Blank B.E., Krantz S.G. Calculus: Single Variable
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29. 3/4 31. 3/5 33. 10
35. 625/4 37. 1 1
ffiffiffi
2
p
=2 39. 25/4
41. 11/30 43. 20/3 45. 8/9
47. 17/999 49. 403/3300 51. 2
53. 3 55. 3/2 63. 1
65. 0 67. 1 69. 3/4
71. 1/6 75. 1/(r 2 1)
2
77. π
2
/24 and π
2
/8
79. The line y 52x/15
83. 1m 85. $1.9 3 10
12
87. Q(1 2 e
2kT
) 93. 0.4587
95. 1.4581 97. 1.1876
Section 8.2
1. No conclusion 3. Divergence
5. Divergence 7. No conclusion
9. No conclusion 11. Divergence
13. No conclusion 15. Divergence
17. No conclusion 19. No conclusion
21. may 23. may
25. may 27. Converges
29. Converges 31. Diverges
33. Diverges 35. Converges
37. Converges 39. Converges
41. Diverges 43. Converges
45. Converges 47. Diverges
49. Converges 51. Converges
53. Converges 55. Divergence
57. Divergence 59. No conclusion
61. Divergence 63. Divergence
65. Divergence 67. No conclusion
69. Converges 71. Diverges
73. Converges 75. (1/24, 7/96)
77.
1
2ð1 1 eÞ
2
;
1 1 3e
2ð1 1 eÞ
3
83. Converges
95. M 5 13, S 5 1.202 97. M 5 7, S 5 0.058
Section 8.3
1. a
n
5 n=ðn
3
1 1Þ# b
n
5 1=n
2
and
P
N
n51
b
n
is a convergent p-series ðp 5 2 . 1Þ.
3. a
n
5 1=ðn 1
ffiffiffi
2
p
Þ
2
# b
n
5 1=n
2
and
P
N
n51
b
n
is a convergent p-series ðp 5 2 . 1Þ.
5. a
n
5
2 1 sinðnÞ
=n
4
# b
n
5 3=n
4
and
P
N
n51
b
n
is a constant multiple
of a convergent p-series ðp 5 4 . 1Þ.
7. a
n
5 ðn 1 2Þ=ð2n
5=2
1 3Þ# b
n
5 1=n
3=2
and
P
N
n51
b
n
is a convergent p-series ðp 5 3=2 . 1Þ.
9. a
n
5 2
n
=ðn3
n
Þ# b
n
5 ð2=3Þ
n
and
P
N
n51
b
n
is a convergent geometric
series with ðpositiveÞ ratio r less than 1.
11. a
n
5 1=ð2n
ffiffiffi
n
p
2 1Þ# b
n
5 1=n
3=2
and
P
N
n51
b
n
is a convergent p-series ðp 5 3=2 . 1Þ.
13. a
n
5 n
3
=ðn
4:01
1 1Þ# b
n
5 1=n
1:01
and
P
N
n51
b
n
is a convergent p-series ðp 5 1:01 . 1Þ.
15. a
n
5 1=n! # b
n
5 2=n
2
and
P
N
n51
b
n
is a constant multiple of a
convergent p-series ðp 5 2 . 1Þ.
17. b
n
5 1=ð2nÞ# a
n
5 1=ð2n 2 1Þ
and
P
N
n51
b
n
is a constant multiple of
the harmonic series.
19. b
n
5 1=n # a
n
5 1=lnð nÞ
and
P
N
n51
b
n
is the harmonic series.
21. b
n
5 1=n # a
n
5 ð2n 1 5Þ=ðn
2
1 1Þ
and
P
N
n51
b
n
is the harmonic series.
23. b
n
5 1=ð4nÞ# a
n
5 1=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
10 1 n
2
p
and
P
N
n51
b
n
is a
constant multiple of the harmonic series.
25. b
n
5 1=2 # a
n
5 ð3
n
1 1Þ=ð3
n
1 2
n
Þ
and
P
N
n51
b
n
diverges by the Divergence Test.
27. b
n
5 1=
ffiffiffi
n
p
# a
n
5 ð3
n
1 nÞ=ð
ffiffiffi
n
p
3
n
1 1Þ
and
P
N
n51
b
n
is a divergent p-series ðp 5 1=2 # 1Þ.
29. Diverges 31. Diverges
33. Converges 35. Converges
37. Converges 39. Diverges
41. Converges 43. Converges
45. Diverges 47. Converges
49. Diverges 51. Diverges
53. Converges 55. Diverges
57. Diverges 59. Converges
61. Converges 63. Converges
65. Converges 67. Converges
69. Converges 71. Converges
Section 8.4
13. Converges absolutely
15. Converges conditionally
17. Converges conditionally
19. Converges absolutely
21. Converges conditionally
23. Converges absolutely
Section 8.4 A-51
25. Converges absolutely
27. Converges absolutely
29. Diverges
31. Diverges
33. Converges absolutely
35. Converges conditionally
37. 9999 39. 200 41. 4
43. 10 45. 8 47. 20.632
49. 20.424 57. 9 59. 18
Section 8.5
1. Convergent 3. Divergent
5. Convergent 7. Divergent
9. Divergent 11. Convergent
13. Converges conditionally
15. Converges absolutely
17. Diverges
19. Converges absolutely
21. Diverges 23. Diverges
25. Convergent 27. Convergent
29. Convergent 31. Convergent
33. Convergent 35. Converges
37. Diverges 39. Diverges
41. Diverges 43. Converges
45. Converges conditionally
47. Diverges
49. Converges conditionally
51. Converges conditionally
53. Diverges
55. Converges absolutely
57. Converges conditionally
59. Diverges
Section 8.6
1. R 5 1/2; I 5 (21/2, 1/2)
3. R 5 1; I 5 (21, 1)
5. R 5 1; I 5 (21, 1)
7. R 5 5; I 5 (25, 5)
9. R 5 0; I 5 {0}
11. R 5 1; I 5 (21, 1]
13. R 5 1; I 5 [21, 1)
15. R 5 1/3; I 5 [21/3, 1/3]
17. R 5 1; I 5 (21, 1)
19. R 5 1/4; I 5 [21/4, 1/4)
21. R 5 1; I 5 [27, 25)
23. R 5
1; I 5 [25, 23]
25. R 5 1/2; I 5 (21,
0)
27. R 5 1; I 5 (26, 24)
29. R 5N; I 5 (2N,N)
31. (1/2, 3/2)
33. (27/3, 25/3)
35. 1 1 4x 1 9x
2
1 16x
3
1 25x
4
; I 5 (21, 1)
37. 1 1
1
ffiffiffi
2
p
ðx 2 1Þ1
1
ffiffiffi
3
p
ðx 2 1Þ
2
1
1
2
ðx 2 1Þ
3
1
1
ffiffiffi
5
p
ðx 2 1Þ
4
; I 5 ½0; 2Þ
39. x 1
1
ffiffiffi
2
p
x
2
1
1
ffiffiffi
3
p
x
3
1
1
2
x
4
; I 5 ½21; 1Þ
41. ðx 2 2Þ1
1
6
ðx 2 2Þ
2
1
1
27
ðx 2 2Þ
3
1
1
108
ðx 2 2Þ
4
;
I 5 ½21; 5Þ
43. 1/2 45. 4 47. N
49. (24, 2) 51. (1/4, 3/4) 53. (21/2, 1/2)
55. ð2
ffiffiffi
3
p
2 e;
ffiffiffi
3
p
2 eÞ
57.
P
N
n51
x 2 c
r
n
where c 5 ð α 1 βÞ=2 and r 5 ðβ 2 αÞ=2
59.
P
N
n51
1
n
x 2 c
r
n
where c 5 ðα 1 βÞ=2 and r 5 ðβ 2 αÞ=2
63. 1 67. 1=
ffiffiffi
2
p
69.
0.5
1
1.5
1
2.5
0 0.5
x
y
y
y
y
y
1
1 x
S
6
(x)
S
2
(x)
S
4
(x)
y
y S
50
(x)
y S
4
(x)
y S
2
(x)
1
1 x
0.9
0.6
0.8
0.8 0.7 0.6
x
A-52 Answers to Selected Exercises
y
y S
20
(x)
y S
4
(x)
y S
2
(x)
1
1 x
20
0.6 0.7 0.8 0.9
x
40
71.
y In(1 x)
y S
6
(x)
y S
4
(x)
y S
2
(x)
0.5
0.5
0.5 0.5 1
x
y
4
2
0.8 0.6
x
y In(1 x)
y S
50
(x)
y S
4
(x)
y S
2
(x)
Section 8.7
1.
P
N
n50
2
n
x
n
; R 5 1=2
3.
P
N
n50
2
2ðn11Þ
x
n
; R 5 2
5.
P
N
n50
ð21Þ
n
4
2ðn11Þ
x
n
; R 5 4
7. 1 1
P
N
n51
2x
2n
; R 5 1
9.
P
N
n50
ð21Þ
n
x
4n13
; R 5 1
11.
P
N
m51
ð21Þ
m21
4
m
m
x
m
; R 5 1=4
13.
P
N
m51
ð21Þ
m21
m
x
2m
; R 5 1
15.
P
N
m51
ð21Þ
m21
mð2m 1 1Þ
x
2m 1 1
; R 5 1
17.
P
N
n50
ð21Þ
n
x
4n 1 2
2n 1 1
; R 5 1
19.
P
N
n50
ð21Þ
n
ð2n 1 1Þð4n 1 3Þ
x
4n13
; R 5 1
21.
P
N
n50
ðx 2 5Þ
n
; R 5 1
23.
P
N
n50
ð21Þ
2
n 1 1
ðx 1 3Þ
n
; R 5 2
25.
P
N
n50
ð21Þ
n
2
n
3
n 1 1
ðx 1 1Þ
n
; R 5 3=2
27.
1
2
2
ffiffiffi
3
p
2
ðx 2 π=3 Þ2
1
4
ðx 2 π=3 Þ
2
1
ffiffiffi
3
p
12
ðx 2 π=3 Þ
3
1
1
48
ðx 2 π=3 Þ
4
29. 2 1
1
4
ðx 2 4Þ2
1
64
ðx 2 4Þ
2
1
1
512
ðx 2 4Þ
3
31. 6 1 9ðx 2 1Þ1 6ðx 2 1Þ
2
2 ðx 2 1Þ
3
1 ðx 2 1Þ
4
33.
1
8
2
3
16
ðx 2 2Þ1
3
16
ðx 2 2Þ
2
2
5
32
ðx 2 2Þ
3
35. 3ðx 1 2Þ2
9
2
ðx 1 2Þ
2
1 9ðx 1 2Þ
3
2
81
4
ðx 1 2Þ
4
37. x 1
1
3
x
3
39.
1
2
2
1
4
ðx 2 1Þ
2
1
1
4
ðx 2 1Þ
3
41. 2 1
1
12
ðx 2 3Þ2
1
288
ðx 2 3Þ
2
1
5
20736
ðx 2 3Þ
3
43.
P
N
n50
ðn 1 1Þx
n
45.
P
N
n50
ð3n 1 2Þx
n
47.
x
2
1 2 x
49.
2x
ð1 2 xÞ
3
51.
x
3
ð1 2 x
2
Þ
2
53. a
n
5 1/n!
55. a
0
5 a
1
5 1, a
n
5 2/n! for n $ 2
57. a
0
5 0, a
1
5 1, a
n
5 2/n! for n $ 2
65. a. 1 1 x 1
1
2
x
2
1
1
2
x
3
1
1
8
x
4
1
1
40
x
5
1
1
240
x
6
1
1
1680
x
7
40
30
20
10
10
y
x
11323
0
y 3 x 2e
x
y S
3
(x)
Section 8.7 A-53
11323
40
30
20
10
0
10
y
x
y 3 x 2e
x
y S
7
(x)
Section 8.8
1. 1:1051708 ::: ; error :
ð0:1Þ
5
120
e
s
3. 0:29552025 ; error : 2
sinðsÞ
720
ð0:3Þ
6
5. 0:877583 ::: ; error :
sinðsÞ
6
ð0:5 2 π=6Þ
3
7. 1:09863 :::; error : 2
1
4s
4
ð3 2 eÞ
4
9. 0:983 :::; error :
3s
2
2 1
3ð1 1 s
2
Þ
3
ð1:5 2
ffiffiffi
3
p
Þ
3
11. 2:049375; error :
1
16ð1 1 sÞ
5=2
ð0:2Þ
3
13. a. 1:184 b.
15
16 4!
ð0:4Þ
4
; or 0 :001
15. a. 2 0:19 b.
8
3!
ð0:1Þ
3
; or 0:0013 :::
17. a. 2:26 b.
1
3!
ð0:2Þ
3
; or 0:0013 :::
19. a. 0:9164 b.
1
3e
3
ðe 2 2:5Þ
3
; or 0:00017 :::
21. a. 0:4 b.
2
3!
ð0:4Þ
3
; or 0:021 :::
23.
P
N
n50
ð21Þ
n11
2
2n21
ð2n 2 1Þ!
x
2n21
25.
P
N
n50
ð21Þ
n11
2
2n21
ð2n 2 1Þ!
x
2n11
27.
P
N
n50
ð21Þ
n
n!
c
n
x
n
where c
n
5 5 2
n
if n is even and c
n
5 4 3
n
if n is odd
29.
P
N
n50
ð21Þ
n
n!
x
2n
31.
P
N
n50
ð21Þ
n
x
2n11
33.
P
N
n50
1=2
n
x
3n
35. 1 1
P
N
n51
ð21Þ
n11
ð2n 2 2Þ!
ðn 2 1Þ!n!2
2n21
ðx 2 1Þ
n
37.
P
N
n50
ðn 1 1Þe
n!
ðx 2 1Þ
n
39.
P
N
n51
ð21Þ
n 1 1
n
ðx 2 1Þ
n
41.
P
N
n50
ð21Þ
n
27ln
n
ð3Þ
n!
ðx 1 3Þ
n
43. 1 1
3
4
x 2
3
32
x
2
1
5
128
x
3
45. 1 2
1
2
x 1
3
8
x
2
2
5
16
x
3
47. 1 2
3
2
x 1
15
8
x
2
2
35
16
x
3
49. 21/6 51. 21/2 53. 22
55. 1/2 57. 21/3 59. 22
61.
P
N
n51
2
2n21
ð21Þ
n11
ð2nÞ!
x
2n
63. 21/3 67. 1/2 69. e 71. e
2
2 1
73. Maclaurin series of tanðxÞup to the x
5
term :
x 1
1
3
x
3
1
2
15
x
5
75. Maclaurin series of ð1 1 uÞ
21=2
:
P
N
n50
ð21Þ
n
ð2nÞ!
2
2n
ðn!Þ
2
u
n
;
Maclaurin series of arcsinðxÞ:
P
N
n50
ð2nÞ!
ð2n 1 1Þð2
2n
Þðn!Þ
2
x
2n11
83. 0.2955 85. 0.1489
87. s 5 1.07766 . . .
y f(x
0
) T
3
(x
0
)
1.07766
6e-06
7e-06
1.2 1.3 1.4
(x
0
c)
4
y
f
(4)
(t)
4!
89. s 5 3.23405 . . .
A-54 Answers to Selected Exercises
y f(x
0
) T
3
(x
0
)
3.2
3.23405
0.002
0.0021
3.3 3.4 3.5
(x
0
c)
4
y
f
(4)
(t)
4!
91. 0.01392
93. 0.00188
95. 0.46128
97. 0.33311
99.
sin(tan(x)) tan(sin(x))
arcsin(arctan(x)) arctan(arcsin(x))
y
1.0002
1.0004
1.0006
0.01 0.02
REVIEW EXERCIS ES FOR CHAPTER 8
1. S
1
¼ 3; S
2
¼ 7; S
3
¼ 23=2; S
4
¼ 163=10; S
5
¼ 213=10
3. 5/2
5. 27/4
7. 1389/2200
9. Converges conditionally
11. Diverges
13. Converges conditionally
15. Diverges
17. Converges absolutely
19. Converges absolutely
21. Converges conditionally
23. Converges absolutely
25. Converges absolutely
27. Converges conditionally
29. Converges absolutely
31. Converges absolutely
33. Diverges
35. Converges absolutely
37. Converges conditionally
39. Converges absolutely
41. Converges absolutely
43. Converges absolutely
45. a) conclusive; b) inconclusive
47. a) inconclusive; b) inconclusive
49. a) conclusive; b) conclusive
51. 0.224 . . .
53. lower estimate: 2; upper estimate: 3
55. (25/3, 21)
57. [21, 1]
59. (21, 1]
61. (23/2, 1/2]
63. [23, 0)
65. (22/3, 4/3]
67.
X
N
n¼0
ð1Þ
n
2
3
nþ1
x
n
69.
X
N
n¼0
27
81
nþ1
x
4n
71.
X
N
n¼0
ð1Þ
n
x
2ðnþ1Þ
73.
40 2 10 lnð2Þ1 ln
2
ð2Þ
=32,
2
1 2 s
ln
2
ð2Þ
3 2 s lnð2Þ
=384
75.
360 1 12
ffiffiffi
3
p
π 1 π
2
=144,
2 sinðsÞπ
3
=1296
77. a) 2.100008049 b) 8.6235 3 10
26
79. a) 3.166666667 b) 0.000645287
81. 0.002667
83. 0.1996
85. 1
87. 6
Review Exercises for Chapter 8 A-55
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INDEX
A
abscissa, 11
absolute convergence, 663664
addition and scalar multiplication,
681682
power series, 675678
Root Test, 670671
absolute difference, Newton-Raphson
accuracy, 351
absolute error, 7, 29
absolute extrema
applied problems, 307315
curve sketching, 330
absolute maximum, 290292
curve sketching, 328
absolute minimum, 290292
curve sketching, 328330
absolute value function, 3637
definition and symbol, 34
acceleration, 164165
antidifferentiation, 361363
higher derivatives, 231
linear velocity law, 614615
Newton’s Law, 581583
accuracy
Newton-Raphson calculations,
349352
Taylor polynomials, 702704
Achilles (Zeno), 81
acoustic theory, harmonic series, 636637
addition
associativity of, 377
constants for, 190191
power series, 681682
Addition Formula, 68, 70
d’Alembert, Jean le Rond, 161
Algebra (Wallis), 373374
algebraic manipulations
indefinite integrals, 360
l’Ho
ˆ
pital’s Rule for indeterminate
forms, 342345
numbers, 80
partial fractions for linear factors, 499
periodic function simplification, 332
Rolle’s Theorem and, 373
Simpson’s Rule, 452454
summation notation, 377
algorithm, Newton-Raphson method,
348349
alternating series test, 661665
power series of functions, 688
689
Ampe
`
re,
Andr
´
e Marie, 373
Amthor, A., 80
Analyst, The (Berkeley), 161
analytic geometry, 81
historical background, 8182
principles of, 80
antiderivative(s)
defined, 357
of functions, 358
Fundamental Theorem of Calculus,
408411
improper integrals with infinite inte-
grands, 514515
integral calculations, 400402
integration
by parts, 470471
reversal, 402403
of trigonometric functions, 436437
linear differential equations, 608609
logarithmic differentiation and inte-
gration, 426
order properties of integrals, 404405
power series of functions and,
688689
powers of x and, 358360
Riemann integrals, 394396
separable differential equations,
591598
substitution of integrals and, 431432
trigonometric and exponential func-
tions, 360361
velocity and acceleration, 361363
antidifferentiation
defined, 357358
power series, 682683
powers of x, 358360
trigonometric and exponential
functions, 360361
velocity and acceleration, 361363
Apollonius, 275, 371
approximate derivative rule, 196
second derivatives, 232233
approximations
basic properties, 78
differential, 250
251
of
functions, 246251
graphing calculators, 334336
midpoint approximation, 447449
numerical integration techniques,
446449
order properties of integrals, 404405
Simpson’s Rule, 452454
sine and cosine derivatives, 185186
Taylor series, 708
trapezoidal rule, 449452
volume slicing, 538539
arbitrary bases
logarithms with, 423424
power series, 679681
arbitrary constants
antiderivatives, 359
differential equation solutions, 589
trigonometric substitution, 491
arccosine, 255
Archimedes, 8081, 465466, 534
arc length and, 625
arc length
Archimedes and, 625
calculation methods, 552553, 620
examples of, 553554
parametric curves, 554556
rectification theory and, 625 626
surface area, 556559
arcsine
antiderivative of, 360361
transcendental functions, 254256
area
approximation of, 379381, 460
axes role reversal, 443444
calculation of, 440444, 461
formulas for, 465
function, 408
Fundamental Theorem of Calculus
computation of, 396397
integration and, 376385
precise definition of, 381384
Riemann sums and difference of,
390392
summation notation, 377
sums for, 378379
surface area, 556559
of
triangle, 383384
between two curves, 441443
argument of a function, 34
Aristotle, 81
infinite series and, 718
Arithmetica Infinitorum (Wallis), 467
arithmetic-geometric mean, 534
arithmetic operations
combining functions, 49
continuity theorems, 110111
Arrow (Zeno), 81
I-1
arrow notation, 35
Assayer, The (Galileo), 82
associated principal angle, sine and
cosine, 67
associativity of addition, summation, 377
astronomical measurements, 43
asymptotes
curve sketching, 327328
graphing calculators, 335
horizontal, 121124
infinite limits and, 118124
periodic functions, 332
skew-asymptotes, 332333
vertical asymptotes, 119121
atomic measurements, integration by
parts, 472473
average, calculation of, 567
average value
of function, 561569, 620
integrals, 405406
probability theory, 567
random variables, 565566
average velocity, 164165
axes, 1011. See also x-axis; y-axis
Method of Cylindrical Shells rotation,
546548
rotation about
noncoordinate axis, 544545
x- and y-axes, 540542
axis of symmetry, 15
B
backward difference quotient, 195
Barrow, Isaac, 466, 534
base point
differentials and function
approximation, 246248
power series
alternating series test, 689
arbitrary base points, 679681
basic functions
differentiation of, 196
summary of derivatives, 227, 266
Basic Rule, closed intervals, 309311
Beaugrand, Jean de, 372
Bell, E. T., 534
Berkeley, George, 161, 277
Bernoulli, Daniel, 161
Bernoulli, Jakob, 627, 720
best line approximation, 248250
differential approximation, 251
Big Bang theory, 43
binomial series, 707
biological applications
differential equations, 614615
implicit differentiation, 285286
logarithmic differentiation, 225226
Mean Value Theorem, 289290
Simpson’s Rule, 454455
Bode’s law, 534535
Bohr radius, 472473
Bolzano, Bernard, 161162
bond valuation, Newton-Raphson
method, 353354
bounded intervals, 46
alternating series test, 661665
integrals on, 521523
Riemann sums and, 392
bounded sequences, 139, 156
Boyle, Robert, 82
Briggs, Henry, 719
C
calculators
computer algebra systems, 9
graphing calculators, 334336
calculus. See also differential calculus;
Fundamental Theorem of Calcu-
lus; integral calculus
approximations in, 78
differential, 161, 278, 396
history of, 8182, 533535
integral calculus, 375, 396
logarithmic functions and exponential
equations, 417426
real number systems, 2
of variations, 628
cancellations
alternating series, 661665
of summands, 379
Cantor, Georg, 80, 82
capital gain, bond valuation, 353354
Carath
´
edory, Constantin, 278
cardiac output measurements, Simpson’s
Rule, 454455
carrying capacity constant, logistic
growth, 596598
Cartesian equations, 12
plane(s), ordered pair notation, 11
Caspar, Max, 625
catastrophic cancellation, 8
catenary, 626627
catenoid, 628
Cauchy, Augustin-Louis, 161162
Cauchy-Riemann equations, 468
Cavalieri, Bonaventura, 373, 466467
celestial mechanics, 534
center of mass, 572577, 620621
mathematical equations, 574577
moments in, 572574
two point systems, 572
central difference quotient, 195196
parametric curves, 555556
Certaine Errors of Navigation Corrected
(Wright), 533
cgs measurement system, 580
Chain Rule, 210216
applications, 214215
chain rule-power rule, 213
exponential function derivatives,
215216
functional notation, 212
Fundamental Theorem of Calculus,
411412
gravitational force and, 595596
history of, 278
implicit
differentiation, 237239
inverse functions, 220227
trigonometric functions, 258259
logarithms
differentiation and integration, 426
natural logarithm, 419420
multiple compositions, 214
Newton and, 467
proof of, 278279
related rates problem, 215, 282285
separable differential equations,
591598
substitution method and, 432
transcendental functions, 256
two-function composition
differentiation, 210212
change, rates of, 164175
instantaneous rate of, 168170
change of variable. See substitution
chemical kinetics, differential equations
in, 594595
choice of points, Riemann integral,
386392
Cicero, 625
circle(s)
distance formula, 1213
endpoint solutions, 310311
equation for, 13
surface area, 556559
tangent lines and, 171173,
203204
circumference, defined, 2
Clerselier, Claude, 372373
closed intervals, 5
absolute maxima/mimima, 308311
coefficient(s)
binomial, 707
constant coefficients, 611612
of inequality, 451452
of partial fractions, decomposition
and, 509512
in power series, 689691
collapsing series, 635636
collapsing sum, 378
I-2 Index
combining functions, 4961
arithmetic operations, 49
composition of, 5152
even and odd functions, 58
inverse functions, 5256
graph of, 56
pairing functions-parametric curves,
5960
parameterized curves and graphs, 61
polynomial functions, 5051
vertical and horizontal translations, 57
common denominator
distinct linear factors, partial
fractions, 500
l’Ho
ˆ
pital’s Rule for indeterminate
forms, 344345
partial fractions for linear factors,
498499
repeated linear factors, 502503
comparison tests, 653658
absolute convergence, 664
convergence, 653655
divergence, 655657
Root Test and, 671
Comparison Theorem for Integrals over
Unbounded Intervals, 525526
Comparison Theorem for Unbounded
Integrands, 518519
completeness property of real
numbers, 82
continuous functions and, 111112
Intermediate Value Theorem, 162
Monotone Convergence Property
and, 139
nested intervals, 111
Completing the Square, Method of,
1315
composition
continuity theorems, 110111
of functions, 5152
compound interest, 147149
computer algebra systems, 9
average value of function, 563564
bond valuation, 354
graphing calculators, 334336
Newton-Raphson method, 352353
tables of integrals and, 435
concavity
critical point testing, 323 324
curve sketching, 327328
defined, 320
extrema classification, 323
324
natural
logarithms, 420
points of inflection, 322323
second derivative test, 321325
conchoid, 371372
cones, volume approximation, 538539
conic sections, 15
conjugation, l’Ho
ˆ
pital’s Rule, 344345
consecutive squares, sum of, 378
constant coefficients, linear equations,
611612
constant magnitude, work calculations,
580585
constant multiple rule, 196
constants
addition, subtraction, and
multiplication by, 190191
antiderivatives, 357, 359
decay constant, 150151
derivatives of, 190
Hubble’s constant, 42
of integration, 362363
antiderivatives, 357
spring constant, 583584
constant temperature, Newton’s laws for
temperature change, 612613
continuity, 105115
arithmetic operations and composition
theorems, 110111
continuous extensions, 108109
continuous functions, advanced prop-
erties, 111114
defined, 105106
differentiability and, 182183
equivalent formulation, 106108
intermediate value theorem and,
112114, 161162
of inverse functions, 220223
one-sided, 109110
continuous compounding, 148149
continuous derivatives, parametric
curves, 555556
continuous extensions, 108109
continuous functions
advanced properties, 111114
area between two curves, 441443
center
of mass, 574577
closed intervals, 308311
first derivative test for local extrema,
301303
Fundamental Theorem of Calculus
computation of, 396397
of integrals, 400402
integrals, value of, 405406
integral test and, 647648
integration reversal, 402403
limits calculations, 135136
natural logarithms, 420
precise definition of area, 382384
real numbers, 111112
Riemann integrals, 393396
continuous second derivatives, Newton-
Raphson accuracy, 351
continuous variables, 1
contrapositive, law of, 643644
convergence
absolute convergence, 663664
alternating series test, 661 665
arbitrary base points, power series,
679681
comparison test for, 653655,
664665
Comparison Theorem for Unbounded
Intervals and, 525526
improper integrals, 515, 521523
infinite sequences, 631633
infinite series, 634635
integral test of, 646648, 649650
interior singularity of integrands, 516
interval of, 674678
Taylor series approximation, 708
Limit Comparison Test and, 657658
limits of sequences and, 128129
monotone convergence property,
138140
nonnegative terms in series, 645
power series, 674
683
p-series,
648649
Ratio Test for, 667669
Root Test, 670671
series, 637638
special sequences, 130
tail end of series, 645646
Taylor series expansions, power series,
705706
unbounded integrands, endpoints
at, 517
unevaluated proof, 517518, 525526
convexity, conchoids, 371372
corners, vertical tangent lines and,
174175
cosecant, 68
higher power of, 481482
integration of, 436437
inverse, 258259
periodic function, 331332
square of, 480481
cosine(s)
antiderivative of, 360361
conversion to, 485
derivatives of, 184186, 203204
higher power of, 481482, 529
hyperbolic functions, 260262
integrals involving, 483485
integral values, 406
integration of, 436437
inverse sine, 254256
Newton-Raphson method, 348349
odd powers of, 482483, 529
periodic function, 331332
Index I-3
cosine(s) (continued )
Riemann integral, 396
square of, 480481
trigonometry, 6568
cotangent, 68
integration of, 436437
countable numbers, 80
coupon rate, bond valuation,
353354
Cours d’Analyse (Cauchy), 162
critical points
closed intervals, 308311
curve sketching, 327328, 330
first derivative test, 301
maxima and minima, 301303
points of inflection and, 322323
Second Derivative test of, 323324
transcendental functions, 314315
cubic polynomials, Newton-Raphson
algorithm, 373374
Cureau de la Chambre, Marin, 372373
curve(s)
arc length calculations, 552553
area between two curves, 441443
axes role reversal, 443444
Folium of Descartes, 276277
graphing functions, 327336
Lorenz curve, 451
Newton-Raphson method, 348349
normal lines to, 173174
parametric, 554556
precise definition of area, 381384
sketching strategies, 327331
tangent lines to, 171173, 180181
cycling phenomenon, Newton-Raphson
method, 352
cycloid, 467
Cylindrical shells, method of, volume
calculations, 545548, 620
D
data, functions from, 4044
Debye function, 593
decay constant, exponential decay,
150151
decimal expansions
floating point, 8
geometric series, 639
infinite series, 629630, 633634
irrational numbers, 7
midpoint approximation, 448449
nonterminating, 23
nonterminating, nonrepeating, 3
Taylor series accuracy, 702704
terminating, 23
types of, 23
decomposition
Heaviside’s method, 501502
partial fractions, 500, 509512
telescoping series, 636
decreasing functions
concavity, 321322
curve sketching, 327328
derivatives, 299300
identification, 298299
integral test and, 647648
profit maximization, 313314
transcendental functions, 314315
decreasing sequences, 138
Dedekind, Julius, 82
definite integral
Fundamental Theorem of
Calculus, 411
point rule, 447449
Riemann integrals, 393396
substitution for, 430432
table of, 433435
degree of accuracy, limits, 8788
demand curve, 313
demand equation, 313314
demand function, 313314
Democritus, 465
De Moivre, Abraham, 720
denominator
common denominator, l’Ho
ˆ
pital’s
Rule, 344345
distinct linear factors, partial fractions
for, 499
500
Heaviside’s
method, 501502
infinite sequences, 632633
l’Ho
ˆ
pital’s Rule for indeterminate
forms, 339342
quadratic terms in, rational functions,
506508
density function theory
population density function, 568569
probability density function, 565566
denumerable numbers, 80
derivative(s). See also antiderivative(s)
applications, 281
basic properties, 163
chain rule, 210212
of constant, 190
defined, 178180
differentiability graphing and, 183186
e (number) and, 422423
of expressions, 186
of functions
basic functions, 196
derived functions, 181
exponential functions, 204206
increasing and decreasing functions,
299300
increasing/decreasing functions,
299300
inverse functions, 220227
polynomial functions, 203
higher derivatives, 230234, 241242
hyperbolic functions, 262263
local extrema test, 301303
of logarithms, 223225
Mean Value Theorem and, 294295
Newton-Raphson algorithm, 373374
notations, 180181
power series
coefficients, 689691
differentiation and antidifferentia-
tion, 682683
powers of x and, 200203
Product Rule and, 191
192
Quotient
Rule, 193194
Reciprocal Rule and, 192193
related rates applications, 282286
trigonometric functions, 203204
Desargues, G
´
erard, 276277
Descartes, Ren
´
e, 8182, 371, 467
arc length and, 625
law of refraction and, 372373
tangent problem solutions, 275277
diagonal square, irrational numbers
and, 2
Dichotomy (Zeno), 81
difference of areas, Riemann sums,
390392
difference quotient, 194196
differentiability
continuity and, 182183
defined, 180181
graphic investigation of, 183186
infinite, 682683
of inverse functions, 220223
l’Ho
ˆ
pital’s Rule for indeterminate
forms, 339341
Mean Value Theorem applications
and, 294295
Reverse Product Rule, 470471
differential approximation, 246251, 271
differential calculus, 396
history of, 278
infinitesimals and, 161
differential equations
catenary formula, 627
initial value problems, 590593
integral applications, 537
linear first order equations, 606615
logistic growth and, 596598
physical sciences applications,
593596
power series and, 692693
slope fields, 589590
I-4 Index