Therefore, by the Bounded Increasing Partial Sums Theorem, ∞k=1akconverges.. So Sngrows without bound and ∞k=1akdiverges.. Now we show that the behavior of the integral can be determined
Trang 1Proofs to Accompany Chapter 30, Series 1131
lim
n→∞Sn≤ lim
n→∞a1+
n
1 f (x) dx = a1+ A The sequence of partial sums is bounded and increasing Therefore, by the Bounded Increasing Partial Sums Theorem, ∞k=1akconverges.
Suppose 1∞f (x) dx = ∞ Refer to Equation (H.1) to obtain
n−1
1 f (x) dx ≤ a1+ a2+ · · · + an= Sn
Taking the limit as n → ∞ gives
∞ ≤ lim
n→∞Sn.
So Sngrows without bound and ∞k=1akdiverges.
Now we show that the behavior of the integral can be determined by that of the series.
Because f (x) > 0, decreasing, and continuous on [1, ∞), limb→∞
b
1 f (x) dx is either finite or grows without bound Therefore, if we can find an upper bound, the integral converges If it has no upper bound, it diverges.
Suppose ∞k=1akconverges Denote its sum by S From Equation (H.1) we know
n
1 f (x) dx ≤ a1+ a2+ · · · + an−1
lim
n→∞
n
1 f (x) dx ≤ lim
n→∞
n−1
k=1
ak= S.
If limn→∞1nf (x) dx is bounded, so too is limb→∞1bf (x) dx (given the hypothe-ses).
Suppose ∞k=1ak diverges Because the terms are all positive, we know limn→∞nk=1ak= ∞ From Equation (H.1) we know
a2+ · · · + an≤
n 1
f (x) dx
lim
n→∞
n
k=2
ak≤ lim
n→∞
n 1
f (x) dx.
We conclude that limb→∞1bf (x) dx = ∞; the improper integral diverges.
Trang 3Abel, Niels, 382, 1078n
Absolute convergence
conditional, 953
explanation of, 952–953
implies convergence, 1128–1129
Absolute maximum point, 347
Absolute maximum value, 252, 347
Absolute minimum point, 347
Absolute minimum value, 347, 352
Absolute value function
explanation of, 61–62
maximum and minimum and, 350
Absolute values
analytic principle for working with, 66
elements of, 65
functions and, 129
geometric principle for working with,
66–68
Acceleration
due to force of gravity, 150n, 406–407
explanation of, 76
Accruement, 746
Accumulation, 746
Addition
of functions, 101–103
principles of, 1059–1061
Addition formulas, 668–669, 709
Additive integrand property, 738
Algebra
equations and, 1053
explanation of, 1051–1052
exponential, 247, 309–312
exponents and, 1053–1054
expressions and, 1052–1053, 1056–1070
order of operations and, 1054
solving equations using, 1071–1083 (See
alsoEquations)
square roots and, 1054–1055
Alternating series
error estimate, 954–955
explanation of, 953
Alternating series test, 953–956, 977
Amount added, 746 Amplitude definition of, 603 modifications to, 604, 606
Ancient Egyptians, 95, 97n, 627, 854
Angles complementary, 630
of depression, 631
of elevation, 631 initial side of, 619 measurement of, 619–622 right, 620
terminal side of, 619–620 trigonometric functions of, 622–623 vertex of, 619
Antiderivatives definition of, 761, 762 for integrand, 805–806 list of basic, 783–786 table of, 789–790 use of, 763, 764, 766, 770, 771 Antidifferentiate, 805
Approximations constant, 920
of definite integrals, 805–816, 820–825 Euler’s method and, 1022
examples of, 828–829 higher degree, 923–924 linear, 159, 163 local linearity and, 279–282 net change, 715–718 Newton’s method, 1121–1125 polynomial, 693–694, 919–931 second degree, 921–922 successive, 170, 176, 208, 715–718 tangent line, 282–284, 920–921, 937 Taylor polynomial, 924–931 third degree, 922–923 Arbitrarily close, 249 Arbitrarily small, 249, 250 Arccosine, 646, 647
Archimedes, 97n, 245, 1079n
Arc length definite integrals and, 865–867 definition of, 594, 708 explanation of, 621–622 Arcsine, 646, 647 Arctangent, 646 Area
of circle, 17
of oblique angles, 662–663
of oblique triangles, 662–664 slicing to find, 843–845 Area function
amount added, accumulation, accruement and, 745 characteristics of, 747–755 definition of, 745 explanation of, 743–744 general principles of, 744–745 Astronomy, 1107
Asymptotes horizontal, 64–65, 407–409, 411 overview of, 64–65
vertical, 64, 407, 408, 617 Autonomous differential equations explanation of, 997
qualitative analysis of solutions to, 1002–1014
Average rate of change calculation of, 170 definition of, 75 explanation of, 73–76, 176 Average value
definition of, 777
of functions, 775–780 Average velocity
to approximate instantaneous velocity, 208
calculation of, 171–176 explanation of, 171 over time interval, 76, 77
Avogadro’s number, 150n
1133
Trang 41134 Index
Babylonians, 95, 97n, 620
Bacterial growth, 306, 324–325
Balance line, 603, 604
Balance value
change in, 606
definition of, 603
explanation of, 604
Base, 1053
Benchmark, 53
Bernoulli, Johann, 17n
Bessel, Freidrich, 961
Bessel function, 961, 976
Bhaskara, 95
Binomial series
explanation of, 946–947
use of, 948–949
Bottle calibration, 4, 20–21, 29n, 32,
209–210
Bounded Increasing Partial Sums Theorem,
965, 966, 1131
Bounded Monotonic Convergence
Theorem, 965
Boyle, Robert, 546n
Boyle’s Law, 546n
Braces, 18n
Brahe, Tycho, 1107
Briggs, Henry, 447
Calculators See also Graphing calculators
irrational numbers and, 96, 97
limits and, 267–269
logarithms using, 440–443, 450
order of operations on, 1054
roots on, 1058
trigonometric functions on, 605n, 608,
647, 919
use of graphing, 87–89
Calculus
differential, 111n, 806
Fundamental Theorem of, 757–758,
761–771 (See also Fundamental
Theorem of Calculus)
historical background of, 211, 245
integral, 806
Calibration, bottle, 4, 20–21, 29n, 32,
210–211
Cardano, Girolamo, 95, 382
Carrying capacity, 150n, 1008–1010
Cellular biology, 984
Chain Rule
application of, 517–519, 523–527,
539–543, 551, 785, 805, 1028
definition of, 517, 709
derivatives and, 521–522, 692, 706
interpretation of, 517
in reverse, 787–796
substitution to reverse, 792–794
Charles, Jacques, 546n
Charles’s Law, 546n
Circle area of, 17 explanation of, 1100, 1102 illustration of, 1099 symmetry of unit, 623–624
Circumference formula, 17n
Closed form of sum, 564 Closed intervals, 18 Coefficients differential equations with constant, 1045–1049
explanation of, 1053 leading, 379, 1063
of the xk term, 380 Cofunction identity, 669 Common log of x, 442 Comparison Tests, 969–972, 977 Comparison Theorem, 912 Complementary angles, 630
Completeness Axiom, 965n
Completing the square, 231–232, 1075 Composite functions
Chain Rule to differentiate, 788 derivatives of, 513–517, 521–527 Composition, of functions, 108–113 Concave down, 52, 195
Concave up, 51, 52, 195 Concavity
cubic and, 376 second derivative and, 356–358 Conditional convergence, 953 Conic sections
astronomy and, 1107 degenerate, 1099 from geometric viewpoint, 1100–1101 overview of, 1099–1100
reflective properties of, 1106–1107 relating geometric and algebraic representations of, 1102–1105 Conjecture, 335
Constant factor property, 738 Constant function, 8 Constant Multiple Rule, 291–292
Constant of proportionality, 306n
Constant rate of change calculating net change in case of, 712–714
linear functions and, 143–144, 169
Constant solutions See Equilibrium
solutions Continuity definition of, 270, 273 Extreme Value Theorem and, 271 Intermediate Value Theorem and, 271 Sandwich Theorem and, 272–273 Continuous functions
average value of, 775–776
of closed interval, 352 explanation of, 54–55
invertible, 426 Convergence absolute, 952–953, 1128–1129 alternating series and, 954–956 conditional, 953
explanation of, 566, 568, 569, 904 improper integrals and, 904–907
in infinite series, 574
of Maclaurin series, 944
of power series, 945–946, 956, 1129 radius of, 945, 956–958
of Taylor series, 944n, 947
Convergence tests basic principles of, 964–965 Comparison Test and, 969–971 Integral Test and, 965–969, 972 Limit Comparison Test and, 971–972 Ratio and Root tests and, 972–976 summary of, 977
transition to, 962 use of, 975 Convergence Theorem for Power Series, 1129
Converse, 30n
Cosecant, 629
Cosine functions See also Trigonometric
functions definition of, 594 domain and range of, 598 graphs of, 597–598, 603–609 symmetry properties of, 598–599 Cosines, Law of, 658–662, 709 Cotangent, 629
Coterminal angles, 620 Critical points concavity and, 356–358
of cubics, 375–376 extreme values and, 349–353
of polynomials, 386–387 Cubic formula, 382 Cubics
characteristics of, 375–377 explanation of, 373 function as, 373–377 roots of, 381–382 Simpson’s Rule applied to, 823 Curves
calculating slope of, 169–177 slicing to find area between, 843–845 solution, 501–503
Decomposition
of functions, 119–121 into partial fractions, 898–901 Definite integrals
arc length and, 865–867 area between curves and, 843–850 average value of function and, 775–780 definition of, 728
Trang 5Index 1135
evaluation of, 763
explanation of, 725–727
finding exact area and, 727–729
finding mass when density varies and,
827–838
fluid pressure and, 871–874
Fundamental Theorem of Calculus used
to compute, 770
integration by parts and, 882–883
interpretations of, 729, 743
numerical methods of approximating,
805–816, 820–825
properties of, 738–741
qualitative analysis and signed area and,
731–736
substitution in, 794–796
work and, 867–871
Definitions, 3
Degrees
of angles, 620
radians vs., 685
Demand function, 104
Density, 827–838
Dependent variables, 5
Derivatives
of bx, 473–475
calculation of, 190–192
of composite functions, 513–517,
521–527
Constant Multiple Rule and, 290–293
of cubic, 374–375
definition of, 187
explanation of, 188–190, 481, 523
of exponential functions, 334–338
historical background of, 211
instantaneous rate of change and,
209–211, 711, 983
of inverse trigonometric functions,
703–706
limit definition of, 190, 191
local linearity and, 279–284
of logarithmic functions, 467–471
meaning and notation and, 208–209
modeling using, 288–289
Product Rule and, 292–295
properties of, 273, 291–292
of quadratics, 219–221
qualitative interpretation of, 194–201
Quotient Rule and, 297–298
second, 221, 288–289, 356–358
of sums, 290–292
theoretical basis of applications of,
1087–1093
as tool for adjustment, 282
of trigonometric functions, 683–685, 708
working with limits and, 272
of xn for n, 295–297
Derivative tests
first, 351, 358
second, 357, 358 Descartes, Rene, 61
Differential calculus, 111n, 806
Differential equations applications for, 984–988 autonomous, 997 with constant coefficients, 1045–1049 definition of, 498
explanation of, 498–500 first order, 988, 1018–1022 homogeneous, 1045–1049 modeling with, 503–506, 983–990 nonseparable, 1022
obtaining information from, 988–990 population growth and, 503–506, 984 power series and, 959–961
qualitative analysis of solutions to autonomous, 1002–1014 radioactive decay and, 503, 504 second order, 988, 1045–1049 separable, 1018–1022, 1030 solutions to, 498–503, 988, 991–997, 1002–1014, 1018–1022
systems of, 1024–1040 Differentiation
of composite functions, 513–517 examples of, 535–538
flawed approaches to, 536
implicit, 541–554 (See also Implicit
differentiation) logarithmic, 538–541
of logarithmic functions and exponential functions, 476–481
polynomials and, 385, 386
of power series, 956–959 Product Rule and, 292–294
of trigonometric functions, 683–698, 703–706
Differentiation formulas, 523–524 Direct Comparison Test, 969 Directrix, 1100, 1101
Dirichlet, Lejeune, 17n
Discontinuities points of, 407–410 removable and nonremovable, 407 Discontinuous functions, 54–55 Distributive Law, 1056, 1065 Divergence
explanation of, 566, 568, 904
of harmonic series, 953 improper integrals and, 904–907
in infinite series, 574 nth Term Test for, 574, 964, 977 Division
of fractions, 1061–1063
of functions, 103–105
of polynomials, 382–383
by zero, 1053
Domain
of function, 5 natural, 17
of trigonometric functions, 598 Dominance property, 738 Double-angle formula, 669, 709 Double root, 380
Dummy variables, 784 Einstein’s theory of special relativity, 938 Ellipses
explanation of, 1100–1103 illustration of, 1099 reflecting properties of, 1106 Endpoint reversal property, 738, 744 Epidemic models, 1024–1025, 1034–1038 Epidemiology, 986, 1024–1025, 1034–1038 Equal Derivatives Theorem, 1088, 1092–1093
Equations See also Differential equations
explanation of, 1053 exponentiation to solve, 449–455 expressions vs., 537–538 fitted to sinusoidal graph, 606–607 fundamental principle for working with, 1053
growth, 306 higher degree, 1078–1080 linear, 159, 1071–1073
of lines, 144–145, 148–150 logarithms to solve, 449–457 quadratic, 1073–1078 with radicals, 1081–1083 solutions to, 1071–1083 trigonometric, 651–655 Equilibrium, stable, 106–108 Equilibrium solutions definition of, 1004 example of, 1004–1005 explanation of, 989 stability and, 1006–1008
Erd¨os, Paul, 688n
Error bounds, 823–825
Euler, Leonhard, 17n, 492n
Euler’s Formula, 1048 Euler’s method, 1022 Even functions, 65 Existence and Uniqueness Theorem, 994 Exponential algebra
application of, 247, 309 basics of, 309–312 Exponential functions derivative of, 334–338 explanation of, 312–314, 497 food decay and, 303, 326–327 generalized power functions and, 524 graphs of, 462–463
growth of money in bank account and,
303, 320–322, 325, 503
Trang 61136 Index
Exponential functions (continued)
laws of logarithmic and, 444–447
manipulation of, 314–316
modeling with, 559
radioactive decay and, 322–324
writing and rewriting, 324–326
Exponential growth
examples of, 306–307
explanation of, 303–306
Exponentiation, 449–455
Exponent Laws, 1058
explanation of, 309, 310
list of, 481
Exponents
explanation of, 1053–1054
multiplication and working with,
1057–1059
Expressions
addition and subtraction, 1059–1061
division and complex fractions,
1061–1063
equations vs., 537–538
explanation of, 1052–1053
factoring, 1064–1070
multiplying and factoring, 1056–1057
multiplying and working with exponents,
1057–1059
terminology and, 1063–1064
working with, 1056–1070
Extraneous roots, 1082
Extrema
analysis of, 346–348
definition of, 347
global, 351–354
local, 351, 353, 391–392
Extreme, absolute, 353
Extreme Value Theorem
continuity and, 271
explanation of, 349, 352, 1087, 1089,
1090
Factorial notation, 694
Factoring
difference of perfect squares, 1068–1069
explanation of, 1056–1057
out common factor, 1065–1066
quadratics, 1066–1068
use of, 1064–1065, 1069–1070
Finite geometric sum, 561, 563
First derivative, 288–289
First derivative test, 351, 358
First order differential equations
explanation of, 988
solutions to separable, 1018–1022, 1030
Fixed costs, 102, 103
Flipping, 126–129
Fluid pressure, 871–874
Focus, 1100
fog, 108
Folium of Descartes, 543, 544 Food decay, 303, 326–327 Foot-pound, 868 Fractions adding and subtracting, 1059–1061 complex, 1061–1063
division of, 1061–1063 historical background of, 95 integration using partial, 898–901 multiplication of, 1057
simplification of, 1057 Free fall, 237–240 Functional notation examples of, 9–10, 17 explanation of, 8–9
Functions See also specific types of
functions absolute value and, 66–68, 350 addition and subtraction of, 101–103 altered, 126–131
area, 743–755 asymptotes and, 64–65 average value of, 775–780 Bessel, 961, 976 composite, 513–517, 519–525 composition of, 108–113 constant, 8
continuous/discontinuous, 54–57, 352, 426
decomposition of, 119–121 definition of, 5
derivatives as, 187–211 (See also
Derivatives) domain and range of, 5, 17–21 equality of, 17
even, 65 examples of, 61–64 explanation of, 2–3, 5–9 exponential, 303, 312–328
to fit parabolic graph, 233–235
graphs of (See Graphs)
increasing/decreasing features of, 51–54 inverse, 32, 110, 421–426, 429–432, 434 invertible, 423–426, 429–434, 440 linear, 143–151
locally linear, 142
logarithmic, 439–457, 462–466 (See
alsoLogarithmic functions)
as machine, 6–7
as mapping, 6–8
of more than one variable, 33 multiplication and division of, 103–105 names of, 5
odd, 65 overview of, 1–2 portable strategies for problem solving and, 21–28
positive/negative features of, 49–51 quadratic, 217–240
rational, 407–417 reciprocal, 62, 130–131 representations of, 15–17 sinusoidal, 603–604 solution to differential equation as, 988 Fundamental Theorem of Calculus definite integrals and, 761–763 explanation of, 757–758, 770–771 power of, 762, 767–770
use of, 763–767, 1097 Galois, ´Evariste, 1078n Galois theory, 1078n
Generalized power functions, 524 Generalized power rule, 524 Generalized Ratio Test, 975–976 General solutions, 500–502, 996
Geometric formulas, 17n, 1086
Geometric series applications of, 579–586
to compute finite geometric sums, 570 convergence and, 977
definition of, 566 explanation of, 566–570, 962, 964 Geometric sums
applications of, 579–586 examples of, 563–564 finite, 561, 563 geometric series to compute finite, 570 introductory example of, 559–563 Global extrema, 351–354
Global maximum point, 347 Global maximum value, 347, 352 Global minimum point, 347 Global minimum value, 347, 352
Graphing calculators See also Calculators
Taylor polynomials on, 937
trigonometric functions on, 605n, 608
use of, 87–89, 382 Graphs
of derivative functions, 196–201
of functions, 28–33, 62, 63, 65, 84, 87–89
of invertable functions, 425–426
of linear functions, 144–147
of logarithmic functions, 462–466
of periodic functions, 596
of polynomials, 384–385, 391–399 position, 84–85
of quadratics, 219, 221, 223, 231–235 rate, 85–87
of rational functions, 410–417
of reciprocal functions, 130–131
of squaring functions, 63 tips for reading, 85, 87
of trigonometric functions, 597–598, 603–609, 616–618, 647, 708 Gravitational attraction, 406
Gregorian calendar, 53n
Trang 7Index 1137
Guy-Lussac, Joseph, 546n
Guy-Lussac’s Law, 546n
Happiness index, 51
Height, 77
Herodotus, 95n
Higher degree equations, 1078–1080
Hindus, 95n
Hipparchus, 627
Hooke’s Law, 1045
Horizontal asymptotes
explanation of, 64–65
rational functions and, 407–409, 411
Horizontal line test, 33, 426
Horizontal shift, 607–608, 669
Hypatia, 95n
Hyperbolas
explanation of, 1103
illustration of, 1099
reflective properties of, 1107
Hypotenuse, 628
Identity function, 61
Implicit differentiation
explanation of, 541–545
particular variables and, 546–548
process of, 545–546
related rates of change and, 550–554
Improper integrals
explanation of, 903
infinite interval of integration and,
903–907
method of comparison and, 912–914
methods to approach, 908–912
unbounded and discontinuous integrands
and, 907–908
use of, 903
Increasing and Bounded Partial Sums Test,
574
Increasing/Decreasing Theorem, 1088
Indefinite integrals
definition of, 783
using integration by parts, 878–882
Independent variables, 5
Indeterminate forms, 490n
Induction, 296, 1095–1097
Induction hypothesis, 296
Inequalities, Intermediate Value Theorem
and, 55–56
Infinite series
definition of, 566
general discussion of, 572–574
geometric, 566–570
Inflation, 487
Inflection point, 357, 376
Initial condition, 502
Initial conditions, 501–503
Instantaneous rate of change
calculation of, 170–178, 208
derivative and, 209–211, 711, 983 explanation of, 170, 176, 523 velocity and, 170, 173 Instantaneous velocity average velocity to approximate, 208 explanation of, 170, 173–174 limit and, 245–246
Integers, 1063 Integral calculus, 806 Integrals
definite (See Definite integrals)
improper, 903–914 indefinite, 783–785, 878–882 substitution to alter form of, 798–802 trigonometric, 886–890
Integral Test for convergence of series, 1129–1131 explanation of, 965–968, 977 proof of, 968
use of, 968–969, 972 Integrands
antiderivative of, 805–806 definition of, 783 explanation of, 729, 770 expressed as sum, 798 unbounded and discontinuous, 907–908 Integration
infinite interval of, 903–907
of power series, 956–959 solving differential equations by, 995
Integration by parts See also Product Rule
definite integrals and, 882–883 explanation of, 877–878 formula for, 878, 879, 882 indefinite integrals and, 878–879 repeated use of, 879–882
use of, 805, 846n
Interest rates, 487–489, 494–495, 579–586 Intermediate Value Theorem
continuity and, 271 explanation of, 55 inequalities and, 55–56 Interval notation, 18–19 Interval of convergence explanation of, 945–946
Taylor series and, 944n, 947
Intervals closed, 18 increase and decrease on, 51–54 open, 18, 270
Inverse functions arriving at expression for, 429–432 definition of, 423
explanation of, 32, 110, 421–426, 440 graphs and, 425
horizontal line test and, 426 interpreting meaning of, 434 slicing and, 846–850 Inverse sine, 646
Inverse trigonometric functions derivatives of, 703–706 explanation of, 645–649, 708 Irrational numbers, 1063–1064 definition of, 87
historical background of, 95, 96 nature of, 88–89, 96–99
proof of, 88n
working with, 247
Isosceles triangles, 635n
Joule, 868
Julian calendar, 53n
Kepler, Johannes, 447, 1107 Kepler’s Laws, 961 al-Khowarizmi, 95 Kinetic energy, 938 Lagrange, Joseph, 211 Lambert, Johann, 97 Law of Cooling (Newton), 326, 985–986, 1002–1004, 1025–1028
Law of Cosines explanation of, 658–659, 709 proof of, 660–661
use of, 661–662 Law of Sines explanation of, 659–660, 709 proof of, 664
use of, 664–665 Leading coefficients, 379 Least common denominator (LCD), 1059, 1060
Left- and right-hand sums explanation of, 727, 728, 820 limits and, 258–262, 270, 728 net change and, 718–722 use of, 806–808, 812–815 Legs, 628
Leibniz, Gottfried Wilhelm, 209, 211, 245,
292n, 953
Leibniz’s notation, 209, 211, 221
Lemmas, 688n
L’Hˆopital’s Rule definition of, 1112 proof of, 1114–1116
use of, 490n, 1112–1114, 1117, 1118
Limit Comparison Test, 971–972, 977
Limit principle, 490, 492n, 493n
Limits application of, 245–246 approaches to, 265–269 computation of, 251–255, 489–491 definition of, 250–255, 524 explanation of, 246–250 function of, 491–495 left- and right-handed, 258–262, 270 one-sided, 259–260
Trang 81138 Index
Limits (continued)
principles for working with, 272–273
trigonometric functions and, 688–691
two-sided, 262
Linear approximations, 159, 163–164
Linear equations
definition of, 1071
simultaneous, 159
solutions to, 1071–1073
Linear functions
characteristics of, 143–144
definition of, 144
explanation of, 373
graphs of, 144–147
piecewise, 159, 161
zero of, 381
Linearity
differential calculus and local, 111n
intuitive approach to local, 142–143
Linear models, 159–164
Lines
equations of, 144–145, 148–150
graphs of, 147
parallel, 148
perpendicular, 148
point-slope form of, 149
secant, 75
slope-intercept form of, 149
slope of, 145–150
vertical, 147–148
Lithotripsy, 1106–1107
Liu Hui, 95
Local extrema, 351, 353, 391–392
Local Extremum Theorem, 1087–1090
Local linearity
derivatives and, 279–284
differential calculus and, 111n
explanation of, 142–143, 1114
use of, 143
Local maximum point, 347
Local maximum value, 347, 354
Local minimum point, 347
Local minimum value, 347, 354
Logarithmic differentiation
definition of, 538
to find y1, 538–539
use of, 539–540
Logarithmic functions
definition of, 440, 442
derivative of, 467–475
explanation of, 440–443
graphs of, 462–466
historical background of, 447
introductory example of, 439–440
laws of exponential and, 444–447
Logarithmic laws, 445, 481
Logarithms
calculator use and, 440–443, 450
converted from one base to another, 450–454
definition of, 441–442 differentiation and, 476–480 properties of, 444–448 solving equations using, 449–457 summary of, 480–481
uses for, 447–448 Logistic growth model, 898, 1009, 1019–1020, 1024
Long division, 382–383 Lower bound, 716, 717 MacLaurin, Colin, 920 Maclaurin series alternation series and, 953 binomial series and, 946, 947 convergence and, 944 explanation of, 941, 948 procedure for finding, 941–942, 957, 961 Marginal cost, 210
Mass, 827–838 Mathematical induction, 296, 1095–1097 Mathematical models, 2
Mayans, 95n Mean Value Theorem, 346, 866n, 1087,
1090–1093 Midpoint sum, 807–811, 813–815 Modeling
with derivatives, 288–289 with differential equations, 503–506, 983–990
of population interactions, 1038–1040 Money, growth of, 303, 320–322, 325 Multiplication
with exponents, 1057–1059 with expressions, 1056–1057 with fractions, 1057 with functions, 103–105 Napier, John, 447 Natural domain, 17 Natural log of x, 442 Negative numbers, 95 Net change
with constant rate of change, 712–714 difference between left- and right-hand sums and, 718–722
explanation of, 712 with nonconstant rate of change, 715–718
overview of, 711–712 Newton, 868
Newton, Isaac, 211, 245, 1107 Newtonian physics, 938–939 Newton-meter, 868 Newton’s Law of Cooling, 326, 985–986, 1002–1004, 1025–1028
Newton’s method applications for, 1124–1125 for approximating root of f, 1123–1124 explanation of, 1121–1122
use of, 1122–1123 Newton’s Second Law, 868, 1034 Nonremovable discontinuities, 407 Nonseparable first order differential equations, 1022
Notation factorial, 694 functional, 8–10, 17 interval, 18–19 summation, 575–577, 769 nth partial sum, 566 nth Term Test for Divergence, 574, 964, 977
Nullclines, 1030–1032, 1035 Number line, 18
Numbers irrational, 87, 88, 96–99 rational, 87, 95, 96 real, 87, 88 Oblique triangles area of, 663–664 explanation of, 657 Law of Cosines and Law of Sines and, 658
Obtuse angles, 662–663 Odd functions, 65 Open interval, 18, 270 Operator notation, 209 Optimization analysis of extrema and, 341–348 application of, 361–364 concavity and second derivative and, 356–358
extrema of f and, 348–354 overview of, 341
Order, of differential equations, 988 Order of operations, 1054
Parabolas definition of, 219 with derivatives, 220–221 explanation of, 1100, 1101 graphs of, 231–235, 342–343 illustration of, 1099 reflecting properties of, 1106 turning point of, 221 vertex of, 221, 223–225 Parabolic arc, 1107 Parabolic graphs, 233–235 Parallel lines, 148
Parameter, 126n
Partial fraction decomposition, 898–901 Partial sums, 964–966
Particular solutions, 498, 501, 502
Trang 9Index 1139
Perfect squares
equations with, 1075
factoring difference of, 1068–1069
Period, 603, 604
Periodic functions, 596
Periodicity-based identity, 669
Periodicity-reducing identity, 669
Perpendicular lines, 148
Piecewise linear function, 161
Points, 347
Point-slope form of lines, 149
Polynomial approximations
explanation of, 919–920
of sin x around x = 0, 920–924
Taylor, 924–931
of trigonometric functions, 693–694
Polynomials
characteristics of and differentiation of,
383–387
critical points of, 386–387
cubics and, 373–377
degree of, 380, 1063
of even and odd degrees, 385
explanation of, 373, 379–380, 385–386,
1063
graphs of, 384–385, 391–399
long division of, 383
overview of, 373
zeros of, 380–383, 391
Population growth
differential equations and, 503–506, 984
growth equations and, 306
instantaneous rate of change and, 210
logistic, 1008–1011, 1019–1020, 1024
Population interactions, 1038–1040
Position versus time, 210
Positive integers, 295–297, 1063
Power-reducing formula, 669
Power series
convergence and, 945–946, 956, 1129
definition of, 945
differential equations and, 959–961
differentiation and integration of,
956–959
manipulating, 956
Uniqueness Theorem and, 945
Predictions, 139–143
Present value, 582–585
Probability density function, 903
Production cost versus amount produced,
210
Product Rule See also Integration by parts
differentiating f (x) · g(x), 292–294
explanation of, 294
proof of, 295
use of, 297, 513, 543, 548, 692, 805
Proof by induction, 1095–1097
Proportionality
constant of, 306n
direct, 16 explanation of, 306
Ptolemy, 95n, 627
Pythagoras, 95, 97 Pythagorean identities explanation of, 599, 669, 709 trigonometric identities and, 886, 887 use of, 668
Pythagoreans, 89, 95–97 Pythagorean Theorem historical background of, 95 Law of Cosines and, 658 use of, 554, 599, 631, 635, 637, 672 Quadratic equations
explanation of, 1073–1077 factoring, 1077–1078 Quadratic formula, 1074, 1076–1077, 1080 Quadratics
calculus perspective to, 217–221 definition of, 219
disguised, 1080–1081 examples of, 217–219 explanation of, 373 factoring, 1066–1068 free fall and, 237–240 graphs of, 219, 221, 223, 231–235 noncalculus perspective to, 223–225 overview of, 217
use of quadratic formula to solve, 1076–1077
zero of, 381 Qualitative analysis, 1002–1014 Quotient Rule
application of, 337, 522, 692, 1115 explanation of, 297–298
Radians converted to degrees, 621 converting degrees to, 621 definition of, 620 degrees vs., 685 Radicals, 1081–1083 Radioactive decay differential equations and, 503, 504 rate of, 322–324
Radio waves, 605n
Radius of convergence, 945 Range
of functions, 5, 17–21
of trigonometric functions, 598 Rate of change
average, 73–76, 170, 176 constant, 143–144, 169, 712–714 decomposition to find, 121 implicit differentiation and, 550–554 instantaneous, 170–178, 209–211, 523,
711, 983 interpreting slope as, 153
nonconstant, 715–718 predictions and, 140–142
of quadratics, 219 Rational functions asymptotes and, 407–409 decomposition of, 898–900 discontinuity and, 407–410 explanation of, 406–407 graphs of, 410–417 Rational numbers, 1063 definition of, 87 historical background of, 95, 96 Ratio Test, 973–974, 977 Real numbers, 87, 1063 Real number system historical background of, 95–96 irrationality and, 96–99 Reciprocal functions explanation of, 62 graphs of, 130–131 Reduction formula derivation of, 880–882, 888 use of, 888–890
Relation, 988n
Relative maximum point, 347 Relative minimum point, 347 Removable discontinuities, 407 Revolution, 856–862
Richter scale for earthquakes, 448 Riemann, Bernhard, 727, 955 Riemann sums
explanation of, 727–729
limit of, 728n, 831, 1097
to obtain approximations of definite integrals, 806
use of, 828, 830, 832, 866 Right angles, 620
Right triangles applications for, 627–628 definitions of, 628–630, 707
45◦, 45◦, 635–637
30◦, 60◦, 637–643 trigonometry of, 631–633
Rolle, Michel, 1089n
Rolle’s Theorem, 1087, 1089–1091, 1127, 1128
Roots of multiplicity, 380 Root Test, 975, 977 Sandwich Theorem, 272–273, 757, 944 Secant, 629
Secant line, 75, 208 Second derivative, 221 Second derivative test, 357, 358 Second order differential equations with constant coefficients, 1045–1049 explanation of, 988
Separable differential equations, 1018– 1022
Trang 101140 Index
Sequences, 964, 965, 1129–1131
Series
absolute and conditional convergence
and, 952–953
alternating, 953–956
binomial, 946–949
convergence tests and, 962, 964–977,
1129–1131
geometric, 566–570, 579–586, 962, 964
Maclaurin, 941–942, 944, 946–948, 953,
957, 961
power, 945–946, 956–961
Taylor, 941–944, 947–949, 962
Shifting, 126–129
Shrinking, 126–129
Signed area, 727, 732
Similar triangles, 628
Simple root, 380
Simpson’s Rule
explanation of, 820–821
requirements for, 822–823
use of, 821–823
Sine functions See also Trigonometric
functions
definition of, 594
domain and range of, 598
graphs of, 597–598, 603–609
symmetry properties of, 598–599
Sines, Law of, 659–660, 664–665, 709
Singh, Jai, 627
Sinusoidal functions, 603–604
Sinusoidal graphs, 606–607
Slicing
to find area between two curves, 843–845
to find mass when density varies,
827–838
to find volume, 853–856
to help with definite integrals, 846–850
Slope
of f, 194–196
of line, 145–150
modeling and interpreting, 153–155
of tangent line, 208, 684
Slope function
explanation of, 161
of f (x), 188, 189
Slope-intercept form of lines, 149
Solution curves
examples of, 1005, 1006, 1010, 1027
explanation of, 501–503
Solutions
to differential equations, 498–503, 988,
991–997, 1002–1014
equilibrium, 989, 1004–1008
to trigonometric equations, 651–655
Soper, H E., 1037
Splitting interval property, 738
Square roots, 1054–1055
Squaring function explanation of, 61 graph of, 63 Squeeze Theorem, 272, 757 Stable equilibrium, 106–108 Standard position, 620 Stationary points, 350 Stifel, Michael, 96 Stretching, 126–129 Substitution
to alter form of integral, 798–802
in definite integrals, 794–796 explanation of, 787
mechanics of, 790–791 Taylor series and, 947–949 trigonometric, 890–894 used to reverse Chain Rule, 792–794 Subtraction
of functions, 101–103 principles of, 1059–1061 Subtraction formulas, 668–669 Successive approximations, 715–718 Summation notation
examples of, 576–577 explanation of, 575–576 use of, 769
Sum Rule, 291, 292 Symmetry even and odd, 65, 386–387 polynomials and, 386–387
of trigonometric functions, 598–599, 623–624
Symmetry-based identity, 669 Symmetry property, 738 Tangent functions definition of, 615 geometric interpretation of, 615 graph of, 615–618
period of, 616 Tangent line
to f at x = a, 176 slope of, 208, 684 Tangent line approximations, 282–284, 920–921, 937
Tartaglia, Niccolo Fontana, 95, 382 Taylor, Brook, 694, 920
Taylor expansions, 947 Taylor polynomials centered at x = 0, 925–928, 936, 948 centered at x = b, 928–931, 934, 941 definition of, 925
examples of, 935–939 Taylor remainders, 934–935, 942 Taylor series
convergence and, 944n, 947, 949
explanation of, 941–944 obtaining new, 947–949 substitution and, 947–949
Taylor’s Inequality, 935 Taylor’s Theorem convergence and, 962 definition of, 935 proof of, 1127–1128 use of, 934–935, 937, 942 Temperature change, 73–74
Theon of Alexandria, 95n
Theorem on Differentiation of Power Series, 956, 959
Total cost function, 102, 105, 406 Trajectories, 1027–1036 Trapezoidal sums explanation of, 808 use of, 812–815, 820 Triangles
historical background of, 627 oblique, 657
perspective on, 627–628 similar, 628
solving, 631 trigonometry of general, 657–665 trigonometry of right, 627–633, 635–643
(See also Right triangles)
Trigonometric equations explanation of, 651 solutions to, 651–655
Trigonometric functions See also specific
functions
of angles, 622–723
on calculators, 605n, 608, 647
definitions of, 594 differentiation of, 683–698, 703–706 domain and range of, 598
f (x) = tan x, 615–618 graphs of, 597–598, 603–609, 616–618,
647, 708 inverse, 645–649, 703–706, 708 periodicity and, 596
polynomial approximations of, 693–694 properties of, 594–595
summary of, 707–709 symmetry properties of, 598–599, 623–624
Trigonometric identities addition formulas and, 668–669 explanation of, 599–600, 709 summary of, 669
trigonometric integrals and, 886–887 use of, 600–601, 667, 697
Trigonometric integrals explanation of, 886 miscellaneous, 890 sin x and cos x, 886–888 tan x and sec x, 888–890 Trigonometric substitution, 890–894 Trigonometry
angles and arc lengths and, 619–624 applied to general triangle, 657–665