in/out, 82
Graph of y ax
n
, 270
Greatest integer function, 145, 163
catalog, 170
H
Half-angle identities, 538–539
Harmonic sum, 385
Harper, 588
Heron’s formula, area of a triangle,
617–618
Hidden behavior, functions, 167
Higher-degree equations, 26
Holes, rational graph, 294–295
Homogeneous system, 800
Horizon, measurements on, 624
Horizontal asymptotes, 289, 296, 297, 917
Horizontal lines, 58
Horizontal Line Test, 218
Horizontal shifts, 181, 677, 705
Hyperbola, 670, 686–687
applications, 695–700
graphing techniques, 693–694
parametric equations, 715–716
vertical and horizontal shifts, 690–693
I
Identities, 499–500
addition and subtraction, 524, 526
graphs and, 473
matrix and inverses, 810–812
reducing equations, 560–562
strategies and proof techniques, 516–521
Identity function, 891–892
Identity function limit, 891
Imaginary axis, 626
Imaginary numbers, 322
Inconsistent system, 776
Increasing functions, 167
Increasing on an interval, function, 167
Independent system, 800
Indirect equation solving, 97–99
Induction hypothesis, 867
Inductive step, 867
Infinite limits, 914–915
Infinite Limit Theorem, 919–921
Infinite series, 852–856
sum of, 853
Infinity limits, 918–922
Infinity symbol (insert symbol), 6
Initial point, vectors in the plane, 640
Initial side, 428
Initial side angle, 586
Input-output analysis, 823–824
Instantaneous rates of change,
estimating, 228
Integer part function, 151
Integers, 2
Intercepts, 297
arc, 435
rational graph, 294
Intermediate Value Theorem, 910–911
Intersecting lines, 670
Intersection finder, 99
Interval notation, 5–6
Inverse cosine function, 548–549
Inverse functions, 228
Inverse sine function, 545–546
Inverse tangent function, 551
Inverse variation, 34
Irrational exponents, 348
Irrational numbers, 3, 17
K
Kinetic energy, 927
Kth partial sum, 834
L
Lahr, Bob, 734
Latus rectum, 703
Laurent, Theresa, 849
Law of Cosines, 597–598
Law of Sines, 606–616
Leading coefficient, 251
Least squares regression line, 123
Left, continuous from, 907
Lemniscate, 751
Length, vectors in the plane, 640
Limaçon, 751
Limits
and continuity, 879–929
of difference quotients, 895–896
formal definition of, 899
of functions, 880–883
from the left, 886
nonexistence, 883–885
from the right, 886
infinity, 914–918
at infinity, 917
infinity properties, 918–922
one-sided, 885–887
of polynomial functions
constant, 891–892
identity function, 891–892
properties, 890–891
of rational functions, 892–894
Limit Theorem, 894
Line, 670
general form, 60
horizontal distance (change in x), 54
parallel lines, 60–61
perpendicular lines, 61–62
point-slope form of the equation of, 58
rates of change, slope as, 62–64
slope-intercept form, 56–59
slope of a line, 54–56
vertical distance (change in y), 54
Linear combination of i and j, 646
Linear equation, 19
large systems of, 792–793
applications, 801–806
elementary operations, 793–795
Gauss-Jordan method, 798–800
matrix methods, 795–798