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Cambridge University Press 978-0-521-86744-3 - Calculus: Third Edition Michael Spivak Index More information

INDEX

AalbmcndoE, 273 Archimedian property Abel, Niels Henrik, 404, 513 for the rational numbers, 574 Abel summable, 514 for the real numbers, 136 Abel's formula for summation by parts, Archimedian spiral, 85, 246 388 Arcsec, 317, 379 Abel's Lemma, 389 Arcsin, 307 Abel's test, 488 addition formula for, 314 Abel's Theorem, 513 derivative of, 308 Absolute value, 11 Taylor series for, 509 of a complex number, 525 Arctan, 308 Absolutely convergent, 473, 547 addition formula for, 314 Absolutely summable, 473 derivative of, 308 Acceleration, 159 Taylor polynomial for, 407, 414 remainder term for, 422 Acta Eruditorum, 146 Addition, 3 Area, 250, 255 associative law for, 9 Arg cosh, 350 commutative law for, 9 Arg sinh, 350 of complex numbers, 522 Arg tanh, 350 Argument, 527 geometric interpretation of, 526 Argument function, 533 of vector-valued functions, 242 discontinuities of, 537 of vectors, 75 Argument of the hyperbolic Addition formula functions, 350 for arcsin, 314 Arithmetic mean, 33 for arctan, 314 Arrow, 75, 76 for cos, 311 \x arrow sin(x2)", 45 for sin, 310, 311 Associative law for tan, 314 for addition, 9 Additive identity of vectors, 76 existence of, 9 for multiplication, 9 for vectors, 76 Average velocity, 150 Additive inverses Axis existence of, 9 horizontal, 57 Algebra, Fundamental Theorem of, imaginary, 525 373, 529, 539, 558 real, 524 Algebraic functions, 359 vertical, 57 Algebraic number, 435 Algebraist's real numbers, 588 Almost lower bound, 140 Almost upper bound, 140 Bacon, Francis, vi Analyst's real numbers, 588 Basic properties of numbers, 3 Angle, 300 \Bent graphs", 147 directed, 300 Bernoulli, 146, 565 Antidiagonal, 239 Bernoulli numbers, 563 Arabic numerals, multiplication of, 8 Bernoulli polynomials, 566 Arc length, 275, 281 Bernoulli's inequality, 32 Arccos, 308 Big game hunting, mathematical derivative of, 308 theory of, 543 Archimedes, 136, 139, 260 Binary operation, 571

659

660 Index

Binomial coefˇcient, 27, 429 Closed rectangle, 538 Binomial series, 487, 510 Closure under addition, 9 Binomial theorem, 28 Closure under multiplication, 9 Bisection argument, 140, 543 Commutative law Bohr, Harold, 390 for addition, 9 Bolzano-Weierstrass Theorem, 451, 461, of vectors, 76 543 for multiplication, 9 Bound Comparison test, 467, 468 almost lower, 140 Comparison Theorem, Sturm, 320 almost upper, 140 Complete induction, 23 greatest lower, 132 Complete ordered ˇeld, 574, 593 least upper, 131, 574 Completing the square, 17, 375 lower, 132 Complex analysis, 556 upper, 131 Complex function Bounded above, 120, 131, 450, 574 continuous, 536 Bounded below, 132, 450 differentiable, 541 Bourbaki, Nicholas, 146 graph of, 533 limit of, 533 nondifferentiable, 542 Taylor series for, 554 Cardioid, 89, 247 Complex nth root, 527 Cartesian coordinates, 84 Complex numbers, 517, 522 Cauchy, 278 absolute value of, 525 Cauchy Condensation Theorem, 488 addition of, 522 Cauchy criterion, 466 geometric interpretation of, 526 Cauchy form of the remainder, 417, geometric interpretation of, 525 419 imaginary part of, 522 Cauchy Mean Value Theorem, 201 inˇnite sequence of, 546 Cauchy product, 486, 505 inˇnite series of, 546{548 Cauchy sequence, 452, 562 logarithm of, 561 equivalence of, 589 modulus of, 525 Cauchy-Hadamard formula, 560 multiplication of, 522 Cauchy-Schwarz inequality, 278 geometric interpretation Cavalieri, 272 of, 526{527 Cesaro summable, 486 real part of, 522 Chain Rule, 172 ff. Complex plane, 524 proof of, 176 Complex power series, 548 Change, rate of, 150 circle of convergence of, 550 Characteristic (of a ˇeld), 576 radius of convergence of, 550 Circle, 65 Complex-valued functions, 532 \f circle g", 44 Composition of functions, 44 unit, 66 Concave function, 217 Circle of convergence, 550 Conditionally convergent series, 474 Classical notation Cone, 80 for derivatives, 152{154, 160, 165, generating line of, 80 184, 238 surface area of, 399 for integrals, 262 Conic sections, 80; see also Ellipse, Cleio, 183 Hyperbola, Parabola Closed interval, 57 Conjugate, 525, 530

Index 661

Conjugate function, 532 derivative of, 307 Constant function, 43 Cubic equation, general solution, Construction of the real numbers, 519{520 578 ff. Curve Continued fraction, 455 parameterized, tangent line of, 243 Continuous, uniformly, 142 parametric representation of, 241 Continuous at a, 113, 536 reparameterization of, 244 Continuous function, 113, 116, 537 Cycloid, 247 nowhere differentiable, 157, 501 Continuous on (a, b), 116 Continuous on [a,b], 116 Contraction, 459 Darboux's Theorem, 211 Contraction lemma, 459 De Moivre's Theorem, 527 Converge Decimal expansion, 73, 485 pointwise, 494 Decreasing function, 192 uniformly, 494, 498 Decreasing sequence, 450 Convergent sequence, 446, 546 Dedekind, Richard, 38 Convergent series, 465, 547 Deˇned implicitly, 238 absolutely, 473, 547 Deˇnite integral, 361 conditionally, 474 DEFINITION,47 Convex function, 216 Deˇnition, recursive, 23 strictly, 226 Degree (of a polynomial), 42 weakly, 226 Degree measurement, 63, 301{302 Convex subset of the plane, 226, 544 Delicate ratio test, 486 Cooling, Newton's law of, 352 Delicate root test, 486 Coordinate Dense, 138 ˇrst, 57 Derivative, 147 ff., 149 second, 57 classical notation for, 152{154, 160, Coordinate system, 57 165, 184, 238 cartesian, 84 higher-order, 159 origin of, 57 \inˇnite", 156 Coordinates left-hand, 154 polar, 84 Leibnizian notation for, see Derivative, \Corner", 60 classical notation for Cos, 300, 303, 318{319, 554 logarithmic, 348 addition formula for, 311 \negative inˇnity", 156 derivative of, 170, 304 of f , 149 inverse of, see Arccos of f at a, 149 Taylor polynomials for, 407 of vector-valued function, 243 remainder term for, 420 right-hand, 154 Cosh, 349 Schwarzian, 182 Cosine, hyperbolic, 349 second, 159 Cot, 307 Schwarz, 431 derivative of, 307 Derivative of quotient, incantation for, Countable, 442 169 Counting numbers, 21 Descartes, Rene,Ã 84 Critical point, 187 Determinant, 79 Critical value, 187 of vector-valued functions, 243 Csc, 307 Diagonal, 230

662 Index

Difference operator, 564 Entire function, 558 Differentiable, 149, 541 Epsilon, 18 Differential equation, 289, 297, 318, Equal up to order n, 412 320, 352, 357, 432 Equality, order of, 412 initial conditions for, 433 Equations, differential, see Differential Differentiation, 166 ff. equations implicit, 238 Equivalent Cauchy sequences, 589 logarithmic, 348 Etymology lesson, 82 Differentiation operator, 564 Euler, 565 Dini's Theorem, 515 Euler's number, 456 Directed angle, 300 Euler-Maclaurin Summation Formula, Dirichlet's test, 488 566 Disc method, 397 Even function, 51, 196 Discontinuities of a nondecreasing Even number, 25 function, 443 Eventually inside, 546 Discontinuity, removable, 119 Exhaustion, method of, 139 Disraeli, Benjamin, 2 Exp, 340 ff., 554 Distance, 58, 525 classical approach to, 354 shortest between two points, 275 elementary deˇnition of, 461 Distributive law, 9 Taylor polynomials for, 407 Diverge, 446, 547 remainder term for, 422 Division, 6 Expansion, decimal, 73, 485 Division by zero, 6 Extension of a function, 113{114 Domain, 40, 41, 47, 591 Dot product of vectors, 78 of vector-valued functions, 243 Double intersection, 163 Factorial, 23 Double root, 183 Factorials, table of, 428 Durege,Â 38 Factorization into primes, 31 Fibonacci, 32 Fibonacci Association, 32 Fibonacci Quarterly, The,32 e, 340 Fibonacci sequence, 32, 512, 563 irrationality of, 425 Field, 571 relation with π, 441, 555 characteristic of, 576 transcendentality of, 437 complete ordered, 574, 593 value of, 341, 422 ordered, 573 Eccentricity of ellipse, 87 First coordinate, 57 Elementary function, 359 First Fundamental Theorem of Calculus, Ellipse, 66, 82 282 axes of, 87 Fixed point of a function, 458 eccentricity of, 87 Focus point, 66, 86 equation in polar coordinates, 86 Force, as vector, 76 focus point of, 66, 86 Four leaf clover, 88 major axis of, 87 Fourier series, 315, 317, 320 minor axis of, 87 Fraction, continued, 455 Ellipsoid of revolution, 400 Function, 39, 47 Empty collection, 23 absolute value, 532

Index 663

Function (continued) polynomial, 42 argument, 533 positive part of, 51 discontinuities of, 537 power, 60 complex valued, 532 product of, 43 composition of, 44 quotient of, 43 concave, 217 rational, 42 conjugate, 532 real part, 532 constant, 43 real-valued, 532 continuous, 113 ff. \reasonable", 68, 116, 178 convex, 216 regulated, 515 critical point of, 187 step, 275 critical value of, 187 decreasing, 192 strict maximum point of, 215 derivative of, 147 ff. sum of, 43 differentiable, 149, 541 trigonometric, 300 ff. elementary, 359 value at x,40 entire, 558 vector-valued, 241 even, 51, 196 Fundamental Theorem of Algebra, 373, exponential, 340{341 529, 539, 558 extension of, 113{114 Fundamental Theorem of Calculus ˇxed point of, 458 First, 282 from A to B, 591 Second, 286 from real numbers to the plane, 241 graphs of, 57{65, 195, 533 hyperbolic, 349 identity, 43 imaginary part, 532 increasing, 192 integrable, 255 Gabriel, 402 integral of, 255 Galileo, 146, 162 inverse, 228 ff. Gamma function, 390, 437 linear, 58 Generating line, of a cone, 80 local maximum point of, 186 Geometric mean, 33 local minimum point of, 186 Geometric series, 466 local strict maximum point of, 215 Global property, 121 logarithm, 338, 343 Goes to, \x goes to sin(x2)", 45 maximum point of, 185 Graph of polynomial function, 194 maximum value of, 185 Graph sketching, 193{198 minimum value of, 185 Graphs, 57{65, 85 ff., 90{91, 196, 533 most general deˇnition of, 591 negative part of, 51 Gravitation, 327 nondecreasing, 240 Greatest lower bound, 132 nonincreasing, 240 Grin and bear it, 381{382 nonnegative, 51 Gronwall's inequality, 353 notation for, 40, 45 Grow odd, 51, 196 at the same rate as, 358 periodic, 71, 162, 296 faster than, 358

664 Index

Hadamard, 560 Liouville's, 441 Half-life (of radioactive substance), 352 Schwarz, 17, 32, 278 Hermite, 436 triangle, 71 High-school student's real numbers, 589 Young's, 273 Higher-order derivatives, 159 Inˇmum, 132 Hilbert, 436 \Inˇnite" derivative, 156 Horizontal axis, 57 Inˇnite intervals, 57 Hyperbola, 67, 82 Inˇnite product, 326, 391 equation in polar coordinates, 88 Inˇnite products, 489 Hyperbolic cosine, 349 Inˇnite sequence, 445, 546 Hyperbolic functions, 349 Inˇnite series, 465 Hyperbolic sine, 349 multiplication of, 479{481 Hyperbolic spiral, 313 Inˇnite sum, 426, 464 Inˇnite trumpet, 402 Inˇnitely many primes, 32 \Inˇnitely small", 153, 261 Inˇnity, 57 Identity minus, 57 additive, 9 In ection point, 222 multiplicative, 9 Initial conditions for differential equa- Identity function, 43 tions, 433 Identity operator, 564 Instantaneous speed, 150 Imaginary axis, 525 Instantaneous velocity, 150 Imaginary part function, 532 Integer, 25 Imaginary part of a complex number, Integrable, 255 522 Integral, 255 Implicit differentiation, 238 classical notation for, 262 Implicitly deˇned, 238 deˇnite, 361 Improper integral, 298{299, 391{393 improper, 298{299, 391{393 Incantation for derivative of quotient, indeˇnite, 361 169 short table of, 361{362 Increasing at a, 214 Leibnizian notation for, see Integral, Increasing function, 192 classical notation for Increasing sequence, 450 lower, 292 Indeˇnite integral, 361 Mean Value Theorem for, 274 Indeˇnite integrals, short table of, Second Mean Value Theorem for, 361{362 387 Induction, mathematical, 21 upper, 292 complete, 23 Integral form of the remainder, 417, Inductive set of real numbers, 34 418 Inequalities, 9 Integral sign, 255 in an ordered ˇeld, 574 Integral test, 471 Inequality Integration Bernoulli's, 32 by parts, 362 ff. Cauchy-Schwarz, 278 by substitution, 365 ff. geometric-arithmetic mean, 33 limits of, 255 Gronwall's, 353 of rational functions, 373 ff. Jensen's, 225 Interest (ˇnance), 351

Index 665

Intermediate Value Theorem, 122, 129, Lemma, 100 133, 296 Lemniscate, 89 Interpolation, Lagrange, 49 Length, 275, 281 Intersection of sets, 43 L'Hopital,^ Marquis de, 146 Interval, 56 L'Hopital's^ Rule, 201, 210{211 closed, 57 Limit, 90 ff., 96, 533 inˇnite, 57 at inˇnity, 105 open, 56; see also Nested Intervals Theorem \does not exist", 99 Inverse from above, 104 additive, 9 from below, 104 multiplicative, 9 of a sequence, 446 Inverse of a function, 228 ff. of vector-valued function, 243, 249 Inverse square law, 327 uniqueness of, 98 Inverses of trigonometric functions, Limit of integration, 255 see Trigonometric functions Irrational numbers, 25 Limit point, 462, 543 Isomorphic ˇelds, 592 Limit superior, 141, 460 Isomorphism, 592 Lindemann, 440 Line, real, 56 Line, tangent, see Tangent line Linear functions, 58 Jensen's inequality, 225 Liouville, 441 Johnson, Samuel, 597 Liouville's inequality, 441 Jump, 60 Liouville's Theorem, 558 Lipschitz of order α, 207 Local maximum point of function, 186 higher-order derivative test for, 411 Kepler, 327 second derivative test for, 198 Kepler's laws of planetary motion, 327 Local minimum point of function, 186; see also Local maximum point Local property, 107, 121, 164 Local strict maximum point, 215 Lagrange form of the remainder, 417, 418 Log, 338, 343 Lagrange interpolation formula, 49 Taylor polynomials for, 407 Large negative, 64 remainder term for, 423 Least upper bound, 131 ff., 574 Logarithm Least upper bound property, 133 classical approach to, 354 Lebesgue, see Riemann-Lebesgue Napierian, 354 Lemma of a complex number, 561 Left-hand derivative, 154 to the base 10, 336 Leibniz, 153, 261 Logarithmic derivative, 348 Leibniz's formula, 182 Leibniz's Theorem, 474 Lower bound, 132 Leibnizian notation for derivatives, 153{ almost, 140 154, 165, 184, 238 greatest, 132 for higher order derivatives, 160 Lower integral, 292

666 Index

Lower limit of integration, 255 Multiplicative inverses, existence of, 9 Lower sum, 251 Multiplicity (of a root), 128 Lowest terms, 73

Napier, 355 Maclaurin, 566 Napierian logarithm, 354 Major axis of ellipse, 87 Natural numbers, 21, 34 Mass, rate of change of, 150 Negative, large, 64 Mathematical induction, 21 \Negative inˇnity", derivative, 156 Maximum of two numbers, 16 Negative number, 9 Maximum point of a function, 185 Negative numbers, product of two, 7 local, 186; see also Local maximum Negative part of a function, 51 point Nested Interval Theorem, 140 local strict, 215 Newton, 153, 273, 327 strict, 215 Newton's law of cooling, 352 Maximum value of function, 185 Newton's laws of motion, 159 Mean Newton's method, 457 arithmetic, 33 Nondecreasing function, 240 geometric, 33 Nondecreasing sequence, 450 Mean Value Theorem, 190, 191 Nondifferentiable complex functions, Cauchy, 201 542 for integrals, 274 Nonincreasing function, 240 Second, 387 Nonincreasing sequence, 450 Method of exhaustion, 139 Nonnegative function, 51 Minimum of function, 185 Nonnegative sequence, 467 Minimum of two numbers, 16 Norm, 78, 249 Minimum point of a function, local, Notational nonsense, 564 186; see also Local minimum point Nowhere differentiable continuous Minor axis of ellipse, 87 function, 501 Minus inˇnity, 57 nth root, 71, 527 Miriﬁci logarithmonum canonis description, existence of, 123, 527, 544 355 primitive, 531 Modulus of a complex number, 525 Null set, 23 Mollerup, Johannes, 390 Number Multiplication, 5 algebraic, 435 associative law for, 9 complex, 517, 522 closure under, 9 counting, 21 commutative law for, 9 even, 25 of arabic numerals , 8 imaginary, 517 of complex numbers, 522 irrational, 25 geometric interpretation, 526{527 natural, 21, 34 of function and vector-valued func- odd, 25 tion, 242 prime, 31 of inˇnite series, 479{481 rational, 25 of number and vector, 77 real, 25, 525, 579 of vectors , 77 transcendental, 435 Multiplicative identity, existence of, 9 Numbers, basic properties of, 3

Index 667

Odd function, 51, 196 Polynomials, Bernoulli, 566 Odd number, 25 Pope, Alexander, 327 One-one function, 227 Position, rate of change of, 150 Open interval, 56 Positive element of R, 583 \Or", 6 Positive elements of an ordered Order of equality, 412 ˇeld, 573 Ordered ˇeld, 573 Positive number, 9 complete, 574 Positive part of a function, 51 Ordered pair, 47 (footnote), 54 Power functions, 60 Origin (of a coordinate system), 57 Power series, complex, 548 Power series centered at a, 502, 555 Power series representation, uniqueness of, 512 Pair, 46 Powers of 2, table of, 428 ordered, 47 (footnote), 54 Prime number, 31 Parabola, 60, 82 characteristic of a ˇeld, 576 area under, 260 inˇnitely many of, 32 equation in polar coordinates, 88 unique factorization into, 31 Parallelogram, 76 Primitive, 359 Parameterized curve, tangent line of, Primitive nth root, 531 243 , 273 Parametric representation of a curve, Principia Product, 5 241 Cauchy, 486, 505 Partial fraction decomposition, 374 inˇnite, 326, 391, 489 Partial sums, 464 of function and vector-valued func- Partition, 251 tion, 242 Parts of functions, 43 Abel's formula for summation by, 388 integration by, 362 ff. of number and vector, 77 Pascal's triangle, 27 of two negative numbers, 7 \Peak", 61 of vectors, 77 Peak point, 451 Pyramid Period of a function, 71, 162, 296 surface area of, 398 Periodic function, 71, 162, 296 volume of, 402 Perpendicularity of lines, 70 Pythagorean theorem, 25, 58 Petard, H, 543 π, 302 Pig, yellow, v, 371 Archimedes' approximation of, 139 Pigheaded, 183 irrationality of, 323 Plane, 58 relation to e, 441, 555 complex, 524 transcendentality of, 440 Planetary motion, Kepler's laws of, 327 value of, 429 Point, 56 Viete'sÂ product for 2/π, 326 Point of contact, 217 Wallis' product for π/2, 391 Point-slope form of equation of a line, 59, 70 Polar coordinates, 84 ff. Polynomial function, 42 graph of, 61, 194 Quaternions, 577 multiplicity of roots, 128, 183 Quotient of functions, 43

668 Index

Rabbits double, 183; see also nth roots growth of population, 32 Root test, 485 Radian measure, 63, 301{302 delicate, 486 Radioactive decay, 352 Radius of convergence of complex power series, 550 Rate of change of mass, 150 Same sign, 12 Rate of change of position, 150 Scalar, 78 Ratio test, 469 Scalar product of vectors, 78 delicate, 486 Schwarz, H. A., 17, 215 Rational functions, 42 Schwarz inequality, 17, 32, 278 integration of, 373 ff. Schwarz second derivative, 431 Rational numbers, 25 Schwarzian derivative, 182 Real axis, 524 Sec, 307 Real line, 56 derivative of, 307 Real number (formal deˇnition), 579 inverse of, see Arcsec Real numbers, 25 Secant line, 148 algebraist's, 588 Second coordinate, 57 analyst's, 588 Second derivative, 159 Archimedian property of, 136 Schwarz, 431 construction of, 578 ff. Second derivative test for maxima and high-school student's, 589 minima, 198 inductive set of, 34 Second Fundamental Theorem of Real part function, 532 Calculus, 286 Real part of a complex number, 522 Second Mean Value Theorem for Real-valued function, 532 Integrals, 387 Rearrangement of a sequence, 476 Sequence \Reasonable" function, 68, 116, 147, absolutely summable, 473 178 Cauchy, 452 Rectangle, closed, 538 complex, 562 Recursive deˇnition, 23 equivalence of, 589 Reduction formulas, 373 complex numbers, 546 Regulated function, 515 convergent, 446 Remainder term for Taylor poly- pointwise, 494 nomials, 415 uniformly, 494 Removable discontinuity, 119 decreasing, 450 Reparameterization, 244 divergent, 446 Revolution Fibonacci, 32, 512, 563 ellipsoid of, 400 increasing, 450 solid of, 397 inˇnite, 445 Riemann sum, 279 limit of, 446 Riemann-Lebesgue Lemma, 317, 387 nondecreasing, 450 Right-hand derivative, 154 nonincreasing, 450 Rising Sun Lemma, 141 nonnegative, 467 Rolle, 183 rearrangement of, 476 Rolle's Theorem, 190 summable, 465 Root Series multiplicity of, 128 absolutely convergent, 473 Root of a polynomial function, 50 conditionally convergent, 474

Index 669

Series (continued) Substitution convergent, 465 integration by, 365 ff. Fourier, 315, 317, 320 world's sneakiest, 382 geometric, 466 Substitution formula, 365 power, 502, 548 Subtraction, 5 Taylor, 503 Sum Set, 22 ˇnite, 3{4 empty, 23 inˇnite, 426, 464 Sets lower, 251 intersection of, 43 of an inˇnite sequence, 465 notation for, 43{44 of an inˇnite sequence of complex Shadow point, 141 numbers, 546 Shell method, 398 of functions, 43 Sigma, 24 of vector-valued functions, 242 Sign, 12 of vectors, 75 Simpson's rule, 396 partial, 464 Sin, 43, 300, 303, 318{319, 554 sigma notation for, 24 addition formula for, 310, 311 upper, 251 derivative of, 170, 304 Sum of squares, 543 inverse of, see Arcsin Summable, 465, 547 Taylor polynomials for, 406 Abel, 514 remainder term for, 420 absolutely, 473 Sine, hyperbolic, 349 Cesaro, 486 Sine function, 43 uniformly, 498 Sinh, 349 Summation by parts, Abel's formula for, Sketching graphs, 193{198 388 Skew ˇeld, 577 Supremum, 132 Slope of a straight line, 58 Surface area Solid of revolution, 397 of cone, 399 Speed, instantaneous, 150 of pyramid, 398 Spiral of solid of revolution, 397 Archimedian, 85 Swift, Jonathan, 570 hyperbolic, 313 Symmetry in graphs, 196 Square root, 12, 518 existence of, 122 Square root function, 537{538 Square root in a ˇeld, 576 Tan, 307 Squaring the circle, 440 derivative of, 307 Step function, 275 inverse of, see Arctan Stirling's Formula, 568 Taylor series for, 564 Straight line Tangent, hyperbolic, 349 analytic deˇnition, 58 Tangent line, 147, 149 shortest distance between two points, of parameterized curve, 243 275 point of contact of, 217 slope of, 58 \Tangent line", vertical, 156 Strict maximum point, 215 Tanh, 349 Strictly convex, 226 Taylor polynomial, 406 ff. Sturm Comparison Theorem, 320 remainder term of, 415, 417, 418; Subsequence, 451 see also speciﬁc functions

670 Index

Taylor series, 503, 554 Vanishing condition, 466 Taylor's Theorem, 417 Vector-valued functions, 241 Torus, 400 Vector-valued functions Transcendental number, 435 determinant of, 243 Trapezoid rule, 394 derivative of, 243 Triangle inequality, 71 dot product of, 243 Trichotomy law, 9 limit of, 243, 249 Trigonometric functions, 300, see also multiplication of function by, 242 cos, cot, csc, sec, sin, tan sum of, 242 integration of, 372{373 Vectors, 75 inverses of, 307; see also Arccos, addition of, 75 arcsec, arcsin, arctan as forces, 76 Trumpet dot product of, 78 inˇnite, 402 multiplication by numbers, 77 Two-time differentiable, 159 multiplication of, 77 scalar product of, 78 Velocity average, 150 instantaneous, 150 Vertical axis, 57 Uniform limit, 494 Viete,Â Francois, 326 Uniformly continuous function, 142 Volume, 397{398 Uniformly convergent sequence, 494 Uniformly convergent series, 498 Uniformly distributed sequence, 462 Uniformly summable, 498 Uniqueness of factorization into primes, 31 Wallis' product, 391 of limits, 98 Weakly convex, 226 of power series representations, 512 Weierstrass, see Bolzano-Weierstrass Theorem Unit circle, 66 Weierstrass -test, 499 Upper bound, 131, 574 M Well-ordering principle, 23 almost, 140 Wright, 383 least, 131 Upper integral, 292 Upper limit of integration, 255 Upper sum, 251

Young's inequality, 273

\Valley", 61 Value absolute, see Absolute value Zahl, 25 Value of f at x Zero, division by, 6