无穷小带来什微积分么大礼包

无穷小带来什微积分么大礼包？
    当今，超实无穷小在我国普通高校的校园中四处游荡，散发微积分“大礼包”
，帮助男、女同学学习微积分。
   微积分大礼包里面有40余个微积分重要定理、600多张精心绘制的示意图片、800多道练习题以及完备的微积分名词索引（微积分大百科全书）。
    坦率地说，学微积，用手机，是无穷小的大功劳。
袁萌  陈启清 4月10日
附件：无穷小带来的大礼包，
电子版微积分教材的名词索引，一千余条。
Index
Absolute convergence, 521 trapezoidal, 226 surface, 825 Absolute value, 12 within E, 284 surface of revolution, 328 of a complex number, 876 Arc, 323, 365 under a curve, 188 Absolute value function, 12 length, 324, 366, 420 Argand diagram, 876 Acceleration, 94 arccos, 384 Argument, complex number, vector, 565, 623 Arccosecant, 384 876 Addition formulas for sine derivative, 387 Associative Law, 905 and cosine, 372 Arccosine, 384 inner product, 597 Addition Property, 190, 303 derivative, 387 scalar multiples, 570 variables integral, 392 vector product, failure, 602 two, 712, 720 arccot, 384 vector sum, 568 three, 760 Arccotangent, 384 Asymptote Adiabatic process, 490 derivative, 387 of a hyperbola, 270 Almqst parallel vectors, 632 arccsc, 384 vertical, 251 Alternating series, 517 Archimedes, 902 Average slope, 22, 168 harmonic, 520, 546 arcsec, 384 Average speed, 339 Alternating Series Test, 518 Arcsecant, 384 Average value, 337 Amplitude, 886 derivative, 387 Average velocity, 169, 339 Angle, 77, 365 integral, 473 Axioms between vectors, 573 arcsin, 384 for hyperreal numbers, 906 three dimensions, 588 Arcsine, 384 for real numbers, 905 Angular measure, 367 derivative, 387 Axis Anticommutative Law for integral, 392 coordinates vector products, 604 power series, 560 plane, 3 Antiderivative, 192 arctan, 384 space, 585 Approximation Arctangent, 384 ellipse, 264 by alternating series, 518 derivative, 387 hyperbola, 268 arctan x, 541 integral, 472 parabola, 257 by derivatives, 286 power series, 536 e, 442, 551 Area Barrow, Isaac, 902 In x, 541 below a curve, 188 Basis vectors by Newton's method, 290 between two curves, 220, plane, 570 pi, 542 304 space, 587 by power series, 541 double integral, 722 Bell-shaped curve, 537 by Riemann sums, 178 geometric figures, 115 Berkeley, Bishop, 903 Simpson's, 230 by Green's Theorem, 818 Binomial series, 558 sin x, 552 infinite, 353 Binomial Theorem, 559 by Taylor's Formula, 547 polar, 421 Bound variable, 119, 178
A 57
A 58 INDEX
Boundary oriented surface, 834 plane region, 689 solid region, 757, 838 Bounded open region, 694 Box cylindrical, 770 rectangular, 758 spherical, 776
Cantor, Georg, 903 Capital Accumulation, 452 Cardioid, 751 Catenary, 449 Cauchy, A. L., 903 Cauchy Convergence Test (Criterion), 500, 504 Cauchy Divergence Test, 505 Cauchy equivalence, 911 Cauchy sequence, 911 Center circle, 5 ellipse, 264 hyperbola, 268 sphere, 641 Center of gravity, 344 Center of mass double integral, 745 triple integral, 765 two dimensions, 346 wire, 344 Centroid, 346 polar region, 430 Chain Rule for continuous functions, 130 variables one, 86 two, 672 Change of variables, 210, 484 Changing base of logarithms, 439 Characteristic polynomial, 882 Circle, 5 area, 115 polar equation, 410 Circular cylinder, 644 Circular helix, 616 Circulation dimensions two, 820 three, 836 Circumference, 115, 324 Circumscribed cylinder, 721 Circumscribed rectangle, 189 Circumscribed rectangular box, 759 cis, 876 Closed interval, 2 hyperreal, 908
Closed rectangle, 713 Closed region, 689 hyper real, 700 space, 757 Closure Laws, 905 Commodity vector, 565 Common logarithm, 436 Commutative Law, 905 inner product, 597 vector product, failure, 602 vector sum, 568 Comparison test, 512, 524 Completeness axiom, 905 Completing the square, 274, 479 Complex number, 874 conjugate, 875 exponent, 879 plane, 874 Complex valued differential equation, 879 Complex valued function, 879 Component directed line segment, 564 vector, 565 valued function, 615 Composition of functions, 86 continuity, 130, 653 derivative, 86, 672 Concave downward, 152 Concave upward, 152 Conditional convergence, 521 Cone area, 327 elliptic, 642 volume, 115 Conic section, 264 Conjugate axis, 268 Conjugate complex, 875 Conservative field, 807 Constant, II Constant function, II derivative of, 61 integral of, 184 Constant of integration, 199 Constant on an interval, 151 Constant Rule derivative, 61 double integral, 732 integral, 200 series, 508 Continuity, 125 of compositions, 130, 653 of differentiable functions, 127 E, o condition, 288 of an integral, 192 on an interval, 132 law of, 902 on a set, 653
two variables, 651 vector, 635 Continuously compounded interest, 443, 497 Contour, 645 Contour map, 644 Convergence improper integral, 352 increasing sequence, 511 interval of, 529 positive term series, 511 radius of, 530 sequence, 493 series, 502 summary of tests, 525 Coordinates cylindrical, 769 polar, 407 rectangular plane, 3 space, 585, 639 spherical, 775 cos, 78, 368 Cosecant function, 370 derivative, 376 cosh, 449 power series, 536 Cosine, 78, 368 derivative, 79, 374 integral, 377 power series, 558 Cosines, Law of, 573 cot, 370 Cotangent function, 370 derivative, 376 coth, 450 Critical damping, 887 Critical point, 136 interior, 137 two variables, 689 Critical Point Theorem, 136 two variables, 689 Cross product, 600 esc, 370 csch, 450 Curl dimensions two, 819 three, 832 Curve length, 320 parametric, 321 polar coordinates, 425 in space, 622 Curve sketching using derivatives, 156 ellipse, 266 hyperbola, 269 using limits, 251 parabola, 261 polar, 415
INDEX A 59
rotation of axes, 281 on an interval, 132 infinite, 719 translation of axes, 275 two variables, 710 Dummy variable, 119, 178 Cycloid, 616 Differential, 55 Cylinder exact, 806 e, 83, 443 area, 327 form, 806 irrationality of, 563 circular, 644 second, 94 as a limit, 443 quadric, 641 total, 662 Element volume, 115, 309 vector, 633 of area, 736 Cylinder Property, 712, 721 Differential equation of a set, 2 Cylindrical box, 770 complex valued, 879 of volume Cylindrical coordinates, exact, 810 cylindrical, 772 769 first order, 461, 846 rectangular, 764 Cylindrical Integration homogeneous, 852, 882 spherical, 777 Formula, 771 linear, 857 Elementary function, 481 Cylindrical region, 770 second order, 461, 881 Ellipse, 264 Cylindrical shell method, 313 separable variables, 848 sketching method, 266 Differentiation, 45 Ellipsoid, 642 Damped oscillation, 887 logarithmic, 470 Elliptic cone, 642 Damping, critical, 887 rules for, 68 Elliptic cylinder, 641 Decreasing on an interval, 152 Direct Test, 137 Elliptic paraboloid, 642 Definite integral, 183 Directed curve, 795 Empty set, 3 Degenerate conic section, 263, Directed line segment, 564 Endpoint, 2 273 Direction angle, 574, 588 Epidemic, 463 Degree, angular unit, 367 Direction cosine, 574, 588 t, o condition Deleted neighborhood, 243 Direction, real, 630 continuity, 288 Demand function, 114 Direction vector derivative, 295 De Moivre's Formula, 878 of a line, 578, 590 limits, 283, 286 Density of a plane, 606 t, N condition dimensions Directional derivative, 786 for infinite limits, 499 one, 341 second, 793 for sequences, 499 two, 342 three dimensions, 791 Equality of mixed partials, three, 764 Directrix, parabola, 257 703 of a wire, 364 Disc method, 309 three variables, 705 Dependent variable, 8, 45 Discontinuity, 125 Equation, 906 Derivative, 45 Discriminant, 875 differential, 461, 846 directional, 786 test, 273 linear, 20 exponential function, 83, two variables, 273 parametric, 90 443 Displacement vector, 565 point-slope, 16 higher, 94 Distance second degree, 272 higher partial, 703 plane, 4 two-point, 18 hyperbolic functions, 449 between reals, 13, 282 Error estimate, 228, 296 inverse trig functions, 387 space, 641 for derivatives, 296 In, 84, 455 Distributive Law, 905 for Simpson's Rule, 231 mixed partial, 703 scalar multiples, 570 and Taylor's Formula, 550 partial, 656 products for trapezoidal sums, 228 polar coordinates, 414 inner, 597 Escape velocity, 491 power rule, 63, 76 vector, 602 Euler approximation, 865 power series, 534 Divergence Euler's Formula, 879 rational function, 68 dimensions Even function, 218 rules for, 68 two, 819 Exact differential, 806 sec and esc, 376 three, 833 equation, 810 second, 94 Theorem, 839 Existence Theorem, differensecond partial, 703 Divergent tial equations, 867 sin and cos, 374 improper integral, 352 exp, 443 tan and cot, 376 sequence, 494 Explosion, 851, 871 vector, 620 series, 502 Exponent Determinant, 600 Domain, 10 complex, 879 Difference, vector, 569 Dot product, 593 inequalities, 432 Differentiable function, 45 Double integral, 711,719 real, 432 continuity of, 127 Double Riemann sum, 715 rules, 432
A60 INDEX
Exponential function, 83, 443 with base, 82, 432 continuity, 433 derivative, 83, 443 integral, 444 power series, 536 Exponential spiral, 448 Expression, real, 28, 906 Extension Axiom, 906 Extension Principle, 27 Extreme Value Theorem, 164 two dimensions, 689
Factor Theorem, 477 Factorial, n!, 493 Fermat, P., 902 Finite hyper real number, 30 partition, 160 Riemann sum, 178 set, 3 vector, 627 First Derivative Test, 174 First order differential equation, 461 Fluid flow, 820 Flux, 821 Focus ellipse, 264 hyperbola, 268 parabola, 257 Force vector, 565 Forcing term, 892 Formula, 45, 906 Frequency, 886 Frustum, area, 328 Fubini's Theorem, 726 Function, 45 hyperreal, 27 implicit, 97 inverse, 70, 381 linear, 17 real, 6 two variables, 14 vector-valued, 615 Fundamental Theorem of Calculus, 193
Galileo, G., 902 Gauss' Theorem, 839 General solution, 461, 846 Generalized Mean Value Theorem, 552 Geometric series, 503 formula, 446 Godel, Kurt, 904 grad f, 787 Gradient, 787 Graph, 7 dimensions
one, 7 two, 640 of an equation, 5 infinite sequence, 492 polar coordinates, 408 space, 639 Gravity dimensions one, 491 two, 809 three, 843, 844 Greatest integer function, 130 Green's Theorem, 815 vector form, 820 Growth rate, 51 populations, 51, 462, 491 sequences, 497
Half-life, 463 Half-open interval, 2 Harmonic series, 505 alternating, 546 Helix, circular, 616 Hewitt, E., 904 Higher derivative, 94 partial, 703 Higher differential, 94 Homogeneous differential equation, 852 second order, 8 82 solution, 852, 886 Homogeneous function, 710 Horizontal axis, 3 Horizontal ellipse, 266 Horizontal parabola, 261 Hyperbola, 268 graphing method, 270 Hyperbolic cosine, 449 derivative, 449 integral, 449 Hyperbolic cylinder, 641 Hyperbolic function, 449 Hyperbolic identities, 449 Hyperbolic paraboloid, 643 Hyperbolic sine, 449 derivative, 449 integral, 449 Hyperboloid, 643 Hyperinteger, 160 Hyperrational number, 431 Hyperreal function, 27 Hyper real interval, 161, 908 Hyperreal number, 27 Hyperreal region, 697 Hyperreal vector, 627
i, 874 Identity function, 12 Identity Law, 905 vector sum, 568
Imaginary number, 874 Imaginary part, 874 Implicit differentiation, 97 Implicit function, 97 derivative of, 97, 680 two variables, 680 Implicit Function Theorem, 680 Improper integral, 352 Incompressible field, 822 Increasing function, 151 Increasing sequence, 511 Increment, 45 two variables, 662 vector, 633 Increment Theorem, 54, 299 variables two, 664 three, 670 Indefinite integral, 199 Ind_ependent variable, 8, 45 Indeterminate form, 31 Induction, Principle of, 64 Inequalities for exponents, 432 Inequality, 906 Inequality Rule double integral, 733 series, 508 Inertia, moment of dimensions two, 748 three, 767 Infinite hyperreal number, 24 examples, 912 Infinite partial sum, 502 Infinite partition, 161 Infinite Riemann sum, 181 double, 719 Infinite sequence, 492 Infinite series, 502 Infinite Sum Theorem, 303 double integral, 736 triple integral, 764 Infinite telescope, 24 Infinite vector, 627 Infinitely close, 24, 35 compared to, 302 two dimensions, 651 vectors, 627 Infinitesimal, 24, 28 examples, 913 microscope, 24 vector, 627 Inflection, point of, 156 Initial condition, 462, 847, 881 Initial point directed line segment, 564 smooth curve, 795 Initial value problem first order, 847
INDEX A61
second order, 881 double, 725 Local maximum, 156 Inner product, 594 triple, 761 Local minimum, 156 derivative, 625 Locus rules, 597 Judicious guessing, 893 of an equation, 7 Inscribed cylinder, 721 Justification of a definition, polar coordinates, 408 Inscribed rectangle, 189 304 space, 639 Integers, 3 Kepler, J., 902 Logarithm Integral changing base, 439 arcsec and arccsc, 473 Kinetic energy, 784 common, 436 arcsin and arccos, 392 Kline, Morris, 907 natural, 83, 454 arctan and arccot, 472 Law of continuity, 902 rules, 437 definite, 183 Law of Cosines, 573 Logarithmic differentiation, double, 711, 719 Least squares, method of, 702 470 exponential function, 443 Leibniz, Gottfried Wilhelm, Logarithmic function, 436 improper, 352 902 Logistic function, 464 indefinite, 199 Length Lower endpoint, 2 iterated, 724 arc, 366 line, 795 curve, 320 MacLaurin series, 555 In, 456 parametric, 321 MacLaurin's Formula, 550 partial, 798 space, 622 Magnitude, 567 power series, 534 polar coordinates, 425 Major axis, ellipse, 264 rational functions, 477 real, 630 Malcev, A., 904 sin and cos, 377 vector, 566 Map, 209 sinh and cosh, 449 Level curve, 645 Marginal price vector, 624 surface, 829 Level surface, 791 Marginal revenue, 114 tan, 471 !'Hospital's Rule Marginal values, 51 triple, 760 for 0/0, 243 Mass Integral Test, 514 for oo/oo, 246 dimensions Integration Limit, 117 one, 341 change of variables, 210, and curve sketching, 251 two, 342 484 E, o condition, 283, 286 double integral, 743 coordinates finite, 117 triple integral, 764 cylindrical, 771 from the left, 123 Mass-spring system, 888 polar, 751 from the right, 123 Maximum, 135 spherical, 777 infinite, 237 closed region, 691 methods of, 481 one-sided, 123 local, 156 by parts, 391, 394 of a Riemann sum, 188, 316 open region, 698 substitution, 210, 484 rules, 121 two variables, 688 trigonometric, 403, 485 sequence, 493 value, 135 Interest, 442, 451 and standard parts, 120 Mean Value Theorem, 168 Interior point, 137, 689 two variables, 656 generalized, 552 critical, 137 Limit Comparison Test, 513, for Integrals, 338 Intermediate Value Theorem, 524 Member of a set, 2 162 Line, 16 Method of least squares, 702 Intermediate variable, 671 equation of, 16, 18, 20 Methods of integration, 481 Interval, 2 polar equation, 409 Midpoint, 582, 592 of convergence, 529 space, 589 Minimum, 135 hyperreal, 161, 908 vector equation, plane, 578 local, 156 Inverse function, 70, 381 vector equation, space, 589 two variables, 688 hyperbolic, 461 vertical, 16 value, 135 trigonometric, 384 Line integral, 795 Minor axis, ellipse, 264 Inverse Function Rule, 72 Linear differential equation, Mixed partial derivative, 703 Inverse Function Theorem, 857 Model theory, 904 386 second order, 892 Moment Inverse Laws, 905 solution, 858, 862, 897 dimensions Inverse relation, 3 81 Linear equation, 20 one, 343 Inverse square law, 491 Linear function, 17, 60 two, 345 Irrotational field, 822 In, 83, 454 double integral, 744 Iterated integral, 724 integral, 456 triple integral, 765 Iterated Integral Theorem power series, 536 wire, 344
A62 INDEX
Moment of inertia (second moment), dimensions two, 748 three, 767
Natural extension, 906 Natural logarithm, 81, 454 derivative, 84, 455 integral, 456 Negative infinite, 24, 30 Negative infinitesimal, 28 Neighborhood, 243 deleted, 243 Newton, Sir Isaac, 902 Newton's First Law, 567 Newton's Law of Cooling, 490 Newton's method, 290 Newton's Second Law, 570 Nonstandard analysis, 904 Nonuniqueness, differential equation, 872 Norm of a vector, 566 Normal curve, 537 Normal vector curve, 606 surface, 828 nth derivative, 94 nth differential, 94
Odd function, 218 One-to-one function, 382 Open interval, 2 Open rectangle, 794 Open rectangular solid, 803 Open region, 694 Ordered pair, 3 Oriented surface, 828 Origin, 3 in polar coordinates, 412 Orthogonal vectors, 597 Oscillation damped, 887 simple, 886 Overdamping, 888
pseries, 515 Parabola, 257 graphing method, 261 Parabolic cylinder, 641 Paraboloid elliptic, 642 hyperbolic, 643 Parallel line and plane, 609 Parallel planes, 609 Parallel, test for, 597 Parallel vectors, 597 Parallelogram, 583 area, 604 Parametric curve, 90
Parametric equations, 90 of a line, 579, 590 polar coordinates, 411 and vector equations, 579, 615 Partial derivative, 656 higher, 703 implicit function, 686 mixed, 703 second, 703 three variables, 659 Partial fraction, 477 Partial integral, 798 derivative of, 811 Partial second derivative, 703 Partial sum, 502 Particular solution, 846 Partition finite, 160 infinite, 161 of a rectangle, 715 of a rectangular box, 759 Partition point, 161, 176 Path Independence Theorem, 806 Perimeter, 115 Period, 886 Periodic function, 368 Perpendicular line and plane, 609 Perpendicular planes, 609 Perpendicular, test for, 597 Perpendicular vectors, 597 Phase shift, 886 Pi, 1r, 366 Piecewise continuous, 357 Piecewise smooth, 800 Plane, 4, 604 Point, 3 Point of inflection, 156 Point-slope equation, 16 Polar coordinates, 407 area, 421 Polar equation circle, 410 line, 409 Polar form, complex numbers, 876 Polar Integration Formula, 751 Polar rectangle, 7 50 Polar region, 750 Polynomial, 60 function, 60 Population growth, 434, 442, 463 Position, 203 Position vector, 565 of a curve, 615 of a line, 578, 590
of a plane, 606 of a point, 576, 589 Positive infinite, 24, 30 Positive infinitesimal, 28 Positive term series, 511 Potential function, 805 existence, 806 method for finding, 807 three variables, 814 Potential energy, 807 Power Rule derivatives, 63 general form, 89 integrals, 200, 471 negative exponents, 66 rational exponents, 76 Power series, 528 multiplication of, 563 operations on, 535 steps for finding, 557 table, 556 Present value, 454 Pressure, 364 Price vector, 565 Principle of Induction, 64 Principle of Superposition, 863, 896 Prism, volume, 115 Product function, 648 inner, 594 vector, 600 Product Law, order axiom, 905 Product Rule derivative, 62 vector derivative, 625 Profit, 137 Pyramid, volume, 306 Pythagoras, Theorem of, 5
Quadrant, 4 Quadratic formula, 103, 477 complex, 875 Quadric cylinder, 641 Quadric surface, 640 Quotient function, 652 Quotient Rule, derivatives, 66
R, I R*, 24 Radian, 78, 367 Radioactive decay, 463 Radius of circle, 5 of convergence, 530 of sphere, 641 Range, 10 Rate of growth, 51, 434, 442, 463
INDEX A63
Ratio Test, 524 Root, 2 Set, 2 Rational function, 60, 474 Root Axiom, 905 Side condition, 692 differentiation, 68 Rotation of axes, 280 Simple closed curve, 801 integration, 477 equations, 277 Simple oscillation, 886 Rational number, 3 Simpson's approximation, 230 Rational term, 60 Saddle point, 690 Simpson's Rule, 231 Real direction, 630 Scalar, 566 sin, 77, 368 Real expression, 906 Scalar Associative Law, 570 Sine, 77, 368 Real length, 630 Scalar equation derivative, 79, 374 Real line, 1 of a line, 577 integral, 377 Real number, 1 of a plane, 604 power series, 558 Real part, 874 Scalar multiple, 570 sinh, 449 Real statement, 907 derivative, 627 derivative, 449 Real term, 906 Scalar product, 593 integral, 449 Real vector, 630 Scalar triple product, 604 power series, 537 Rearrangement, series, 523 Schwartz' Inequality Skolem, Thoralf, 903 Reciprocal function, 8 integrals, 236 Slide rule, 438 Rectangle vectors, 637 Slope area, 115 sec, 370 average, 22, 168 closed, 713 Secant function, 370 curve, 25, 43 open, 794 derivative, 376 directional, 786 polar, 750 integral, 471 function, 43 Rectangle Property, 188 Secant line, 295 implicit function, 680 Rectangular box, 758 sech, 450 as a limit, 119 Rectangular coordinates Second degree equation line, 16 plane, 3 variables polar coordinates, 413 space, 585, 639 two, 272 Slope-intercept equation, 17 Rectangular solid, 115, 758 three, 640 Smooth curve, 319, 795 Reduction formula Second derivative, 94 piecewise, 800 secant and cosecant, 399 partial, 703 space, 803 sine and cosine, 398 Second Derivative Test, 138 Smooth function, 662, 824 tangent and cotangent, 397 two variables, 707 continuity of, 666 Region Second differential, 94 Smooth surface, 662, 824 below a curve, 175 Second order differential Snail, 448 between two curves, 218, equation, 461 Solenoidal field, 822 717 Second partial derivative, 703 Solid region, 757 bounded, 694 Sector, 365 Solution, differential closed, 689 area, 115, 324, 366, 420 equation, 461, 846 cylindrical, 770 Separable variables, 465, 848 Speed, 623 hyperreal, 697 solution method, 848 Sphere, 641 open, 694 Sequence surface area, 115, 333 polar, 750 graph of, 492 volume, 115, 317 solid, 757 increasing, 511 Spherical box, 776 spherical, 775 infinite, 492 Spherical coordinates, 775 unbounded, 694 of partial sums, 502 Spherical Integration under a curve, 175 Series, 502 Formula, 777 Related rates, 112 absolutely convergent, 521 Spherical region, 775 Relativity, theory of, 239 alternating, 517 Spiral Resonance, 897 conditionally convergent, of Archimedes, 411, 474 Resonant frequency, 900 521 exponential, 448 Revenue, 114 convergent, 502 vertical, 616 Riemann sum, 176, 178 divergent, 502 Spring constant, 888 double, 715 of functions, 528 Square function, 8 finite, 178 infinite, 502 Square root, 2 infinite, 181 positive term, 511 st, 36 infinite double, 719 sum, 502 Standard part, 36 triple 759 summary of tests, 525 function, 910 Right-handed coordinates, 585 tail, 508 principle, 36, 908 Robinson, Abraham, 904 Taylor, 554 rules, 37 Rolle's Theorem, 165 telescoping, 506 vector, 629
A64 INDEX
Statement, real, 907 in space, 621 dependent, 8, 45 Steady state part of the and vector derivatives, 620 dummy, 119, 178 solution, 899 Tangent plane, 666 independent, 8, 45 Steepest ascent, 788 of an implicit function, 686 intermediate, 671 Steepest descent, 788 Tangent vector, 621 point, 578 Stokes' Theorem, 834 tanh, 450 vector, 578, 589 Straight line, 16 Tarski, Alfred, 904 Variation of constants, 857 in space, 589 Taylor's Formula, 550 Vector, 565 Strictly between, 2 Taylor polynomial, 549 continuous, 635 Strictly within, 282 Taylor remainder, 549 derivative, 620 Subinterval, 160 Taylor series, 554 difference, 569 Subrectangle, 715 table of, 556 differential, 633 Substitution, integration by, Telescoping series, 506 dimensions 210, 484 Term, 9 n, 586 trigonometric, 402 real, 906 two, 565 Sum Terminal point three, 586 function, 648 directed line segment, 564 direction, 578, 606 partial, 502 smooth curve, 795 equation, 615 Riemann, 178 Third partial derivative, 704 of a line, 578, 589 series, 502 Topographic map, 644 of a plane, 606 trapezoidal, 226 Torus, 336 field, 805 vector, 567 Total differential, 662 finite, 627 Sum Law, order axiom, 905 three variables, 669 function, 615 Sum Rule Transfer Axiom, 908 h yperreal, 627 derivatives, 61 Transfer Principle, 28 increment, 633 double integral, 732 Transient part of solution, 899 infinite, 627 integral, 200 Transitive Law, 905 infinitesimal, 627 series, 508 Translation of axes, 275 negative, 569 use in integration, 483 Transverse axis, 268 normal, 606, 828 vector derivative, 625 Trapezoidal approximation, position, 565 Surface 226 product, 600 area, 328, 825 Trapezoidal Rule, 228 derivative, 627 area, polar, 428 Trapezoidal sum, 226 real, 630 as a boundary, 834 Triangle, area, 115 rules, 625 contour map of, 644 Triangle inequality, 568 sum, 567 implicit, 686 Trichotomy Law, 905 valued function, 615 integral, 829 Trigonometric functions, 365 variable, 578, 589 level, 791 Trigonometric identities, 372 Velocity, 23 oriented, 828 Trigonometric substitution, average, 169 quadric, 640 402 and integration, 203 of revolution, 327 Triple integral, 760 vector, 565, 622 sketching method, 645 Triple Riemann sum, 759 Vertex, parabola, 257 smooth, 662, 824 Two-point equation, 18 Vertical axis, 3 tangent plane of, 666 Vertical ellipse, 266 topographic map of, 644 U1traproduct, 903 Vertical line, 16 Symmetric region, 784 equivalence, 912 Vertical parabola, 259 Symmetry and center of mass, Unbounded open region, 694 Volume, 305 347 Undefined term, 6 below a surface, 723 System of formulas, 45 Undetermined coefficients, between two surfaces, 738 893 cylindrical shell method, Tail rule, 508 Uniqueness Theorem 313 Tail, series, 508 differential equations, 870 disc method, 309 tan 79, 370 double integrals, 722 as a double integral, 723 Tangent of an angle, 79 Unit circle, 365 element of, 764 Tangent function, 370 Unit hyperbola, 450 function, 711, 722 derivative, 376 Unit vector, 570, 588 geometric figures, 115 integral, 471 Upper endpoint, 2 as the integral of area, 305 Tangent line, 53 solid of revolution, 309, 313 of an implicit function, 686 Variable, 8, 45 as a triple integral, 763, 778 in polar coordinates, 412 bound, 119, 178 under a surface, 723
Water pressure, 364 Wave equation, 708 Weierstrass, Karl, 903 Work against gravity, 348 as an inner product, 595 as a line integral, 796 as a single integral, 349
x-axis, 3 x-component, 564, 586 x-coordinate, 3 (x,y) plane, 3, 639 (x,z) plane, 639 (x,y,z) space, 585, 639
y-axis, 3
INDEX
y-component, 564, 586 y-coordinate, 3 (y,z) plane, 639
z-axis, 585, 639 z-component, 586 Zero of a function, 161 Zero vector, 567
A65
RUlES OF INTEGRATION I du = u + C
I du + dv = I du + I dv (Sum Rule)
TABlE OF INTEGrlAlS
(The constant of integration is left out.) I x-+' x' dx = -- r # -I r +I' I ex dx =ex I sinx dx = -cosx I tanx dx = In lsecxl I secx dx = lnlsecx + tanxl
I sec2 x dx = tanx I tanx secx dx = secx J sin2 x dx = -!x --! sinx cosx I I ll- I I sin"xdx= --sin"- 1 xcosx+-- sin"-2 xdx ll ll ftan"x dx = - 1 - tan"- 1 x- Itan"-2 x dx n - I
I
sec"xdx =
1 -sec"- 1 xsinx + n
2
Isec"-2 xdx
n - I n - I I arcsinx dx = x arcsinx + F- x2
I arctanx dx = x arctanx --!In(! + x2)
I arcsecx dx = x arcsecx - ln(x + JxZ=!J, x :2: I I ~=arcsinx I ~ = arcsecx lx1Jx2 - I
I k du = k I du (Constant Rule)
I u dv = uv - I v du (Integration by Parts)
I [ 1 dx = lnlxl
I lnlxl dx = x lnlxl - x I cosx dx = sinx
I cotxdx = lnlsinxl f cscxdx = -lnlcscx + cotxl
I csc2 x dx = -cot x
Icotxcscxdx = -cscx
I cos2 x dx = tx + t sinx cosx
I
cos"x dx = ~ cos"- 1 x sinx + 11 - 1 Icos"-2 x dx ll ll
I
cot"x dx = -
1 - cot"- 1 x- Icot"~2 x dx n- I
I
csc"x dx = -
1 - csc"- 1 x cosx +
11

2 Icsc"-2 x dJ 11- 1 n- 1 I arccosxdx = xarccosx- ~ I arccotx dx = x arccotx +-!In(! + x2 ) I arccscx dx = x arccscx + ln(x + JxZ=!J, x :2: I I dx --2 = arctan x 1 + X
ELEMENTARY ALGEBRA OF THE REAL NUMBERS
a+b=b+a a + (b + c) = (a + b) + c a(b + c) = ab + ac a+0=a•1=a a-a=O
-(-a)= a
-(a- b)= b- a
a c ad + be b + d = bd (b,d #- 0)
a 1 =a !" = 1
a"'-" = a"'ja" (a #- 0) a'"b"' = (ab)m Fa = -.:ja, ( -a)m/n = ~' (a> O,n odd) If a < b then a + c < b + c. If a < b, then - b < -a. 1-al = lal lal • lbl = labl ~ = 1~1 (b #- 0). lbl b
The following terms are undefined: a/0, 0°, o-m, Fa (a, m positive, n even)
ab = ba a(bc) = (ab)c a(-b) = (-a)b = -ab a•O = 0 aja = 1 (a #- 0) 1 -=a 1/a (a#- 0) 1 b
ajb a
(a, b #- 0)
a c ac -.- = - (b, d #- 0) b d bd a0 = 1 (a #- 0) 0" = 0 (n > 0) amn = (amj' amJn = F (a > 0) .::fo.::jb = ;fob (a, b > 0)
If a < b and 0 < c, then ac < be. IfO <a< b, then 1/b < !fa.
-b ± jb2 - 4ac Quadratic F omwla: ax2 + bx + c = 0 if and only if x = ----'--2a
ALGEBRA OF HYPERREAL NUMBERS
Notation:
e, [J are positive infinitesimals b, c are positive and finite but not infinitesimal H, K are positive infinite.
The following are infinitesimal: -e, 1/H, ejb, sjH, bjH, E + 0, e - 0, E • 0, b • E, .:ji.
The following are finite but not infinitesimal: -b, ljb, bjc, b + e, b • c, .:jb, b + c b - c is finite (possibly infinitesimal).
The following are infinite: -H, lje, bfe, Hje, Hjb, H + e, H + b, H • b, H. K, .::(ii, H + K
The following can be infinitesimal, finite but not infinitesimal, or infinite: ejb, HjK, He, H - K
STANDARD PARTS
In the following b, c are finite (possibly infinitesimal).
st(b + c) = st(b) + st(c) st(bc) = st(b)st(c) st(jb) = ;rst{b) if b > 0, n > 0 b::::; st(b) b = st(b) if and only if b is real st(s) = 0, st(H) is undefined
TRIGONOMETRIC IDENTITIES
sin2 8 + cos2 0 = l cot>O + l = csc2 0 cos( ~e)= cosO cos(n/2 ~ 8) =sinO cos(O + r{!) = cosOcosr{! ~sinO sin¢
RULES OF EXPONENTS AND LOGARITHMS
Assume a > 0, b > 0.
1-' = l
a-'"= (ax)Y
alogaX = X
log.(xy) = log,x + log,y log.(x'') = y log,x
logby = log.y/log.b
TABLE OF DERIVATIVES
du = (dujdx) dx (Chain Rule) d(u + v) = du + dv (Sum Rule)
d (u) = v du ~ u dv ~ (Quotient Rule) v v2 d(ln u) = duju d(sin u) = cos u du d(tan tt) = sec2 u du d(sec u) = tan u sec u du du d(arcsinu) = ~ v 1 ~ !12 du d(arctanu) = -2 1 + ll du d(arcsecu) = ~ [u[.yu2 ~I
st(b ~ c) = st(b) ~ st(c) st(bjc) = st(b)jst(c) if st(c) oft 0 st(b') = st(b)"1'l if st(b) > 0 b ::::; c if and only if st(b) = st(c) if b s; c, then st(b) s; st(c)
tan 2 0 + I = sec2 e sin(~O) =~sinO sin(n/2 ~ 0) =cosO sin(O + ¢) = siniJcosrfJ +cosO sin¢
axbx = (ab)'
log.(ax) = x log.(xjy) = log.x ~ log.y bx = axlogab
d(ku) = k du (Constant Rule) d(tw) = u dv + v du (Product Rule)
d(u') = ru'- 1 du (Power Rule)
d(e") = e" du d(cos u) = ~sin u dtt d(cot u) = ~csc2 u dtt d(csc 11) = ~cot u esc u d11
du d(arccosu) = ~ --Jl-=7 du d(arccotu) = ~ -1 + 112 du d(arccscu) = ~ ---[u[p-=i
ILES OF INTEGRATION
'tl = ll + c 'u + dv = f du + f dv (Sum Rule)
.BLE OF INTEGRALS
te constant of integration is left out.)
xr+ 1 'dx = -- r oft -1 r + 1' X
Absolute convergence, 521 trapezoidal, 226 surface, 825 Absolute value, 12 within E, 284 surface of revolution, 328 of a complex number, 876 Arc, 323, 365 under a curve, 188 Absolute value function, 12 length, 324, 366, 420 Argand diagram, 876 Acceleration, 94 arccos, 384 Argument, complex number, vector, 565, 623 Arccosecant, 384 876 Addition formulas for sine derivative, 387 Associative Law, 905 and cosine, 372 Arccosine, 384 inner product, 597 Addition Property, 190, 303 derivative, 387 scalar multiples, 570 variables integral, 392 vector product, failure, 602 two, 712, 720 arccot, 384 vector sum, 568 three, 760 Arccotangent, 384 Asymptote Adiabatic process, 490 derivative, 387 of a hyperbola, 270 Almqst parallel vectors, 632 arccsc, 384 vertical, 251 Alternating series, 517 Archimedes, 902 Average slope, 22, 168 harmonic, 520, 546 arcsec, 384 Average speed, 339 Alternating Series Test, 518 Arcsecant, 384 Average value, 337 Amplitude, 886 derivative, 387 Average velocity, 169, 339 Angle, 77, 365 integral, 473 Axioms between vectors, 573 arcsin, 384 for hyperreal numbers, 906 three dimensions, 588 Arcsine, 384 for real numbers, 905 Angular measure, 367 derivative, 387 Axis Anticommutative Law for integral, 392 coordinates vector products, 604 power series, 560 plane, 3 Antiderivative, 192 arctan, 384 space, 585 Approximation Arctangent, 384 ellipse, 264 by alternating series, 518 derivative, 387 hyperbola, 268 arctan x, 541 integral, 472 parabola, 257 by derivatives, 286 power series, 536 e, 442, 551 Area Barrow, Isaac, 902 In x, 541 below a curve, 188 Basis vectors by Newton's method, 290 between two curves, 220, plane, 570 pi, 542 304 space, 587 by power series, 541 double integral, 722 Bell-shaped curve, 537 by Riemann sums, 178 geometric figures, 115 Berkeley, Bishop, 903 Simpson's, 230 by Green's Theorem, 818 Binomial series, 558 sin x, 552 infinite, 353 Binomial Theorem, 559 by Taylor's Formula, 547 polar, 421 Bound variable, 119, 178
A 57
A 58 INDEX
Boundary oriented surface, 834 plane region, 689 solid region, 757, 838 Bounded open region, 694 Box cylindrical, 770 rectangular, 758 spherical, 776
Cantor, Georg, 903 Capital Accumulation, 452 Cardioid, 751 Catenary, 449 Cauchy, A. L., 903 Cauchy Convergence Test (Criterion), 500, 504 Cauchy Divergence Test, 505 Cauchy equivalence, 911 Cauchy sequence, 911 Center circle, 5 ellipse, 264 hyperbola, 268 sphere, 641 Center of gravity, 344 Center of mass double integral, 745 triple integral, 765 two dimensions, 346 wire, 344 Centroid, 346 polar region, 430 Chain Rule for continuous functions, 130 variables one, 86 two, 672 Change of variables, 210, 484 Changing base of logarithms, 439 Characteristic polynomial, 882 Circle, 5 area, 115 polar equation, 410 Circular cylinder, 644 Circular helix, 616 Circulation dimensions two, 820 three, 836 Circumference, 115, 324 Circumscribed cylinder, 721 Circumscribed rectangle, 189 Circumscribed rectangular box, 759 cis, 876 Closed interval, 2 hyperreal, 908
Closed rectangle, 713 Closed region, 689 hyper real, 700 space, 757 Closure Laws, 905 Commodity vector, 565 Common logarithm, 436 Commutative Law, 905 inner product, 597 vector product, failure, 602 vector sum, 568 Comparison test, 512, 524 Completeness axiom, 905 Completing the square, 274, 479 Complex number, 874 conjugate, 875 exponent, 879 plane, 874 Complex valued differential equation, 879 Complex valued function, 879 Component directed line segment, 564 vector, 565 valued function, 615 Composition of functions, 86 continuity, 130, 653 derivative, 86, 672 Concave downward, 152 Concave upward, 152 Conditional convergence, 521 Cone area, 327 elliptic, 642 volume, 115 Conic section, 264 Conjugate axis, 268 Conjugate complex, 875 Conservative field, 807 Constant, II Constant function, II derivative of, 61 integral of, 184 Constant of integration, 199 Constant on an interval, 151 Constant Rule derivative, 61 double integral, 732 integral, 200 series, 508 Continuity, 125 of compositions, 130, 653 of differentiable functions, 127 E, o condition, 288 of an integral, 192 on an interval, 132 law of, 902 on a set, 653
two variables, 651 vector, 635 Continuously compounded interest, 443, 497 Contour, 645 Contour map, 644 Convergence improper integral, 352 increasing sequence, 511 interval of, 529 positive term series, 511 radius of, 530 sequence, 493 series, 502 summary of tests, 525 Coordinates cylindrical, 769 polar, 407 rectangular plane, 3 space, 585, 639 spherical, 775 cos, 78, 368 Cosecant function, 370 derivative, 376 cosh, 449 power series, 536 Cosine, 78, 368 derivative, 79, 374 integral, 377 power series, 558 Cosines, Law of, 573 cot, 370 Cotangent function, 370 derivative, 376 coth, 450 Critical damping, 887 Critical point, 136 interior, 137 two variables, 689 Critical Point Theorem, 136 two variables, 689 Cross product, 600 esc, 370 csch, 450 Curl dimensions two, 819 three, 832 Curve length, 320 parametric, 321 polar coordinates, 425 in space, 622 Curve sketching using derivatives, 156 ellipse, 266 hyperbola, 269 using limits, 251 parabola, 261 polar, 415
INDEX A 59
rotation of axes, 281 on an interval, 132 infinite, 719 translation of axes, 275 two variables, 710 Dummy variable, 119, 178 Cycloid, 616 Differential, 55 Cylinder exact, 806 e, 83, 443 area, 327 form, 806 irrationality of, 563 circular, 644 second, 94 as a limit, 443 quadric, 641 total, 662 Element volume, 115, 309 vector, 633 of area, 736 Cylinder Property, 712, 721 Differential equation of a set, 2 Cylindrical box, 770 complex valued, 879 of volume Cylindrical coordinates, exact, 810 cylindrical, 772 769 first order, 461, 846 rectangular, 764 Cylindrical Integration homogeneous, 852, 882 spherical, 777 Formula, 771 linear, 857 Elementary function, 481 Cylindrical region, 770 second order, 461, 881 Ellipse, 264 Cylindrical shell method, 313 separable variables, 848 sketching method, 266 Differentiation, 45 Ellipsoid, 642 Damped oscillation, 887 logarithmic, 470 Elliptic cone, 642 Damping, critical, 887 rules for, 68 Elliptic cylinder, 641 Decreasing on an interval, 152 Direct Test, 137 Elliptic paraboloid, 642 Definite integral, 183 Directed curve, 795 Empty set, 3 Degenerate conic section, 263, Directed line segment, 564 Endpoint, 2 273 Direction angle, 574, 588 Epidemic, 463 Degree, angular unit, 367 Direction cosine, 574, 588 t, o condition Deleted neighborhood, 243 Direction, real, 630 continuity, 288 Demand function, 114 Direction vector derivative, 295 De Moivre's Formula, 878 of a line, 578, 590 limits, 283, 286 Density of a plane, 606 t, N condition dimensions Directional derivative, 786 for infinite limits, 499 one, 341 second, 793 for sequences, 499 two, 342 three dimensions, 791 Equality of mixed partials, three, 764 Directrix, parabola, 257 703 of a wire, 364 Disc method, 309 three variables, 705 Dependent variable, 8, 45 Discontinuity, 125 Equation, 906 Derivative, 45 Discriminant, 875 differential, 461, 846 directional, 786 test, 273 linear, 20 exponential function, 83, two variables, 273 parametric, 90 443 Displacement vector, 565 point-slope, 16 higher, 94 Distance second degree, 272 higher partial, 703 plane, 4 two-point, 18 hyperbolic functions, 449 between reals, 13, 282 Error estimate, 228, 296 inverse trig functions, 387 space, 641 for derivatives, 296 In, 84, 455 Distributive Law, 905 for Simpson's Rule, 231 mixed partial, 703 scalar multiples, 570 and Taylor's Formula, 550 partial, 656 products for trapezoidal sums, 228 polar coordinates, 414 inner, 597 Escape velocity, 491 power rule, 63, 76 vector, 602 Euler approximation, 865 power series, 534 Divergence Euler's Formula, 879 rational function, 68 dimensions Even function, 218 rules for, 68 two, 819 Exact differential, 806 sec and esc, 376 three, 833 equation, 810 second, 94 Theorem, 839 Existence Theorem, differensecond partial, 703 Divergent tial equations, 867 sin and cos, 374 improper integral, 352 exp, 443 tan and cot, 376 sequence, 494 Explosion, 851, 871 vector, 620 series, 502 Exponent Determinant, 600 Domain, 10 complex, 879 Difference, vector, 569 Dot product, 593 inequalities, 432 Differentiable function, 45 Double integral, 711,719 real, 432 continuity of, 127 Double Riemann sum, 715 rules, 432
A60 INDEX
Exponential function, 83, 443 with base, 82, 432 continuity, 433 derivative, 83, 443 integral, 444 power series, 536 Exponential spiral, 448 Expression, real, 28, 906 Extension Axiom, 906 Extension Principle, 27 Extreme Value Theorem, 164 two dimensions, 689
Factor Theorem, 477 Factorial, n!, 493 Fermat, P., 902 Finite hyper real number, 30 partition, 160 Riemann sum, 178 set, 3 vector, 627 First Derivative Test, 174 First order differential equation, 461 Fluid flow, 820 Flux, 821 Focus ellipse, 264 hyperbola, 268 parabola, 257 Force vector, 565 Forcing term, 892 Formula, 45, 906 Frequency, 886 Frustum, area, 328 Fubini's Theorem, 726 Function, 45 hyperreal, 27 implicit, 97 inverse, 70, 381 linear, 17 real, 6 two variables, 14 vector-valued, 615 Fundamental Theorem of Calculus, 193
Galileo, G., 902 Gauss' Theorem, 839 General solution, 461, 846 Generalized Mean Value Theorem, 552 Geometric series, 503 formula, 446 Godel, Kurt, 904 grad f, 787 Gradient, 787 Graph, 7 dimensions
one, 7 two, 640 of an equation, 5 infinite sequence, 492 polar coordinates, 408 space, 639 Gravity dimensions one, 491 two, 809 three, 843, 844 Greatest integer function, 130 Green's Theorem, 815 vector form, 820 Growth rate, 51 populations, 51, 462, 491 sequences, 497
Half-life, 463 Half-open interval, 2 Harmonic series, 505 alternating, 546 Helix, circular, 616 Hewitt, E., 904 Higher derivative, 94 partial, 703 Higher differential, 94 Homogeneous differential equation, 852 second order, 8 82 solution, 852, 886 Homogeneous function, 710 Horizontal axis, 3 Horizontal ellipse, 266 Horizontal parabola, 261 Hyperbola, 268 graphing method, 270 Hyperbolic cosine, 449 derivative, 449 integral, 449 Hyperbolic cylinder, 641 Hyperbolic function, 449 Hyperbolic identities, 449 Hyperbolic paraboloid, 643 Hyperbolic sine, 449 derivative, 449 integral, 449 Hyperboloid, 643 Hyperinteger, 160 Hyperrational number, 431 Hyperreal function, 27 Hyper real interval, 161, 908 Hyperreal number, 27 Hyperreal region, 697 Hyperreal vector, 627
i, 874 Identity function, 12 Identity Law, 905 vector sum, 568
Imaginary number, 874 Imaginary part, 874 Implicit differentiation, 97 Implicit function, 97 derivative of, 97, 680 two variables, 680 Implicit Function Theorem, 680 Improper integral, 352 Incompressible field, 822 Increasing function, 151 Increasing sequence, 511 Increment, 45 two variables, 662 vector, 633 Increment Theorem, 54, 299 variables two, 664 three, 670 Indefinite integral, 199 Ind_ependent variable, 8, 45 Indeterminate form, 31 Induction, Principle of, 64 Inequalities for exponents, 432 Inequality, 906 Inequality Rule double integral, 733 series, 508 Inertia, moment of dimensions two, 748 three, 767 Infinite hyperreal number, 24 examples, 912 Infinite partial sum, 502 Infinite partition, 161 Infinite Riemann sum, 181 double, 719 Infinite sequence, 492 Infinite series, 502 Infinite Sum Theorem, 303 double integral, 736 triple integral, 764 Infinite telescope, 24 Infinite vector, 627 Infinitely close, 24, 35 compared to, 302 two dimensions, 651 vectors, 627 Infinitesimal, 24, 28 examples, 913 microscope, 24 vector, 627 Inflection, point of, 156 Initial condition, 462, 847, 881 Initial point directed line segment, 564 smooth curve, 795 Initial value problem first order, 847
INDEX A61
second order, 881 double, 725 Local maximum, 156 Inner product, 594 triple, 761 Local minimum, 156 derivative, 625 Locus rules, 597 Judicious guessing, 893 of an equation, 7 Inscribed cylinder, 721 Justification of a definition, polar coordinates, 408 Inscribed rectangle, 189 304 space, 639 Integers, 3 Kepler, J., 902 Logarithm Integral changing base, 439 arcsec and arccsc, 473 Kinetic energy, 784 common, 436 arcsin and arccos, 392 Kline, Morris, 907 natural, 83, 454 arctan and arccot, 472 Law of continuity, 902 rules, 437 definite, 183 Law of Cosines, 573 Logarithmic differentiation, double, 711, 719 Least squares, method of, 702 470 exponential function, 443 Leibniz, Gottfried Wilhelm, Logarithmic function, 436 improper, 352 902 Logistic function, 464 indefinite, 199 Length Lower endpoint, 2 iterated, 724 arc, 366 line, 795 curve, 320 MacLaurin series, 555 In, 456 parametric, 321 MacLaurin's Formula, 550 partial, 798 space, 622 Magnitude, 567 power series, 534 polar coordinates, 425 Major axis, ellipse, 264 rational functions, 477 real, 630 Malcev, A., 904 sin and cos, 377 vector, 566 Map, 209 sinh and cosh, 449 Level curve, 645 Marginal price vector, 624 surface, 829 Level surface, 791 Marginal revenue, 114 tan, 471 !'Hospital's Rule Marginal values, 51 triple, 760 for 0/0, 243 Mass Integral Test, 514 for oo/oo, 246 dimensions Integration Limit, 117 one, 341 change of variables, 210, and curve sketching, 251 two, 342 484 E, o condition, 283, 286 double integral, 743 coordinates finite, 117 triple integral, 764 cylindrical, 771 from the left, 123 Mass-spring system, 888 polar, 751 from the right, 123 Maximum, 135 spherical, 777 infinite, 237 closed region, 691 methods of, 481 one-sided, 123 local, 156 by parts, 391, 394 of a Riemann sum, 188, 316 open region, 698 substitution, 210, 484 rules, 121 two variables, 688 trigonometric, 403, 485 sequence, 493 value, 135 Interest, 442, 451 and standard parts, 120 Mean Value Theorem, 168 Interior point, 137, 689 two variables, 656 generalized, 552 critical, 137 Limit Comparison Test, 513, for Integrals, 338 Intermediate Value Theorem, 524 Member of a set, 2 162 Line, 16 Method of least squares, 702 Intermediate variable, 671 equation of, 16, 18, 20 Methods of integration, 481 Interval, 2 polar equation, 409 Midpoint, 582, 592 of convergence, 529 space, 589 Minimum, 135 hyperreal, 161, 908 vector equation, plane, 578 local, 156 Inverse function, 70, 381 vector equation, space, 589 two variables, 688 hyperbolic, 461 vertical, 16 value, 135 trigonometric, 384 Line integral, 795 Minor axis, ellipse, 264 Inverse Function Rule, 72 Linear differential equation, Mixed partial derivative, 703 Inverse Function Theorem, 857 Model theory, 904 386 second order, 892 Moment Inverse Laws, 905 solution, 858, 862, 897 dimensions Inverse relation, 3 81 Linear equation, 20 one, 343 Inverse square law, 491 Linear function, 17, 60 two, 345 Irrotational field, 822 In, 83, 454 double integral, 744 Iterated integral, 724 integral, 456 triple integral, 765 Iterated Integral Theorem power series, 536 wire, 344
A62 INDEX
Moment of inertia (second moment), dimensions two, 748 three, 767
Natural extension, 906 Natural logarithm, 81, 454 derivative, 84, 455 integral, 456 Negative infinite, 24, 30 Negative infinitesimal, 28 Neighborhood, 243 deleted, 243 Newton, Sir Isaac, 902 Newton's First Law, 567 Newton's Law of Cooling, 490 Newton's method, 290 Newton's Second Law, 570 Nonstandard analysis, 904 Nonuniqueness, differential equation, 872 Norm of a vector, 566 Normal curve, 537 Normal vector curve, 606 surface, 828 nth derivative, 94 nth differential, 94
Odd function, 218 One-to-one function, 382 Open interval, 2 Open rectangle, 794 Open rectangular solid, 803 Open region, 694 Ordered pair, 3 Oriented surface, 828 Origin, 3 in polar coordinates, 412 Orthogonal vectors, 597 Oscillation damped, 887 simple, 886 Overdamping, 888
pseries, 515 Parabola, 257 graphing method, 261 Parabolic cylinder, 641 Paraboloid elliptic, 642 hyperbolic, 643 Parallel line and plane, 609 Parallel planes, 609 Parallel, test for, 597 Parallel vectors, 597 Parallelogram, 583 area, 604 Parametric curve, 90
Parametric equations, 90 of a line, 579, 590 polar coordinates, 411 and vector equations, 579, 615 Partial derivative, 656 higher, 703 implicit function, 686 mixed, 703 second, 703 three variables, 659 Partial fraction, 477 Partial integral, 798 derivative of, 811 Partial second derivative, 703 Partial sum, 502 Particular solution, 846 Partition finite, 160 infinite, 161 of a rectangle, 715 of a rectangular box, 759 Partition point, 161, 176 Path Independence Theorem, 806 Perimeter, 115 Period, 886 Periodic function, 368 Perpendicular line and plane, 609 Perpendicular planes, 609 Perpendicular, test for, 597 Perpendicular vectors, 597 Phase shift, 886 Pi, 1r, 366 Piecewise continuous, 357 Piecewise smooth, 800 Plane, 4, 604 Point, 3 Point of inflection, 156 Point-slope equation, 16 Polar coordinates, 407 area, 421 Polar equation circle, 410 line, 409 Polar form, complex numbers, 876 Polar Integration Formula, 751 Polar rectangle, 7 50 Polar region, 750 Polynomial, 60 function, 60 Population growth, 434, 442, 463 Position, 203 Position vector, 565 of a curve, 615 of a line, 578, 590
of a plane, 606 of a point, 576, 589 Positive infinite, 24, 30 Positive infinitesimal, 28 Positive term series, 511 Potential function, 805 existence, 806 method for finding, 807 three variables, 814 Potential energy, 807 Power Rule derivatives, 63 general form, 89 integrals, 200, 471 negative exponents, 66 rational exponents, 76 Power series, 528 multiplication of, 563 operations on, 535 steps for finding, 557 table, 556 Present value, 454 Pressure, 364 Price vector, 565 Principle of Induction, 64 Principle of Superposition, 863, 896 Prism, volume, 115 Product function, 648 inner, 594 vector, 600 Product Law, order axiom, 905 Product Rule derivative, 62 vector derivative, 625 Profit, 137 Pyramid, volume, 306 Pythagoras, Theorem of, 5
Quadrant, 4 Quadratic formula, 103, 477 complex, 875 Quadric cylinder, 641 Quadric surface, 640 Quotient function, 652 Quotient Rule, derivatives, 66
R, I R*, 24 Radian, 78, 367 Radioactive decay, 463 Radius of circle, 5 of convergence, 530 of sphere, 641 Range, 10 Rate of growth, 51, 434, 442, 463
INDEX A63
Ratio Test, 524 Root, 2 Set, 2 Rational function, 60, 474 Root Axiom, 905 Side condition, 692 differentiation, 68 Rotation of axes, 280 Simple closed curve, 801 integration, 477 equations, 277 Simple oscillation, 886 Rational number, 3 Simpson's approximation, 230 Rational term, 60 Saddle point, 690 Simpson's Rule, 231 Real direction, 630 Scalar, 566 sin, 77, 368 Real expression, 906 Scalar Associative Law, 570 Sine, 77, 368 Real length, 630 Scalar equation derivative, 79, 374 Real line, 1 of a line, 577 integral, 377 Real number, 1 of a plane, 604 power series, 558 Real part, 874 Scalar multiple, 570 sinh, 449 Real statement, 907 derivative, 627 derivative, 449 Real term, 906 Scalar product, 593 integral, 449 Real vector, 630 Scalar triple product, 604 power series, 537 Rearrangement, series, 523 Schwartz' Inequality Skolem, Thoralf, 903 Reciprocal function, 8 integrals, 236 Slide rule, 438 Rectangle vectors, 637 Slope area, 115 sec, 370 average, 22, 168 closed, 713 Secant function, 370 curve, 25, 43 open, 794 derivative, 376 directional, 786 polar, 750 integral, 471 function, 43 Rectangle Property, 188 Secant line, 295 implicit function, 680 Rectangular box, 758 sech, 450 as a limit, 119 Rectangular coordinates Second degree equation line, 16 plane, 3 variables polar coordinates, 413 space, 585, 639 two, 272 Slope-intercept equation, 17 Rectangular solid, 115, 758 three, 640 Smooth curve, 319, 795 Reduction formula Second derivative, 94 piecewise, 800 secant and cosecant, 399 partial, 703 space, 803 sine and cosine, 398 Second Derivative Test, 138 Smooth function, 662, 824 tangent and cotangent, 397 two variables, 707 continuity of, 666 Region Second differential, 94 Smooth surface, 662, 824 below a curve, 175 Second order differential Snail, 448 between two curves, 218, equation, 461 Solenoidal field, 822 717 Second partial derivative, 703 Solid region, 757 bounded, 694 Sector, 365 Solution, differential closed, 689 area, 115, 324, 366, 420 equation, 461, 846 cylindrical, 770 Separable variables, 465, 848 Speed, 623 hyperreal, 697 solution method, 848 Sphere, 641 open, 694 Sequence surface area, 115, 333 polar, 750 graph of, 492 volume, 115, 317 solid, 757 increasing, 511 Spherical box, 776 spherical, 775 infinite, 492 Spherical coordinates, 775 unbounded, 694 of partial sums, 502 Spherical Integration under a curve, 175 Series, 502 Formula, 777 Related rates, 112 absolutely convergent, 521 Spherical region, 775 Relativity, theory of, 239 alternating, 517 Spiral Resonance, 897 conditionally convergent, of Archimedes, 411, 474 Resonant frequency, 900 521 exponential, 448 Revenue, 114 convergent, 502 vertical, 616 Riemann sum, 176, 178 divergent, 502 Spring constant, 888 double, 715 of functions, 528 Square function, 8 finite, 178 infinite, 502 Square root, 2 infinite, 181 positive term, 511 st, 36 infinite double, 719 sum, 502 Standard part, 36 triple 759 summary of tests, 525 function, 910 Right-handed coordinates, 585 tail, 508 principle, 36, 908 Robinson, Abraham, 904 Taylor, 554 rules, 37 Rolle's Theorem, 165 telescoping, 506 vector, 629
A64 INDEX
Statement, real, 907 in space, 621 dependent, 8, 45 Steady state part of the and vector derivatives, 620 dummy, 119, 178 solution, 899 Tangent plane, 666 independent, 8, 45 Steepest ascent, 788 of an implicit function, 686 intermediate, 671 Steepest descent, 788 Tangent vector, 621 point, 578 Stokes' Theorem, 834 tanh, 450 vector, 578, 589 Straight line, 16 Tarski, Alfred, 904 Variation of constants, 857 in space, 589 Taylor's Formula, 550 Vector, 565 Strictly between, 2 Taylor polynomial, 549 continuous, 635 Strictly within, 282 Taylor remainder, 549 derivative, 620 Subinterval, 160 Taylor series, 554 difference, 569 Subrectangle, 715 table of, 556 differential, 633 Substitution, integration by, Telescoping series, 506 dimensions 210, 484 Term, 9 n, 586 trigonometric, 402 real, 906 two, 565 Sum Terminal point three, 586 function, 648 directed line segment, 564 direction, 578, 606 partial, 502 smooth curve, 795 equation, 615 Riemann, 178 Third partial derivative, 704 of a line, 578, 589 series, 502 Topographic map, 644 of a plane, 606 trapezoidal, 226 Torus, 336 field, 805 vector, 567 Total differential, 662 finite, 627 Sum Law, order axiom, 905 three variables, 669 function, 615 Sum Rule Transfer Axiom, 908 h yperreal, 627 derivatives, 61 Transfer Principle, 28 increment, 633 double integral, 732 Transient part of solution, 899 infinite, 627 integral, 200 Transitive Law, 905 infinitesimal, 627 series, 508 Translation of axes, 275 negative, 569 use in integration, 483 Transverse axis, 268 normal, 606, 828 vector derivative, 625 Trapezoidal approximation, position, 565 Surface 226 product, 600 area, 328, 825 Trapezoidal Rule, 228 derivative, 627 area, polar, 428 Trapezoidal sum, 226 real, 630 as a boundary, 834 Triangle, area, 115 rules, 625 contour map of, 644 Triangle inequality, 568 sum, 567 implicit, 686 Trichotomy Law, 905 valued function, 615 integral, 829 Trigonometric functions, 365 variable, 578, 589 level, 791 Trigonometric identities, 372 Velocity, 23 oriented, 828 Trigonometric substitution, average, 169 quadric, 640 402 and integration, 203 of revolution, 327 Triple integral, 760 vector, 565, 622 sketching method, 645 Triple Riemann sum, 759 Vertex, parabola, 257 smooth, 662, 824 Two-point equation, 18 Vertical axis, 3 tangent plane of, 666 Vertical ellipse, 266 topographic map of, 644 U1traproduct, 903 Vertical line, 16 Symmetric region, 784 equivalence, 912 Vertical parabola, 259 Symmetry and center of mass, Unbounded open region, 694 Volume, 305 347 Undefined term, 6 below a surface, 723 System of formulas, 45 Undetermined coefficients, between two surfaces, 738 893 cylindrical shell method, Tail rule, 508 Uniqueness Theorem 313 Tail, series, 508 differential equations, 870 disc method, 309 tan 79, 370 double integrals, 722 as a double integral, 723 Tangent of an angle, 79 Unit circle, 365 element of, 764 Tangent function, 370 Unit hyperbola, 450 function, 711, 722 derivative, 376 Unit vector, 570, 588 geometric figures, 115 integral, 471 Upper endpoint, 2 as the integral of area, 305 Tangent line, 53 solid of revolution, 309, 313 of an implicit function, 686 Variable, 8, 45 as a triple integral, 763, 778 in polar coordinates, 412 bound, 119, 178 under a surface, 723
Water pressure, 364 Wave equation, 708 Weierstrass, Karl, 903 Work against gravity, 348 as an inner product, 595 as a line integral, 796 as a single integral, 349
x-axis, 3 x-component, 564, 586 x-coordinate, 3 (x,y) plane, 3, 639 (x,z) plane, 639 (x,y,z) space, 585, 639
y-axis, 3
INDEX
y-component, 564, 586 y-coordinate, 3 (y,z) plane, 639
z-axis, 585, 639 z-component, 586 Zero of a function, 161 Zero vector, 567
A65
RUlES OF INTEGRATION I du = u + C
I du + dv = I du + I dv (Sum Rule)
TABlE OF INTEGrlAlS
(The constant of integration is left out.) I x-+' x' dx = -- r # -I r +I' I ex dx =ex I sinx dx = -cosx I tanx dx = In lsecxl I secx dx = lnlsecx + tanxl
I sec2 x dx = tanx I tanx secx dx = secx J sin2 x dx = -!x --! sinx cosx I I ll- I I sin"xdx= --sin"- 1 xcosx+-- sin"-2 xdx ll ll ftan"x dx = - 1 - tan"- 1 x- Itan"-2 x dx n - I
I
sec"xdx =
1 -sec"- 1 xsinx + n
2
Isec"-2 xdx
n - I n - I I arcsinx dx = x arcsinx + F- x2
I arctanx dx = x arctanx --!In(! + x2)
I arcsecx dx = x arcsecx - ln(x + JxZ=!J, x :2: I I ~=arcsinx I ~ = arcsecx lx1Jx2 - I
I k du = k I du (Constant Rule)
I u dv = uv - I v du (Integration by Parts)
I [ 1 dx = lnlxl
I lnlxl dx = x lnlxl - x I cosx dx = sinx
I cotxdx = lnlsinxl f cscxdx = -lnlcscx + cotxl
I csc2 x dx = -cot x
Icotxcscxdx = -cscx
I cos2 x dx = tx + t sinx cosx
I
cos"x dx = ~ cos"- 1 x sinx + 11 - 1 Icos"-2 x dx ll ll
I
cot"x dx = -
1 - cot"- 1 x- Icot"~2 x dx n- I
I
csc"x dx = -
1 - csc"- 1 x cosx +
11

2 Icsc"-2 x dJ 11- 1 n- 1 I arccosxdx = xarccosx- ~ I arccotx dx = x arccotx +-!In(! + x2 ) I arccscx dx = x arccscx + ln(x + JxZ=!J, x :2: I I dx --2 = arctan x 1 + X
ELEMENTARY ALGEBRA OF THE REAL NUMBERS
a+b=b+a a + (b + c) = (a + b) + c a(b + c) = ab + ac a+0=a•1=a a-a=O
-(-a)= a
-(a- b)= b- a
a c ad + be b + d = bd (b,d #- 0)
a 1 =a !" = 1
a"'-" = a"'ja" (a #- 0) a'"b"' = (ab)m Fa = -.:ja, ( -a)m/n = ~' (a> O,n odd) If a < b then a + c < b + c. If a < b, then - b < -a. 1-al = lal lal • lbl = labl ~ = 1~1 (b #- 0). lbl b
The following terms are undefined: a/0, 0°, o-m, Fa (a, m positive, n even)
ab = ba a(bc) = (ab)c a(-b) = (-a)b = -ab a•O = 0 aja = 1 (a #- 0) 1 -=a 1/a (a#- 0) 1 b
ajb a
(a, b #- 0)
a c ac -.- = - (b, d #- 0) b d bd a0 = 1 (a #- 0) 0" = 0 (n > 0) amn = (amj' amJn = F (a > 0) .::fo.::jb = ;fob (a, b > 0)
If a < b and 0 < c, then ac < be. IfO <a< b, then 1/b < !fa.
-b ± jb2 - 4ac Quadratic F omwla: ax2 + bx + c = 0 if and only if x = ----'--2a
ALGEBRA OF HYPERREAL NUMBERS
Notation:
e, [J are positive infinitesimals b, c are positive and finite but not infinitesimal H, K are positive infinite.
The following are infinitesimal: -e, 1/H, ejb, sjH, bjH, E + 0, e - 0, E • 0, b • E, .:ji.
The following are finite but not infinitesimal: -b, ljb, bjc, b + e, b • c, .:jb, b + c b - c is finite (possibly infinitesimal).
The following are infinite: -H, lje, bfe, Hje, Hjb, H + e, H + b, H • b, H. K, .::(ii, H + K
The following can be infinitesimal, finite but not infinitesimal, or infinite: ejb, HjK, He, H - K
STANDARD PARTS
In the following b, c are finite (possibly infinitesimal).
st(b + c) = st(b) + st(c) st(bc) = st(b)st(c) st(jb) = ;rst{b) if b > 0, n > 0 b::::; st(b) b = st(b) if and only if b is real st(s) = 0, st(H) is undefined
TRIGONOMETRIC IDENTITIES
sin2 8 + cos2 0 = l cot>O + l = csc2 0 cos( ~e)= cosO cos(n/2 ~ 8) =sinO cos(O + r{!) = cosOcosr{! ~sinO sin¢
RULES OF EXPONENTS AND LOGARITHMS
Assume a > 0, b > 0.
1-' = l
a-'"= (ax)Y
alogaX = X
log.(xy) = log,x + log,y log.(x'') = y log,x
logby = log.y/log.b
TABLE OF DERIVATIVES
du = (dujdx) dx (Chain Rule) d(u + v) = du + dv (Sum Rule)
d (u) = v du ~ u dv ~ (Quotient Rule) v v2 d(ln u) = duju d(sin u) = cos u du d(tan tt) = sec2 u du d(sec u) = tan u sec u du du d(arcsinu) = ~ v 1 ~ !12 du d(arctanu) = -2 1 + ll du d(arcsecu) = ~ [u[.yu2 ~I
st(b ~ c) = st(b) ~ st(c) st(bjc) = st(b)jst(c) if st(c) oft 0 st(b') = st(b)"1'l if st(b) > 0 b ::::; c if and only if st(b) = st(c) if b s; c, then st(b) s; st(c)
tan 2 0 + I = sec2 e sin(~O) =~sinO sin(n/2 ~ 0) =cosO sin(O + ¢) = siniJcosrfJ +cosO sin¢
axbx = (ab)'
log.(ax) = x log.(xjy) = log.x ~ log.y bx = axlogab
d(ku) = k du (Constant Rule) d(tw) = u dv + v du (Product Rule)
d(u') = ru'- 1 du (Power Rule)
d(e") = e" du d(cos u) = ~sin u dtt d(cot u) = ~csc2 u dtt d(csc 11) = ~cot u esc u d11
du d(arccosu) = ~ --Jl-=7 du d(arccotu) = ~ -1 + 112 du d(arccscu) = ~ ---[u[p-=i
ILES OF INTEGRATION
'tl = ll + c 'u + dv = f du + f dv (Sum Rule)
.BLE OF INTEGRALS
te constant of integration is left out.)
xr+ 1 'dx = -- r oft -1 r + 1'
inxdx= -cosx
m x dx = In lsecxl
~cx dx = lnlsecx + tanxl
~c2 x dx = tanx
mx secx dx = secx
n2 xdx = !x- !sinxcosx
n'' x dx = - - sin''- 1 x cosx + -- sin"- 2 x dx 1 n-1s . " " m"x dx = 1 - tan"- 1 x - J tan"- 2 x dx II - 1
~C11X dx = --sec"- 1 x sinx + -- sec"- 2 x dx 1 ll- 2f II - 1 II - 1
rcsinxdx = xarcsinx + ~
:ctan x dx = x arctan x - ! In (I + x2)
:csecx dx = x arcsecx - ln(x + ~), x 2 1
dx . ~ = arcsmx r1- x2 dx ---- = arcsecx :lp-=!
J k du = k J du (Constant Rule) f u dv = uv - f v du (Integration by Parts)
f x-1 dx = lnlxl f lnlxl dx = x lnlxl - x J cosx dx = sinx .f cotxdx = lnlsinxl f cscx dx = -In lcscx + cotxl J csc2 x dx = -cot x f cotx cscx dx = -cscx J cos2 x dx = !x +! sinx cosx
f
cos"xdx = ~cos"-1xsinx + ~fcos"-2xdx n " Jcot"x dx = - - 1-cot''- 1 x- Jcot"-2 x dx n - 1 fcsc"x dx = - - 1 - csc"- 1 x cosx + 11 2 Jcsc"- 2 x dx n-1 n-1 J arccos x dx = x arccos x - fi-=-7 J arccotx dx = x arccotx +! ln(1 + x2) f arccscx dx = x arccscx + ln(x + p-=!), x 2 1 f dx X
Absolute convergence, 521 trapezoidal, 226 surface, 825 Absolute value, 12 within E, 284 surface of revolution, 328 of a complex number, 876 Arc, 323, 365 under a curve, 188 Absolute value function, 12 length, 324, 366, 420 Argand diagram, 876 Acceleration, 94 arccos, 384 Argument, complex number, vector, 565, 623 Arccosecant, 384 876 Addition formulas for sine derivative, 387 Associative Law, 905 and cosine, 372 Arccosine, 384 inner product, 597 Addition Property, 190, 303 derivative, 387 scalar multiples, 570 variables integral, 392 vector product, failure, 602 two, 712, 720 arccot, 384 vector sum, 568 three, 760 Arccotangent, 384 Asymptote Adiabatic process, 490 derivative, 387 of a hyperbola, 270 Almqst parallel vectors, 632 arccsc, 384 vertical, 251 Alternating series, 517 Archimedes, 902 Average slope, 22, 168 harmonic, 520, 546 arcsec, 384 Average speed, 339 Alternating Series Test, 518 Arcsecant, 384 Average value, 337 Amplitude, 886 derivative, 387 Average velocity, 169, 339 Angle, 77, 365 integral, 473 Axioms between vectors, 573 arcsin, 384 for hyperreal numbers, 906 three dimensions, 588 Arcsine, 384 for real numbers, 905 Angular measure, 367 derivative, 387 Axis Anticommutative Law for integral, 392 coordinates vector products, 604 power series, 560 plane, 3 Antiderivative, 192 arctan, 384 space, 585 Approximation Arctangent, 384 ellipse, 264 by alternating series, 518 derivative, 387 hyperbola, 268 arctan x, 541 integral, 472 parabola, 257 by derivatives, 286 power series, 536 e, 442, 551 Area Barrow, Isaac, 902 In x, 541 below a curve, 188 Basis vectors by Newton's method, 290 between two curves, 220, plane, 570 pi, 542 304 space, 587 by power series, 541 double integral, 722 Bell-shaped curve, 537 by Riemann sums, 178 geometric figures, 115 Berkeley, Bishop, 903 Simpson's, 230 by Green's Theorem, 818 Binomial series, 558 sin x, 552 infinite, 353 Binomial Theorem, 559 by Taylor's Formula, 547 polar, 421 Bound variable, 119, 178
A 57
A 58 INDEX
Boundary oriented surface, 834 plane region, 689 solid region, 757, 838 Bounded open region, 694 Box cylindrical, 770 rectangular, 758 spherical, 776
Cantor, Georg, 903 Capital Accumulation, 452 Cardioid, 751 Catenary, 449 Cauchy, A. L., 903 Cauchy Convergence Test (Criterion), 500, 504 Cauchy Divergence Test, 505 Cauchy equivalence, 911 Cauchy sequence, 911 Center circle, 5 ellipse, 264 hyperbola, 268 sphere, 641 Center of gravity, 344 Center of mass double integral, 745 triple integral, 765 two dimensions, 346 wire, 344 Centroid, 346 polar region, 430 Chain Rule for continuous functions, 130 variables one, 86 two, 672 Change of variables, 210, 484 Changing base of logarithms, 439 Characteristic polynomial, 882 Circle, 5 area, 115 polar equation, 410 Circular cylinder, 644 Circular helix, 616 Circulation dimensions two, 820 three, 836 Circumference, 115, 324 Circumscribed cylinder, 721 Circumscribed rectangle, 189 Circumscribed rectangular box, 759 cis, 876 Closed interval, 2 hyperreal, 908
Closed rectangle, 713 Closed region, 689 hyper real, 700 space, 757 Closure Laws, 905 Commodity vector, 565 Common logarithm, 436 Commutative Law, 905 inner product, 597 vector product, failure, 602 vector sum, 568 Comparison test, 512, 524 Completeness axiom, 905 Completing the square, 274, 479 Complex number, 874 conjugate, 875 exponent, 879 plane, 874 Complex valued differential equation, 879 Complex valued function, 879 Component directed line segment, 564 vector, 565 valued function, 615 Composition of functions, 86 continuity, 130, 653 derivative, 86, 672 Concave downward, 152 Concave upward, 152 Conditional convergence, 521 Cone area, 327 elliptic, 642 volume, 115 Conic section, 264 Conjugate axis, 268 Conjugate complex, 875 Conservative field, 807 Constant, II Constant function, II derivative of, 61 integral of, 184 Constant of integration, 199 Constant on an interval, 151 Constant Rule derivative, 61 double integral, 732 integral, 200 series, 508 Continuity, 125 of compositions, 130, 653 of differentiable functions, 127 E, o condition, 288 of an integral, 192 on an interval, 132 law of, 902 on a set, 653
two variables, 651 vector, 635 Continuously compounded interest, 443, 497 Contour, 645 Contour map, 644 Convergence improper integral, 352 increasing sequence, 511 interval of, 529 positive term series, 511 radius of, 530 sequence, 493 series, 502 summary of tests, 525 Coordinates cylindrical, 769 polar, 407 rectangular plane, 3 space, 585, 639 spherical, 775 cos, 78, 368 Cosecant function, 370 derivative, 376 cosh, 449 power series, 536 Cosine, 78, 368 derivative, 79, 374 integral, 377 power series, 558 Cosines, Law of, 573 cot, 370 Cotangent function, 370 derivative, 376 coth, 450 Critical damping, 887 Critical point, 136 interior, 137 two variables, 689 Critical Point Theorem, 136 two variables, 689 Cross product, 600 esc, 370 csch, 450 Curl dimensions two, 819 three, 832 Curve length, 320 parametric, 321 polar coordinates, 425 in space, 622 Curve sketching using derivatives, 156 ellipse, 266 hyperbola, 269 using limits, 251 parabola, 261 polar, 415
INDEX A 59
rotation of axes, 281 on an interval, 132 infinite, 719 translation of axes, 275 two variables, 710 Dummy variable, 119, 178 Cycloid, 616 Differential, 55 Cylinder exact, 806 e, 83, 443 area, 327 form, 806 irrationality of, 563 circular, 644 second, 94 as a limit, 443 quadric, 641 total, 662 Element volume, 115, 309 vector, 633 of area, 736 Cylinder Property, 712, 721 Differential equation of a set, 2 Cylindrical box, 770 complex valued, 879 of volume Cylindrical coordinates, exact, 810 cylindrical, 772 769 first order, 461, 846 rectangular, 764 Cylindrical Integration homogeneous, 852, 882 spherical, 777 Formula, 771 linear, 857 Elementary function, 481 Cylindrical region, 770 second order, 461, 881 Ellipse, 264 Cylindrical shell method, 313 separable variables, 848 sketching method, 266 Differentiation, 45 Ellipsoid, 642 Damped oscillation, 887 logarithmic, 470 Elliptic cone, 642 Damping, critical, 887 rules for, 68 Elliptic cylinder, 641 Decreasing on an interval, 152 Direct Test, 137 Elliptic paraboloid, 642 Definite integral, 183 Directed curve, 795 Empty set, 3 Degenerate conic section, 263, Directed line segment, 564 Endpoint, 2 273 Direction angle, 574, 588 Epidemic, 463 Degree, angular unit, 367 Direction cosine, 574, 588 t, o condition Deleted neighborhood, 243 Direction, real, 630 continuity, 288 Demand function, 114 Direction vector derivative, 295 De Moivre's Formula, 878 of a line, 578, 590 limits, 283, 286 Density of a plane, 606 t, N condition dimensions Directional derivative, 786 for infinite limits, 499 one, 341 second, 793 for sequences, 499 two, 342 three dimensions, 791 Equality of mixed partials, three, 764 Directrix, parabola, 257 703 of a wire, 364 Disc method, 309 three variables, 705 Dependent variable, 8, 45 Discontinuity, 125 Equation, 906 Derivative, 45 Discriminant, 875 differential, 461, 846 directional, 786 test, 273 linear, 20 exponential function, 83, two variables, 273 parametric, 90 443 Displacement vector, 565 point-slope, 16 higher, 94 Distance second degree, 272 higher partial, 703 plane, 4 two-point, 18 hyperbolic functions, 449 between reals, 13, 282 Error estimate, 228, 296 inverse trig functions, 387 space, 641 for derivatives, 296 In, 84, 455 Distributive Law, 905 for Simpson's Rule, 231 mixed partial, 703 scalar multiples, 570 and Taylor's Formula, 550 partial, 656 products for trapezoidal sums, 228 polar coordinates, 414 inner, 597 Escape velocity, 491 power rule, 63, 76 vector, 602 Euler approximation, 865 power series, 534 Divergence Euler's Formula, 879 rational function, 68 dimensions Even function, 218 rules for, 68 two, 819 Exact differential, 806 sec and esc, 376 three, 833 equation, 810 second, 94 Theorem, 839 Existence Theorem, differensecond partial, 703 Divergent tial equations, 867 sin and cos, 374 improper integral, 352 exp, 443 tan and cot, 376 sequence, 494 Explosion, 851, 871 vector, 620 series, 502 Exponent Determinant, 600 Domain, 10 complex, 879 Difference, vector, 569 Dot product, 593 inequalities, 432 Differentiable function, 45 Double integral, 711,719 real, 432 continuity of, 127 Double Riemann sum, 715 rules, 432
A60 INDEX
Exponential function, 83, 443 with base, 82, 432 continuity, 433 derivative, 83, 443 integral, 444 power series, 536 Exponential spiral, 448 Expression, real, 28, 906 Extension Axiom, 906 Extension Principle, 27 Extreme Value Theorem, 164 two dimensions, 689
Factor Theorem, 477 Factorial, n!, 493 Fermat, P., 902 Finite hyper real number, 30 partition, 160 Riemann sum, 178 set, 3 vector, 627 First Derivative Test, 174 First order differential equation, 461 Fluid flow, 820 Flux, 821 Focus ellipse, 264 hyperbola, 268 parabola, 257 Force vector, 565 Forcing term, 892 Formula, 45, 906 Frequency, 886 Frustum, area, 328 Fubini's Theorem, 726 Function, 45 hyperreal, 27 implicit, 97 inverse, 70, 381 linear, 17 real, 6 two variables, 14 vector-valued, 615 Fundamental Theorem of Calculus, 193
Galileo, G., 902 Gauss' Theorem, 839 General solution, 461, 846 Generalized Mean Value Theorem, 552 Geometric series, 503 formula, 446 Godel, Kurt, 904 grad f, 787 Gradient, 787 Graph, 7 dimensions
one, 7 two, 640 of an equation, 5 infinite sequence, 492 polar coordinates, 408 space, 639 Gravity dimensions one, 491 two, 809 three, 843, 844 Greatest integer function, 130 Green's Theorem, 815 vector form, 820 Growth rate, 51 populations, 51, 462, 491 sequences, 497
Half-life, 463 Half-open interval, 2 Harmonic series, 505 alternating, 546 Helix, circular, 616 Hewitt, E., 904 Higher derivative, 94 partial, 703 Higher differential, 94 Homogeneous differential equation, 852 second order, 8 82 solution, 852, 886 Homogeneous function, 710 Horizontal axis, 3 Horizontal ellipse, 266 Horizontal parabola, 261 Hyperbola, 268 graphing method, 270 Hyperbolic cosine, 449 derivative, 449 integral, 449 Hyperbolic cylinder, 641 Hyperbolic function, 449 Hyperbolic identities, 449 Hyperbolic paraboloid, 643 Hyperbolic sine, 449 derivative, 449 integral, 449 Hyperboloid, 643 Hyperinteger, 160 Hyperrational number, 431 Hyperreal function, 27 Hyper real interval, 161, 908 Hyperreal number, 27 Hyperreal region, 697 Hyperreal vector, 627
i, 874 Identity function, 12 Identity Law, 905 vector sum, 568
Imaginary number, 874 Imaginary part, 874 Implicit differentiation, 97 Implicit function, 97 derivative of, 97, 680 two variables, 680 Implicit Function Theorem, 680 Improper integral, 352 Incompressible field, 822 Increasing function, 151 Increasing sequence, 511 Increment, 45 two variables, 662 vector, 633 Increment Theorem, 54, 299 variables two, 664 three, 670 Indefinite integral, 199 Ind_ependent variable, 8, 45 Indeterminate form, 31 Induction, Principle of, 64 Inequalities for exponents, 432 Inequality, 906 Inequality Rule double integral, 733 series, 508 Inertia, moment of dimensions two, 748 three, 767 Infinite hyperreal number, 24 examples, 912 Infinite partial sum, 502 Infinite partition, 161 Infinite Riemann sum, 181 double, 719 Infinite sequence, 492 Infinite series, 502 Infinite Sum Theorem, 303 double integral, 736 triple integral, 764 Infinite telescope, 24 Infinite vector, 627 Infinitely close, 24, 35 compared to, 302 two dimensions, 651 vectors, 627 Infinitesimal, 24, 28 examples, 913 microscope, 24 vector, 627 Inflection, point of, 156 Initial condition, 462, 847, 881 Initial point directed line segment, 564 smooth curve, 795 Initial value problem first order, 847
INDEX A61
second order, 881 double, 725 Local maximum, 156 Inner product, 594 triple, 761 Local minimum, 156 derivative, 625 Locus rules, 597 Judicious guessing, 893 of an equation, 7 Inscribed cylinder, 721 Justification of a definition, polar coordinates, 408 Inscribed rectangle, 189 304 space, 639 Integers, 3 Kepler, J., 902 Logarithm Integral changing base, 439 arcsec and arccsc, 473 Kinetic energy, 784 common, 436 arcsin and arccos, 392 Kline, Morris, 907 natural, 83, 454 arctan and arccot, 472 Law of continuity, 902 rules, 437 definite, 183 Law of Cosines, 573 Logarithmic differentiation, double, 711, 719 Least squares, method of, 702 470 exponential function, 443 Leibniz, Gottfried Wilhelm, Logarithmic function, 436 improper, 352 902 Logistic function, 464 indefinite, 199 Length Lower endpoint, 2 iterated, 724 arc, 366 line, 795 curve, 320 MacLaurin series, 555 In, 456 parametric, 321 MacLaurin's Formula, 550 partial, 798 space, 622 Magnitude, 567 power series, 534 polar coordinates, 425 Major axis, ellipse, 264 rational functions, 477 real, 630 Malcev, A., 904 sin and cos, 377 vector, 566 Map, 209 sinh and cosh, 449 Level curve, 645 Marginal price vector, 624 surface, 829 Level surface, 791 Marginal revenue, 114 tan, 471 !'Hospital's Rule Marginal values, 51 triple, 760 for 0/0, 243 Mass Integral Test, 514 for oo/oo, 246 dimensions Integration Limit, 117 one, 341 change of variables, 210, and curve sketching, 251 two, 342 484 E, o condition, 283, 286 double integral, 743 coordinates finite, 117 triple integral, 764 cylindrical, 771 from the left, 123 Mass-spring system, 888 polar, 751 from the right, 123 Maximum, 135 spherical, 777 infinite, 237 closed region, 691 methods of, 481 one-sided, 123 local, 156 by parts, 391, 394 of a Riemann sum, 188, 316 open region, 698 substitution, 210, 484 rules, 121 two variables, 688 trigonometric, 403, 485 sequence, 493 value, 135 Interest, 442, 451 and standard parts, 120 Mean Value Theorem, 168 Interior point, 137, 689 two variables, 656 generalized, 552 critical, 137 Limit Comparison Test, 513, for Integrals, 338 Intermediate Value Theorem, 524 Member of a set, 2 162 Line, 16 Method of least squares, 702 Intermediate variable, 671 equation of, 16, 18, 20 Methods of integration, 481 Interval, 2 polar equation, 409 Midpoint, 582, 592 of convergence, 529 space, 589 Minimum, 135 hyperreal, 161, 908 vector equation, plane, 578 local, 156 Inverse function, 70, 381 vector equation, space, 589 two variables, 688 hyperbolic, 461 vertical, 16 value, 135 trigonometric, 384 Line integral, 795 Minor axis, ellipse, 264 Inverse Function Rule, 72 Linear differential equation, Mixed partial derivative, 703 Inverse Function Theorem, 857 Model theory, 904 386 second order, 892 Moment Inverse Laws, 905 solution, 858, 862, 897 dimensions Inverse relation, 3 81 Linear equation, 20 one, 343 Inverse square law, 491 Linear function, 17, 60 two, 345 Irrotational field, 822 In, 83, 454 double integral, 744 Iterated integral, 724 integral, 456 triple integral, 765 Iterated Integral Theorem power series, 536 wire, 344
A62 INDEX
Moment of inertia (second moment), dimensions two, 748 three, 767
Natural extension, 906 Natural logarithm, 81, 454 derivative, 84, 455 integral, 456 Negative infinite, 24, 30 Negative infinitesimal, 28 Neighborhood, 243 deleted, 243 Newton, Sir Isaac, 902 Newton's First Law, 567 Newton's Law of Cooling, 490 Newton's method, 290 Newton's Second Law, 570 Nonstandard analysis, 904 Nonuniqueness, differential equation, 872 Norm of a vector, 566 Normal curve, 537 Normal vector curve, 606 surface, 828 nth derivative, 94 nth differential, 94
Odd function, 218 One-to-one function, 382 Open interval, 2 Open rectangle, 794 Open rectangular solid, 803 Open region, 694 Ordered pair, 3 Oriented surface, 828 Origin, 3 in polar coordinates, 412 Orthogonal vectors, 597 Oscillation damped, 887 simple, 886 Overdamping, 888
pseries, 515 Parabola, 257 graphing method, 261 Parabolic cylinder, 641 Paraboloid elliptic, 642 hyperbolic, 643 Parallel line and plane, 609 Parallel planes, 609 Parallel, test for, 597 Parallel vectors, 597 Parallelogram, 583 area, 604 Parametric curve, 90
Parametric equations, 90 of a line, 579, 590 polar coordinates, 411 and vector equations, 579, 615 Partial derivative, 656 higher, 703 implicit function, 686 mixed, 703 second, 703 three variables, 659 Partial fraction, 477 Partial integral, 798 derivative of, 811 Partial second derivative, 703 Partial sum, 502 Particular solution, 846 Partition finite, 160 infinite, 161 of a rectangle, 715 of a rectangular box, 759 Partition point, 161, 176 Path Independence Theorem, 806 Perimeter, 115 Period, 886 Periodic function, 368 Perpendicular line and plane, 609 Perpendicular planes, 609 Perpendicular, test for, 597 Perpendicular vectors, 597 Phase shift, 886 Pi, 1r, 366 Piecewise continuous, 357 Piecewise smooth, 800 Plane, 4, 604 Point, 3 Point of inflection, 156 Point-slope equation, 16 Polar coordinates, 407 area, 421 Polar equation circle, 410 line, 409 Polar form, complex numbers, 876 Polar Integration Formula, 751 Polar rectangle, 7 50 Polar region, 750 Polynomial, 60 function, 60 Population growth, 434, 442, 463 Position, 203 Position vector, 565 of a curve, 615 of a line, 578, 590
of a plane, 606 of a point, 576, 589 Positive infinite, 24, 30 Positive infinitesimal, 28 Positive term series, 511 Potential function, 805 existence, 806 method for finding, 807 three variables, 814 Potential energy, 807 Power Rule derivatives, 63 general form, 89 integrals, 200, 471 negative exponents, 66 rational exponents, 76 Power series, 528 multiplication of, 563 operations on, 535 steps for finding, 557 table, 556 Present value, 454 Pressure, 364 Price vector, 565 Principle of Induction, 64 Principle of Superposition, 863, 896 Prism, volume, 115 Product function, 648 inner, 594 vector, 600 Product Law, order axiom, 905 Product Rule derivative, 62 vector derivative, 625 Profit, 137 Pyramid, volume, 306 Pythagoras, Theorem of, 5
Quadrant, 4 Quadratic formula, 103, 477 complex, 875 Quadric cylinder, 641 Quadric surface, 640 Quotient function, 652 Quotient Rule, derivatives, 66
R, I R*, 24 Radian, 78, 367 Radioactive decay, 463 Radius of circle, 5 of convergence, 530 of sphere, 641 Range, 10 Rate of growth, 51, 434, 442, 463
INDEX A63
Ratio Test, 524 Root, 2 Set, 2 Rational function, 60, 474 Root Axiom, 905 Side condition, 692 differentiation, 68 Rotation of axes, 280 Simple closed curve, 801 integration, 477 equations, 277 Simple oscillation, 886 Rational number, 3 Simpson's approximation, 230 Rational term, 60 Saddle point, 690 Simpson's Rule, 231 Real direction, 630 Scalar, 566 sin, 77, 368 Real expression, 906 Scalar Associative Law, 570 Sine, 77, 368 Real length, 630 Scalar equation derivative, 79, 374 Real line, 1 of a line, 577 integral, 377 Real number, 1 of a plane, 604 power series, 558 Real part, 874 Scalar multiple, 570 sinh, 449 Real statement, 907 derivative, 627 derivative, 449 Real term, 906 Scalar product, 593 integral, 449 Real vector, 630 Scalar triple product, 604 power series, 537 Rearrangement, series, 523 Schwartz' Inequality Skolem, Thoralf, 903 Reciprocal function, 8 integrals, 236 Slide rule, 438 Rectangle vectors, 637 Slope area, 115 sec, 370 average, 22, 168 closed, 713 Secant function, 370 curve, 25, 43 open, 794 derivative, 376 directional, 786 polar, 750 integral, 471 function, 43 Rectangle Property, 188 Secant line, 295 implicit function, 680 Rectangular box, 758 sech, 450 as a limit, 119 Rectangular coordinates Second degree equation line, 16 plane, 3 variables polar coordinates, 413 space, 585, 639 two, 272 Slope-intercept equation, 17 Rectangular solid, 115, 758 three, 640 Smooth curve, 319, 795 Reduction formula Second derivative, 94 piecewise, 800 secant and cosecant, 399 partial, 703 space, 803 sine and cosine, 398 Second Derivative Test, 138 Smooth function, 662, 824 tangent and cotangent, 397 two variables, 707 continuity of, 666 Region Second differential, 94 Smooth surface, 662, 824 below a curve, 175 Second order differential Snail, 448 between two curves, 218, equation, 461 Solenoidal field, 822 717 Second partial derivative, 703 Solid region, 757 bounded, 694 Sector, 365 Solution, differential closed, 689 area, 115, 324, 366, 420 equation, 461, 846 cylindrical, 770 Separable variables, 465, 848 Speed, 623 hyperreal, 697 solution method, 848 Sphere, 641 open, 694 Sequence surface area, 115, 333 polar, 750 graph of, 492 volume, 115, 317 solid, 757 increasing, 511 Spherical box, 776 spherical, 775 infinite, 492 Spherical coordinates, 775 unbounded, 694 of partial sums, 502 Spherical Integration under a curve, 175 Series, 502 Formula, 777 Related rates, 112 absolutely convergent, 521 Spherical region, 775 Relativity, theory of, 239 alternating, 517 Spiral Resonance, 897 conditionally convergent, of Archimedes, 411, 474 Resonant frequency, 900 521 exponential, 448 Revenue, 114 convergent, 502 vertical, 616 Riemann sum, 176, 178 divergent, 502 Spring constant, 888 double, 715 of functions, 528 Square function, 8 finite, 178 infinite, 502 Square root, 2 infinite, 181 positive term, 511 st, 36 infinite double, 719 sum, 502 Standard part, 36 triple 759 summary of tests, 525 function, 910 Right-handed coordinates, 585 tail, 508 principle, 36, 908 Robinson, Abraham, 904 Taylor, 554 rules, 37 Rolle's Theorem, 165 telescoping, 506 vector, 629
A64 INDEX
Statement, real, 907 in space, 621 dependent, 8, 45 Steady state part of the and vector derivatives, 620 dummy, 119, 178 solution, 899 Tangent plane, 666 independent, 8, 45 Steepest ascent, 788 of an implicit function, 686 intermediate, 671 Steepest descent, 788 Tangent vector, 621 point, 578 Stokes' Theorem, 834 tanh, 450 vector, 578, 589 Straight line, 16 Tarski, Alfred, 904 Variation of constants, 857 in space, 589 Taylor's Formula, 550 Vector, 565 Strictly between, 2 Taylor polynomial, 549 continuous, 635 Strictly within, 282 Taylor remainder, 549 derivative, 620 Subinterval, 160 Taylor series, 554 difference, 569 Subrectangle, 715 table of, 556 differential, 633 Substitution, integration by, Telescoping series, 506 dimensions 210, 484 Term, 9 n, 586 trigonometric, 402 real, 906 two, 565 Sum Terminal point three, 586 function, 648 directed line segment, 564 direction, 578, 606 partial, 502 smooth curve, 795 equation, 615 Riemann, 178 Third partial derivative, 704 of a line, 578, 589 series, 502 Topographic map, 644 of a plane, 606 trapezoidal, 226 Torus, 336 field, 805 vector, 567 Total differential, 662 finite, 627 Sum Law, order axiom, 905 three variables, 669 function, 615 Sum Rule Transfer Axiom, 908 h yperreal, 627 derivatives, 61 Transfer Principle, 28 increment, 633 double integral, 732 Transient part of solution, 899 infinite, 627 integral, 200 Transitive Law, 905 infinitesimal, 627 series, 508 Translation of axes, 275 negative, 569 use in integration, 483 Transverse axis, 268 normal, 606, 828 vector derivative, 625 Trapezoidal approximation, position, 565 Surface 226 product, 600 area, 328, 825 Trapezoidal Rule, 228 derivative, 627 area, polar, 428 Trapezoidal sum, 226 real, 630 as a boundary, 834 Triangle, area, 115 rules, 625 contour map of, 644 Triangle inequality, 568 sum, 567 implicit, 686 Trichotomy Law, 905 valued function, 615 integral, 829 Trigonometric functions, 365 variable, 578, 589 level, 791 Trigonometric identities, 372 Velocity, 23 oriented, 828 Trigonometric substitution, average, 169 quadric, 640 402 and integration, 203 of revolution, 327 Triple integral, 760 vector, 565, 622 sketching method, 645 Triple Riemann sum, 759 Vertex, parabola, 257 smooth, 662, 824 Two-point equation, 18 Vertical axis, 3 tangent plane of, 666 Vertical ellipse, 266 topographic map of, 644 U1traproduct, 903 Vertical line, 16 Symmetric region, 784 equivalence, 912 Vertical parabola, 259 Symmetry and center of mass, Unbounded open region, 694 Volume, 305 347 Undefined term, 6 below a surface, 723 System of formulas, 45 Undetermined coefficients, between two surfaces, 738 893 cylindrical shell method, Tail rule, 508 Uniqueness Theorem 313 Tail, series, 508 differential equations, 870 disc method, 309 tan 79, 370 double integrals, 722 as a double integral, 723 Tangent of an angle, 79 Unit circle, 365 element of, 764 Tangent function, 370 Unit hyperbola, 450 function, 711, 722 derivative, 376 Unit vector, 570, 588 geometric figures, 115 integral, 471 Upper endpoint, 2 as the integral of area, 305 Tangent line, 53 solid of revolution, 309, 313 of an implicit function, 686 Variable, 8, 45 as a triple integral, 763, 778 in polar coordinates, 412 bound, 119, 178 under a surface, 723
Water pressure, 364 Wave equation, 708 Weierstrass, Karl, 903 Work against gravity, 348 as an inner product, 595 as a line integral, 796 as a single integral, 349
x-axis, 3 x-component, 564, 586 x-coordinate, 3 (x,y) plane, 3, 639 (x,z) plane, 639 (x,y,z) space, 585, 639
y-axis, 3
INDEX
y-component, 564, 586 y-coordinate, 3 (y,z) plane, 639
z-axis, 585, 639 z-component, 586 Zero of a function, 161 Zero vector, 567
A65
RUlES OF INTEGRATION I du = u + C
I du + dv = I du + I dv (Sum Rule)
TABlE OF INTEGrlAlS
(The constant of integration is left out.) I x-+' x' dx = -- r # -I r +I' I ex dx =ex I sinx dx = -cosx I tanx dx = In lsecxl I secx dx = lnlsecx + tanxl
I sec2 x dx = tanx I tanx secx dx = secx J sin2 x dx = -!x --! sinx cosx I I ll- I I sin"xdx= --sin"- 1 xcosx+-- sin"-2 xdx ll ll ftan"x dx = - 1 - tan"- 1 x- Itan"-2 x dx n - I
I
sec"xdx =
1 -sec"- 1 xsinx + n
2
Isec"-2 xdx
n - I n - I I arcsinx dx = x arcsinx + F- x2
I arctanx dx = x arctanx --!In(! + x2)
I arcsecx dx = x arcsecx - ln(x + JxZ=!J, x :2: I I ~=arcsinx I ~ = arcsecx lx1Jx2 - I
I k du = k I du (Constant Rule)
I u dv = uv - I v du (Integration by Parts)
I [ 1 dx = lnlxl
I lnlxl dx = x lnlxl - x I cosx dx = sinx
I cotxdx = lnlsinxl f cscxdx = -lnlcscx + cotxl
I csc2 x dx = -cot x
Icotxcscxdx = -cscx
I cos2 x dx = tx + t sinx cosx
I
cos"x dx = ~ cos"- 1 x sinx + 11 - 1 Icos"-2 x dx ll ll
I
cot"x dx = -
1 - cot"- 1 x- Icot"~2 x dx n- I
I
csc"x dx = -
1 - csc"- 1 x cosx +
11

2 Icsc"-2 x dJ 11- 1 n- 1 I arccosxdx = xarccosx- ~ I arccotx dx = x arccotx +-!In(! + x2 ) I arccscx dx = x arccscx + ln(x + JxZ=!J, x :2: I I dx --2 = arctan x 1 + X
ELEMENTARY ALGEBRA OF THE REAL NUMBERS
a+b=b+a a + (b + c) = (a + b) + c a(b + c) = ab + ac a+0=a•1=a a-a=O
-(-a)= a
-(a- b)= b- a
a c ad + be b + d = bd (b,d #- 0)
a 1 =a !" = 1
a"'-" = a"'ja" (a #- 0) a'"b"' = (ab)m Fa = -.:ja, ( -a)m/n = ~' (a> O,n odd) If a < b then a + c < b + c. If a < b, then - b < -a. 1-al = lal lal • lbl = labl ~ = 1~1 (b #- 0). lbl b
The following terms are undefined: a/0, 0°, o-m, Fa (a, m positive, n even)
ab = ba a(bc) = (ab)c a(-b) = (-a)b = -ab a•O = 0 aja = 1 (a #- 0) 1 -=a 1/a (a#- 0) 1 b
ajb a
(a, b #- 0)
a c ac -.- = - (b, d #- 0) b d bd a0 = 1 (a #- 0) 0" = 0 (n > 0) amn = (amj' amJn = F (a > 0) .::fo.::jb = ;fob (a, b > 0)
If a < b and 0 < c, then ac < be. IfO <a< b, then 1/b < !fa.
-b ± jb2 - 4ac Quadratic F omwla: ax2 + bx + c = 0 if and only if x = ----'--2a
ALGEBRA OF HYPERREAL NUMBERS
Notation:
e, [J are positive infinitesimals b, c are positive and finite but not infinitesimal H, K are positive infinite.
The following are infinitesimal: -e, 1/H, ejb, sjH, bjH, E + 0, e - 0, E • 0, b • E, .:ji.
The following are finite but not infinitesimal: -b, ljb, bjc, b + e, b • c, .:jb, b + c b - c is finite (possibly infinitesimal).
The following are infinite: -H, lje, bfe, Hje, Hjb, H + e, H + b, H • b, H. K, .::(ii, H + K
The following can be infinitesimal, finite but not infinitesimal, or infinite: ejb, HjK, He, H - K
STANDARD PARTS
In the following b, c are finite (possibly infinitesimal).
st(b + c) = st(b) + st(c) st(bc) = st(b)st(c) st(jb) = ;rst{b) if b > 0, n > 0 b::::; st(b) b = st(b) if and only if b is real st(s) = 0, st(H) is undefined
TRIGONOMETRIC IDENTITIES
sin2 8 + cos2 0 = l cot>O + l = csc2 0 cos( ~e)= cosO cos(n/2 ~ 8) =sinO cos(O + r{!) = cosOcosr{! ~sinO sin¢
RULES OF EXPONENTS AND LOGARITHMS
Assume a > 0, b > 0.
1-' = l
a-'"= (ax)Y
alogaX = X
log.(xy) = log,x + log,y log.(x'') = y log,x
logby = log.y/log.b
TABLE OF DERIVATIVES
du = (dujdx) dx (Chain Rule) d(u + v) = du + dv (Sum Rule)
d (u) = v du ~ u dv ~ (Quotient Rule) v v2 d(ln u) = duju d(sin u) = cos u du d(tan tt) = sec2 u du d(sec u) = tan u sec u du du d(arcsinu) = ~ v 1 ~ !12 du d(arctanu) = -2 1 + ll du d(arcsecu) = ~ [u[.yu2 ~I
st(b ~ c) = st(b) ~ st(c) st(bjc) = st(b)jst(c) if st(c) oft 0 st(b') = st(b)"1'l if st(b) > 0 b ::::; c if and only if st(b) = st(c) if b s; c, then st(b) s; st(c)
tan 2 0 + I = sec2 e sin(~O) =~sinO sin(n/2 ~ 0) =cosO sin(O + ¢) = siniJcosrfJ +cosO sin¢
axbx = (ab)'
log.(ax) = x log.(xjy) = log.x ~ log.y bx = axlogab
d(ku) = k du (Constant Rule) d(tw) = u dv + v du (Product Rule)
d(u') = ru'- 1 du (Power Rule)
d(e") = e" du d(cos u) = ~sin u dtt d(cot u) = ~csc2 u dtt d(csc 11) = ~cot u esc u d11
du d(arccosu) = ~ --Jl-=7 du d(arccotu) = ~ -1 + 112 du d(arccscu) = ~ ---[u[p-=i
ILES OF INTEGRATION
'tl = ll + c 'u + dv = f du + f dv (Sum Rule)
.BLE OF INTEGRALS
te constant of integration is left out.)
xr+ 1 'dx = -- r oft -1 r + 1'
inxdx= -cosx
m x dx = In lsecxl
~cx dx = lnlsecx + tanxl
~c2 x dx = tanx
mx secx dx = secx
n2 xdx = !x- !sinxcosx
n'' x dx = - - sin''- 1 x cosx + -- sin"- 2 x dx 1 n-1s . " " m"x dx = 1 - tan"- 1 x - J tan"- 2 x dx II - 1
~C11X dx = --sec"- 1 x sinx + -- sec"- 2 x dx 1 ll- 2f II - 1 II - 1
rcsinxdx = xarcsinx + ~
:ctan x dx = x arctan x - ! In (I + x2)
:csecx dx = x arcsecx - ln(x + ~), x 2 1
dx . ~ = arcsmx r1- x2 dx ---- = arcsecx :lp-=!
J k du = k J du (Constant Rule) f u dv = uv - f v du (Integration by Parts)
f x-1 dx = lnlxl f lnlxl dx = x lnlxl - x J cosx dx = sinx .f cotxdx = lnlsinxl f cscx dx = -In lcscx + cotxl J csc2 x dx = -cot x f cotx cscx dx = -cscx J cos2 x dx = !x +! sinx cosx
f
cos"xdx = ~cos"-1xsinx + ~fcos"-2xdx n " Jcot"x dx = - - 1-cot''- 1 x- Jcot"-2 x dx n - 1 fcsc"x dx = - - 1 - csc"- 1 x cosx + 11 2 Jcsc"- 2 x dx n-1 n-1 J arccos x dx = x arccos x - fi-=-7 J arccotx dx = x arccotx +! ln(1 + x2) f arccscx dx = x arccscx + ln(x + p-=!), x 2 1 f dx •X
Absolute convergence, 521 trapezoidal, 226 surface, 825 Absolute value, 12 within E, 284 surface of revolution, 328 of a complex number, 876 Arc, 323, 365 under a curve, 188 Absolute value function, 12 length, 324, 366, 420 Argand diagram, 876 Acceleration, 94 arccos, 384 Argument, complex number, vector, 565, 623 Arccosecant, 384 876 Addition formulas for sine derivative, 387 Associative Law, 905 and cosine, 372 Arccosine, 384 inner product, 597 Addition Property, 190, 303 derivative, 387 scalar multiples, 570 variables integral, 392 vector product, failure, 602 two, 712, 720 arccot, 384 vector sum, 568 three, 760 Arccotangent, 384 Asymptote Adiabatic process, 490 derivative, 387 of a hyperbola, 270 Almqst parallel vectors, 632 arccsc, 384 vertical, 251 Alternating series, 517 Archimedes, 902 Average slope, 22, 168 harmonic, 520, 546 arcsec, 384 Average speed, 339 Alternating Series Test, 518 Arcsecant, 384 Average value, 337 Amplitude, 886 derivative, 387 Average velocity, 169, 339 Angle, 77, 365 integral, 473 Axioms between vectors, 573 arcsin, 384 for hyperreal numbers, 906 three dimensions, 588 Arcsine, 384 for real numbers, 905 Angular measure, 367 derivative, 387 Axis Anticommutative Law for integral, 392 coordinates vector products, 604 power series, 560 plane, 3 Antiderivative, 192 arctan, 384 space, 585 Approximation Arctangent, 384 ellipse, 264 by alternating series, 518 derivative, 387 hyperbola, 268 arctan x, 541 integral, 472 parabola, 257 by derivatives, 286 power series, 536 e, 442, 551 Area Barrow, Isaac, 902 In x, 541 below a curve, 188 Basis vectors by Newton's method, 290 between two curves, 220, plane, 570 pi, 542 304 space, 587 by power series, 541 double integral, 722 Bell-shaped curve, 537 by Riemann sums, 178 geometric figures, 115 Berkeley, Bishop, 903 Simpson's, 230 by Green's Theorem, 818 Binomial series, 558 sin x, 552 infinite, 353 Binomial Theorem, 559 by Taylor's Formula, 547 polar, 421 Bound variable, 119, 178
A 57
A 58 INDEX
Boundary oriented surface, 834 plane region, 689 solid region, 757, 838 Bounded open region, 694 Box cylindrical, 770 rectangular, 758 spherical, 776
Cantor, Georg, 903 Capital Accumulation, 452 Cardioid, 751 Catenary, 449 Cauchy, A. L., 903 Cauchy Convergence Test (Criterion), 500, 504 Cauchy Divergence Test, 505 Cauchy equivalence, 911 Cauchy sequence, 911 Center circle, 5 ellipse, 264 hyperbola, 268 sphere, 641 Center of gravity, 344 Center of mass double integral, 745 triple integral, 765 two dimensions, 346 wire, 344 Centroid, 346 polar region, 430 Chain Rule for continuous functions, 130 variables one, 86 two, 672 Change of variables, 210, 484 Changing base of logarithms, 439 Characteristic polynomial, 882 Circle, 5 area, 115 polar equation, 410 Circular cylinder, 644 Circular helix, 616 Circulation dimensions two, 820 three, 836 Circumference, 115, 324 Circumscribed cylinder, 721 Circumscribed rectangle, 189 Circumscribed rectangular box, 759 cis, 876 Closed interval, 2 hyperreal, 908
Closed rectangle, 713 Closed region, 689 hyper real, 700 space, 757 Closure Laws, 905 Commodity vector, 565 Common logarithm, 436 Commutative Law, 905 inner product, 597 vector product, failure, 602 vector sum, 568 Comparison test, 512, 524 Completeness axiom, 905 Completing the square, 274, 479 Complex number, 874 conjugate, 875 exponent, 879 plane, 874 Complex valued differential equation, 879 Complex valued function, 879 Component directed line segment, 564 vector, 565 valued function, 615 Composition of functions, 86 continuity, 130, 653 derivative, 86, 672 Concave downward, 152 Concave upward, 152 Conditional convergence, 521 Cone area, 327 elliptic, 642 volume, 115 Conic section, 264 Conjugate axis, 268 Conjugate complex, 875 Conservative field, 807 Constant, II Constant function, II derivative of, 61 integral of, 184 Constant of integration, 199 Constant on an interval, 151 Constant Rule derivative, 61 double integral, 732 integral, 200 series, 508 Continuity, 125 of compositions, 130, 653 of differentiable functions, 127 E, o condition, 288 of an integral, 192 on an interval, 132 law of, 902 on a set, 653
two variables, 651 vector, 635 Continuously compounded interest, 443, 497 Contour, 645 Contour map, 644 Convergence improper integral, 352 increasing sequence, 511 interval of, 529 positive term series, 511 radius of, 530 sequence, 493 series, 502 summary of tests, 525 Coordinates cylindrical, 769 polar, 407 rectangular plane, 3 space, 585, 639 spherical, 775 cos, 78, 368 Cosecant function, 370 derivative, 376 cosh, 449 power series, 536 Cosine, 78, 368 derivative, 79, 374 integral, 377 power series, 558 Cosines, Law of, 573 cot, 370 Cotangent function, 370 derivative, 376 coth, 450 Critical damping, 887 Critical point, 136 interior, 137 two variables, 689 Critical Point Theorem, 136 two variables, 689 Cross product, 600 esc, 370 csch, 450 Curl dimensions two, 819 three, 832 Curve length, 320 parametric, 321 polar coordinates, 425 in space, 622 Curve sketching using derivatives, 156 ellipse, 266 hyperbola, 269 using limits, 251 parabola, 261 polar, 415
INDEX A 59
rotation of axes, 281 on an interval, 132 infinite, 719 translation of axes, 275 two variables, 710 Dummy variable, 119, 178 Cycloid, 616 Differential, 55 Cylinder exact, 806 e, 83, 443 area, 327 form, 806 irrationality of, 563 circular, 644 second, 94 as a limit, 443 quadric, 641 total, 662 Element volume, 115, 309 vector, 633 of area, 736 Cylinder Property, 712, 721 Differential equation of a set, 2 Cylindrical box, 770 complex valued, 879 of volume Cylindrical coordinates, exact, 810 cylindrical, 772 769 first order, 461, 846 rectangular, 764 Cylindrical Integration homogeneous, 852, 882 spherical, 777 Formula, 771 linear, 857 Elementary function, 481 Cylindrical region, 770 second order, 461, 881 Ellipse, 264 Cylindrical shell method, 313 separable variables, 848 sketching method, 266 Differentiation, 45 Ellipsoid, 642 Damped oscillation, 887 logarithmic, 470 Elliptic cone, 642 Damping, critical, 887 rules for, 68 Elliptic cylinder, 641 Decreasing on an interval, 152 Direct Test, 137 Elliptic paraboloid, 642 Definite integral, 183 Directed curve, 795 Empty set, 3 Degenerate conic section, 263, Directed line segment, 564 Endpoint, 2 273 Direction angle, 574, 588 Epidemic, 463 Degree, angular unit, 367 Direction cosine, 574, 588 t, o condition Deleted neighborhood, 243 Direction, real, 630 continuity, 288 Demand function, 114 Direction vector derivative, 295 De Moivre's Formula, 878 of a line, 578, 590 limits, 283, 286 Density of a plane, 606 t, N condition dimensions Directional derivative, 786 for infinite limits, 499 one, 341 second, 793 for sequences, 499 two, 342 three dimensions, 791 Equality of mixed partials, three, 764 Directrix, parabola, 257 703 of a wire, 364 Disc method, 309 three variables, 705 Dependent variable, 8, 45 Discontinuity, 125 Equation, 906 Derivative, 45 Discriminant, 875 differential, 461, 846 directional, 786 test, 273 linear, 20 exponential function, 83, two variables, 273 parametric, 90 443 Displacement vector, 565 point-slope, 16 higher, 94 Distance second degree, 272 higher partial, 703 plane, 4 two-point, 18 hyperbolic functions, 449 between reals, 13, 282 Error estimate, 228, 296 inverse trig functions, 387 space, 641 for derivatives, 296 In, 84, 455 Distributive Law, 905 for Simpson's Rule, 231 mixed partial, 703 scalar multiples, 570 and Taylor's Formula, 550 partial, 656 products for trapezoidal sums, 228 polar coordinates, 414 inner, 597 Escape velocity, 491 power rule, 63, 76 vector, 602 Euler approximation, 865 power series, 534 Divergence Euler's Formula, 879 rational function, 68 dimensions Even function, 218 rules for, 68 two, 819 Exact differential, 806 sec and esc, 376 three, 833 equation, 810 second, 94 Theorem, 839 Existence Theorem, differensecond partial, 703 Divergent tial equations, 867 sin and cos, 374 improper integral, 352 exp, 443 tan and cot, 376 sequence, 494 Explosion, 851, 871 vector, 620 series, 502 Exponent Determinant, 600 Domain, 10 complex, 879 Difference, vector, 569 Dot product, 593 inequalities, 432 Differentiable function, 45 Double integral, 711,719 real, 432 continuity of, 127 Double Riemann sum, 715 rules, 432
A60 INDEX
Exponential function, 83, 443 with base, 82, 432 continuity, 433 derivative, 83, 443 integral, 444 power series, 536 Exponential spiral, 448 Expression, real, 28, 906 Extension Axiom, 906 Extension Principle, 27 Extreme Value Theorem, 164 two dimensions, 689
Factor Theorem, 477 Factorial, n!, 493 Fermat, P., 902 Finite hyper real number, 30 partition, 160 Riemann sum, 178 set, 3 vector, 627 First Derivative Test, 174 First order differential equation, 461 Fluid flow, 820 Flux, 821 Focus ellipse, 264 hyperbola, 268 parabola, 257 Force vector, 565 Forcing term, 892 Formula, 45, 906 Frequency, 886 Frustum, area, 328 Fubini's Theorem, 726 Function, 45 hyperreal, 27 implicit, 97 inverse, 70, 381 linear, 17 real, 6 two variables, 14 vector-valued, 615 Fundamental Theorem of Calculus, 193
Galileo, G., 902 Gauss' Theorem, 839 General solution, 461, 846 Generalized Mean Value Theorem, 552 Geometric series, 503 formula, 446 Godel, Kurt, 904 grad f, 787 Gradient, 787 Graph, 7 dimensions
one, 7 two, 640 of an equation, 5 infinite sequence, 492 polar coordinates, 408 space, 639 Gravity dimensions one, 491 two, 809 three, 843, 844 Greatest integer function, 130 Green's Theorem, 815 vector form, 820 Growth rate, 51 populations, 51, 462, 491 sequences, 497
Half-life, 463 Half-open interval, 2 Harmonic series, 505 alternating, 546 Helix, circular, 616 Hewitt, E., 904 Higher derivative, 94 partial, 703 Higher differential, 94 Homogeneous differential equation, 852 second order, 8 82 solution, 852, 886 Homogeneous function, 710 Horizontal axis, 3 Horizontal ellipse, 266 Horizontal parabola, 261 Hyperbola, 268 graphing method, 270 Hyperbolic cosine, 449 derivative, 449 integral, 449 Hyperbolic cylinder, 641 Hyperbolic function, 449 Hyperbolic identities, 449 Hyperbolic paraboloid, 643 Hyperbolic sine, 449 derivative, 449 integral, 449 Hyperboloid, 643 Hyperinteger, 160 Hyperrational number, 431 Hyperreal function, 27 Hyper real interval, 161, 908 Hyperreal number, 27 Hyperreal region, 697 Hyperreal vector, 627
i, 874 Identity function, 12 Identity Law, 905 vector sum, 568
Imaginary number, 874 Imaginary part, 874 Implicit differentiation, 97 Implicit function, 97 derivative of, 97, 680 two variables, 680 Implicit Function Theorem, 680 Improper integral, 352 Incompressible field, 822 Increasing function, 151 Increasing sequence, 511 Increment, 45 two variables, 662 vector, 633 Increment Theorem, 54, 299 variables two, 664 three, 670 Indefinite integral, 199 Ind_ependent variable, 8, 45 Indeterminate form, 31 Induction, Principle of, 64 Inequalities for exponents, 432 Inequality, 906 Inequality Rule double integral, 733 series, 508 Inertia, moment of dimensions two, 748 three, 767 Infinite hyperreal number, 24 examples, 912 Infinite partial sum, 502 Infinite partition, 161 Infinite Riemann sum, 181 double, 719 Infinite sequence, 492 Infinite series, 502 Infinite Sum Theorem, 303 double integral, 736 triple integral, 764 Infinite telescope, 24 Infinite vector, 627 Infinitely close, 24, 35 compared to, 302 two dimensions, 651 vectors, 627 Infinitesimal, 24, 28 examples, 913 microscope, 24 vector, 627 Inflection, point of, 156 Initial condition, 462, 847, 881 Initial point directed line segment, 564 smooth curve, 795 Initial value problem first order, 847
INDEX A61
second order, 881 double, 725 Local maximum, 156 Inner product, 594 triple, 761 Local minimum, 156 derivative, 625 Locus rules, 597 Judicious guessing, 893 of an equation, 7 Inscribed cylinder, 721 Justification of a definition, polar coordinates, 408 Inscribed rectangle, 189 304 space, 639 Integers, 3 Kepler, J., 902 Logarithm Integral changing base, 439 arcsec and arccsc, 473 Kinetic energy, 784 common, 436 arcsin and arccos, 392 Kline, Morris, 907 natural, 83, 454 arctan and arccot, 472 Law of continuity, 902 rules, 437 definite, 183 Law of Cosines, 573 Logarithmic differentiation, double, 711, 719 Least squares, method of, 702 470 exponential function, 443 Leibniz, Gottfried Wilhelm, Logarithmic function, 436 improper, 352 902 Logistic function, 464 indefinite, 199 Length Lower endpoint, 2 iterated, 724 arc, 366 line, 795 curve, 320 MacLaurin series, 555 In, 456 parametric, 321 MacLaurin's Formula, 550 partial, 798 space, 622 Magnitude, 567 power series, 534 polar coordinates, 425 Major axis, ellipse, 264 rational functions, 477 real, 630 Malcev, A., 904 sin and cos, 377 vector, 566 Map, 209 sinh and cosh, 449 Level curve, 645 Marginal price vector, 624 surface, 829 Level surface, 791 Marginal revenue, 114 tan, 471 !'Hospital's Rule Marginal values, 51 triple, 760 for 0/0, 243 Mass Integral Test, 514 for oo/oo, 246 dimensions Integration Limit, 117 one, 341 change of variables, 210, and curve sketching, 251 two, 342 484 E, o condition, 283, 286 double integral, 743 coordinates finite, 117 triple integral, 764 cylindrical, 771 from the left, 123 Mass-spring system, 888 polar, 751 from the right, 123 Maximum, 135 spherical, 777 infinite, 237 closed region, 691 methods of, 481 one-sided, 123 local, 156 by parts, 391, 394 of a Riemann sum, 188, 316 open region, 698 substitution, 210, 484 rules, 121 two variables, 688 trigonometric, 403, 485 sequence, 493 value, 135 Interest, 442, 451 and standard parts, 120 Mean Value Theorem, 168 Interior point, 137, 689 two variables, 656 generalized, 552 critical, 137 Limit Comparison Test, 513, for Integrals, 338 Intermediate Value Theorem, 524 Member of a set, 2 162 Line, 16 Method of least squares, 702 Intermediate variable, 671 equation of, 16, 18, 20 Methods of integration, 481 Interval, 2 polar equation, 409 Midpoint, 582, 592 of convergence, 529 space, 589 Minimum, 135 hyperreal, 161, 908 vector equation, plane, 578 local, 156 Inverse function, 70, 381 vector equation, space, 589 two variables, 688 hyperbolic, 461 vertical, 16 value, 135 trigonometric, 384 Line integral, 795 Minor axis, ellipse, 264 Inverse Function Rule, 72 Linear differential equation, Mixed partial derivative, 703 Inverse Function Theorem, 857 Model theory, 904 386 second order, 892 Moment Inverse Laws, 905 solution, 858, 862, 897 dimensions Inverse relation, 3 81 Linear equation, 20 one, 343 Inverse square law, 491 Linear function, 17, 60 two, 345 Irrotational field, 822 In, 83, 454 double integral, 744 Iterated integral, 724 integral, 456 triple integral, 765 Iterated Integral Theorem power series, 536 wire, 344
A62 INDEX
Moment of inertia (second moment), dimensions two, 748 three, 767
Natural extension, 906 Natural logarithm, 81, 454 derivative, 84, 455 integral, 456 Negative infinite, 24, 30 Negative infinitesimal, 28 Neighborhood, 243 deleted, 243 Newton, Sir Isaac, 902 Newton's First Law, 567 Newton's Law of Cooling, 490 Newton's method, 290 Newton's Second Law, 570 Nonstandard analysis, 904 Nonuniqueness, differential equation, 872 Norm of a vector, 566 Normal curve, 537 Normal vector curve, 606 surface, 828 nth derivative, 94 nth differential, 94
Odd function, 218 One-to-one function, 382 Open interval, 2 Open rectangle, 794 Open rectangular solid, 803 Open region, 694 Ordered pair, 3 Oriented surface, 828 Origin, 3 in polar coordinates, 412 Orthogonal vectors, 597 Oscillation damped, 887 simple, 886 Overdamping, 888
pseries, 515 Parabola, 257 graphing method, 261 Parabolic cylinder, 641 Paraboloid elliptic, 642 hyperbolic, 643 Parallel line and plane, 609 Parallel planes, 609 Parallel, test for, 597 Parallel vectors, 597 Parallelogram, 583 area, 604 Parametric curve, 90
Parametric equations, 90 of a line, 579, 590 polar coordinates, 411 and vector equations, 579, 615 Partial derivative, 656 higher, 703 implicit function, 686 mixed, 703 second, 703 three variables, 659 Partial fraction, 477 Partial integral, 798 derivative of, 811 Partial second derivative, 703 Partial sum, 502 Particular solution, 846 Partition finite, 160 infinite, 161 of a rectangle, 715 of a rectangular box, 759 Partition point, 161, 176 Path Independence Theorem, 806 Perimeter, 115 Period, 886 Periodic function, 368 Perpendicular line and plane, 609 Perpendicular planes, 609 Perpendicular, test for, 597 Perpendicular vectors, 597 Phase shift, 886 Pi, 1r, 366 Piecewise continuous, 357 Piecewise smooth, 800 Plane, 4, 604 Point, 3 Point of inflection, 156 Point-slope equation, 16 Polar coordinates, 407 area, 421 Polar equation circle, 410 line, 409 Polar form, complex numbers, 876 Polar Integration Formula, 751 Polar rectangle, 7 50 Polar region, 750 Polynomial, 60 function, 60 Population growth, 434, 442, 463 Position, 203 Position vector, 565 of a curve, 615 of a line, 578, 590
of a plane, 606 of a point, 576, 589 Positive infinite, 24, 30 Positive infinitesimal, 28 Positive term series, 511 Potential function, 805 existence, 806 method for finding, 807 three variables, 814 Potential energy, 807 Power Rule derivatives, 63 general form, 89 integrals, 200, 471 negative exponents, 66 rational exponents, 76 Power series, 528 multiplication of, 563 operations on, 535 steps for finding, 557 table, 556 Present value, 454 Pressure, 364 Price vector, 565 Principle of Induction, 64 Principle of Superposition, 863, 896 Prism, volume, 115 Product function, 648 inner, 594 vector, 600 Product Law, order axiom, 905 Product Rule derivative, 62 vector derivative, 625 Profit, 137 Pyramid, volume, 306 Pythagoras, Theorem of, 5
Quadrant, 4 Quadratic formula, 103, 477 complex, 875 Quadric cylinder, 641 Quadric surface, 640 Quotient function, 652 Quotient Rule, derivatives, 66
R, I R*, 24 Radian, 78, 367 Radioactive decay, 463 Radius of circle, 5 of convergence, 530 of sphere, 641 Range, 10 Rate of growth, 51, 434, 442, 463
INDEX A63
Ratio Test, 524 Root, 2 Set, 2 Rational function, 60, 474 Root Axiom, 905 Side condition, 692 differentiation, 68 Rotation of axes, 280 Simple closed curve, 801 integration, 477 equations, 277 Simple oscillation, 886 Rational number, 3 Simpson's approximation, 230 Rational term, 60 Saddle point, 690 Simpson's Rule, 231 Real direction, 630 Scalar, 566 sin, 77, 368 Real expression, 906 Scalar Associative Law, 570 Sine, 77, 368 Real length, 630 Scalar equation derivative, 79, 374 Real line, 1 of a line, 577 integral, 377 Real number, 1 of a plane, 604 power series, 558 Real part, 874 Scalar multiple, 570 sinh, 449 Real statement, 907 derivative, 627 derivative, 449 Real term, 906 Scalar product, 593 integral, 449 Real vector, 630 Scalar triple product, 604 power series, 537 Rearrangement, series, 523 Schwartz' Inequality Skolem, Thoralf, 903 Reciprocal function, 8 integrals, 236 Slide rule, 438 Rectangle vectors, 637 Slope area, 115 sec, 370 average, 22, 168 closed, 713 Secant function, 370 curve, 25, 43 open, 794 derivative, 376 directional, 786 polar, 750 integral, 471 function, 43 Rectangle Property, 188 Secant line, 295 implicit function, 680 Rectangular box, 758 sech, 450 as a limit, 119 Rectangular coordinates Second degree equation line, 16 plane, 3 variables polar coordinates, 413 space, 585, 639 two, 272 Slope-intercept equation, 17 Rectangular solid, 115, 758 three, 640 Smooth curve, 319, 795 Reduction formula Second derivative, 94 piecewise, 800 secant and cosecant, 399 partial, 703 space, 803 sine and cosine, 398 Second Derivative Test, 138 Smooth function, 662, 824 tangent and cotangent, 397 two variables, 707 continuity of, 666 Region Second differential, 94 Smooth surface, 662, 824 below a curve, 175 Second order differential Snail, 448 between two curves, 218, equation, 461 Solenoidal field, 822 717 Second partial derivative, 703 Solid region, 757 bounded, 694 Sector, 365 Solution, differential closed, 689 area, 115, 324, 366, 420 equation, 461, 846 cylindrical, 770 Separable variables, 465, 848 Speed, 623 hyperreal, 697 solution method, 848 Sphere, 641 open, 694 Sequence surface area, 115, 333 polar, 750 graph of, 492 volume, 115, 317 solid, 757 increasing, 511 Spherical box, 776 spherical, 775 infinite, 492 Spherical coordinates, 775 unbounded, 694 of partial sums, 502 Spherical Integration under a curve, 175 Series, 502 Formula, 777 Related rates, 112 absolutely convergent, 521 Spherical region, 775 Relativity, theory of, 239 alternating, 517 Spiral Resonance, 897 conditionally convergent, of Archimedes, 411, 474 Resonant frequency, 900 521 exponential, 448 Revenue, 114 convergent, 502 vertical, 616 Riemann sum, 176, 178 divergent, 502 Spring constant, 888 double, 715 of functions, 528 Square function, 8 finite, 178 infinite, 502 Square root, 2 infinite, 181 positive term, 511 st, 36 infinite double, 719 sum, 502 Standard part, 36 triple 759 summary of tests, 525 function, 910 Right-handed coordinates, 585 tail, 508 principle, 36, 908 Robinson, Abraham, 904 Taylor, 554 rules, 37 Rolle's Theorem, 165 telescoping, 506 vector, 629
A64 INDEX
Statement, real, 907 in space, 621 dependent, 8, 45 Steady state part of the and vector derivatives, 620 dummy, 119, 178 solution, 899 Tangent plane, 666 independent, 8, 45 Steepest ascent, 788 of an implicit function, 686 intermediate, 671 Steepest descent, 788 Tangent vector, 621 point, 578 Stokes' Theorem, 834 tanh, 450 vector, 578, 589 Straight line, 16 Tarski, Alfred, 904 Variation of constants, 857 in space, 589 Taylor's Formula, 550 Vector, 565 Strictly between, 2 Taylor polynomial, 549 continuous, 635 Strictly within, 282 Taylor remainder, 549 derivative, 620 Subinterval, 160 Taylor series, 554 difference, 569 Subrectangle, 715 table of, 556 differential, 633 Substitution, integration by, Telescoping series, 506 dimensions 210, 484 Term, 9 n, 586 trigonometric, 402 real, 906 two, 565 Sum Terminal point three, 586 function, 648 directed line segment, 564 direction, 578, 606 partial, 502 smooth curve, 795 equation, 615 Riemann, 178 Third partial derivative, 704 of a line, 578, 589 series, 502 Topographic map, 644 of a plane, 606 trapezoidal, 226 Torus, 336 field, 805 vector, 567 Total differential, 662 finite, 627 Sum Law, order axiom, 905 three variables, 669 function, 615 Sum Rule Transfer Axiom, 908 h yperreal, 627 derivatives, 61 Transfer Principle, 28 increment, 633 double integral, 732 Transient part of solution, 899 infinite, 627 integral, 200 Transitive Law, 905 infinitesimal, 627 series, 508 Translation of axes, 275 negative, 569 use in integration, 483 Transverse axis, 268 normal, 606, 828 vector derivative, 625 Trapezoidal approximation, position, 565 Surface 226 product, 600 area, 328, 825 Trapezoidal Rule, 228 derivative, 627 area, polar, 428 Trapezoidal sum, 226 real, 630 as a boundary, 834 Triangle, area, 115 rules, 625 contour map of, 644 Triangle inequality, 568 sum, 567 implicit, 686 Trichotomy Law, 905 valued function, 615 integral, 829 Trigonometric functions, 365 variable, 578, 589 level, 791 Trigonometric identities, 372 Velocity, 23 oriented, 828 Trigonometric substitution, average, 169 quadric, 640 402 and integration, 203 of revolution, 327 Triple integral, 760 vector, 565, 622 sketching method, 645 Triple Riemann sum, 759 Vertex, parabola, 257 smooth, 662, 824 Two-point equation, 18 Vertical axis, 3 tangent plane of, 666 Vertical ellipse, 266 topographic map of, 644 U1traproduct, 903 Vertical line, 16 Symmetric region, 784 equivalence, 912 Vertical parabola, 259 Symmetry and center of mass, Unbounded open region, 694 Volume, 305 347 Undefined term, 6 below a surface, 723 System of formulas, 45 Undetermined coefficients, between two surfaces, 738 893 cylindrical shell method, Tail rule, 508 Uniqueness Theorem 313 Tail, series, 508 differential equations, 870 disc method, 309 tan 79, 370 double integrals, 722 as a double integral, 723 Tangent of an angle, 79 Unit circle, 365 element of, 764 Tangent function, 370 Unit hyperbola, 450 function, 711, 722 derivative, 376 Unit vector, 570, 588 geometric figures, 115 integral, 471 Upper endpoint, 2 as the integral of area, 305 Tangent line, 53 solid of revolution, 309, 313 of an implicit function, 686 Variable, 8, 45 as a triple integral, 763, 778 in polar coordinates, 412 bound, 119, 178 under a surface, 723
Water pressure, 364 Wave equation, 708 Weierstrass, Karl, 903 Work against gravity, 348 as an inner product, 595 as a line integral, 796 as a single integral, 349
x-axis, 3 x-component, 564, 586 x-coordinate, 3 (x,y) plane, 3, 639 (x,z) plane, 639 (x,y,z) space, 585, 639
y-axis, 3
INDEX
y-component, 564, 586 y-coordinate, 3 (y,z) plane, 639
z-axis, 585, 639 z-component, 586 Zero of a function, 161 Zero vector, 567