A Conclusion or an Introduction?

Transcription

1 A Conclusion or an Introduction? It did not seem right to finish with a conclusion since this book is only an introduction to the vast subject of analysis. Instead, we finish by discussing where to go after reading this book. The sources listed in the References are a good starting point for further explorations. The books by Courant and John [6] and Lay [17] discuss real analysis at roughly the same level as this book. However, Courant and John focus more on calculus, while Lay adopts a more abstract approach. The calculus text by Bers [3] also contains some rigorous analysis. The book by Rudin [19] is the classic introductory text in real analysis. It is more abstract than this book and goes further in some aspects of analysis. It is the next book on analysis to read after this book. Thomson, Bruckner, and Bruckner [20] cover much of the same material as Rudin but in a more modern style. The author consulted all of these books frequently while writing this text. Complex analysis, which is the analysis of functions of complex numbers, is one of the most beautiful areas in mathematics. It should be studied right after real analysis. A standard text is Ahlfors [1], which indeed is a classic book in mathematics. Grinstead and Snell [12] is a favorite introduction to probability. Two general textbooks on numerical analysis are Atkinson [2] and Isaacson and Keller [15]. These books also offer material on general analysis not often discussed in standard analysis texts. These two books discuss Newton s method and the solution of ordinary differential equations, but these subjects are sufficiently complicated to warrant their own books. Dennis and Schnabel [9] is a great source for learning about the solution of nonlin-

2 606 A Conclusion or an Introduction? ear equations. The textbook by Braun [4] and the classic book by Henrici [13] discuss many interesting aspects of differential equations and their numerical solution. For a more modern perspective, the reader might consult Eriksson, Estep, Hansbo, and Johnson [10], which explains how many engineers look at the numerical solution of differential equations. As far as the history of mathematics is concerned, there are some good sources [5, 7, 11, 16, 18] listed in the references. All students of mathematics should eventually read the classic books by Kline [16]. Mathematicians should also read Davis and Hersh [8] when they reach that point when they begin to question what they do and why. It is reasonable to guess that those who have found their way through most of this book are likely to have more adventures and discoveries in analysis awaiting them. Whether these lie in mathematics, science, or engineering, whether future investigations consist of proofs or computations, or whether the goal is to model the physical world or to seek out mathematical truths, remember that it is all analysis.

3 References [1] L. Ahlfors, Complex Analysis, McGraw-Hill Book Company, New York, [2] K. Atkinson, An Introduction to Numerical Analysis, John Wiley and Sons, New York, [3] L. Bers, Calculus, Holt, Rinehart, and Winston, New York, [4] M. Braun, Differential Equations and their Applications, Springer- Verlag, New York, [5] R. Cooke, The History of Mathematics. A Brief Course, John Wiley and Sons, New York, [6] R. Courant and F. John, Introduction to Calculus and Analysis, vol. 1, Springer-Verlag, New York, [7] R. Courant and H. Robbins, What is Mathematics?, Oxford University Press, New York, [8] P. Davis and R. Hersh, The Mathematical Experience, Houghton Mifflin, New York, [9] J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, New Jersey, 1983.

4 608 References [10] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations, Cambridge University Press, New York, [11] I. Grattan-Guiness, The Norton History of the Mathematical Sciences, W.W. Norton and Company, New York, [12] C. Grinstead and J. Snell, Introduction to Probability, American Mathematical Society, Providence, [13] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York, [14] E. Hille, Thomas Hakon Gronwall - In Memoriam, Bulletin of the American Mathematical Society, 38 (1932), pp [15] E. Isaacson and H. Keller, Analysis of Numerical Methods, John Wiley and Sons, New York, [16] M. Kline, Mathematical Thought from Ancient to Modern Times, vol. I, II, III, Oxford University Press, New York, [17] S. Lay, Analysis with an Introduction to Proof, Prentice Hall, New Jersey, [18] J. O Connor and E. Robertson, The MacTutor History of Mathematics Archive, School of Mathematics and Statistics, University of Saint Andrews, Scotland, history/. [19] W. Rudin, Principles of Mathematical Analysis, McGraw Hill Book Company, New York, [20] B. Thomson, J. Bruckner, and A. Bruckner, Elementary Real Analysis, Prentice Hall, New Jersey, [21] T. Ypma, Historical development of the Newton-Raphson method, SIAM Review, 37 (1995), pp

5 Index 1 1, 185 I(t), 69, 84 Σ, 62 N, 19 Q, 23, 37, 39, 45 R, 143,17 exp, 372 inf, 454, 18, 48, 302 lim, 102, 103 log, 358 log b, 363 O, 505 o, 505 C q ([a, b]), 570 max, 455 min, 455, 52, 137 sup, 454, 24, 69 e x, 372 absolute area between curves, 350 underneath a curve, 349 absolute value, 24, 69 acceleration, 267, 286 of gravity, 287 accumulation of errors, 119 accuracy, 100 algebraic equation, 12 algorithm, 12 Bisection, 130, 166, 168, 186 Decasection, 173 Fixed Point Iteration, 198, 200, 573 Forward Euler Method, 584 hybrid-newton method, 425 long division, 23 Newton s method, 418 polynomial long division, 79 analytic functions, 538 antiderivative, 301 antidifferentiation, 301 approximation theory, 555 Archimedes, 345 area

6 610 Index between curves, 350 underneath a curve, 348 argument, 52 Arithmetic Properties of the Real Numbers, 140 Arzela, 591 Arzela s Theorem, 591 associative rule for addition, 16 for multiplication, 16 average rate of change, 241 value of a function, 351 Axiom of Completeness, 149 axis, 56 Babylonians, 7 bacteria, 45, 46, 385 Banach, 205, 573 Barrows, 436 basis, 544 coefficients with respect to a, 544 Bernoulli Jacob, 246, 436, 514 Johann, 246, 436, 497, 539 Bernstein, 510 Approximation Theorem, 518 polynomial, 517, 538 big O, 505 binomial coefficient, 510 Binomial Expansion, 511 binomial polynomials, 513 Bisection Algorithm, 130, 166, 168, 186 blows up, 564 Bolzano, 169, 238, 437, 442, 455 Bolzano s Theorem, 169, 444 bounded function, 93 sequence, 112 set of numbers, 88, 454 Brouwer, 20 Cantor, 20, 136, 149, 437 Cauchy, 88, 106, 169, 238, 437, 442, 599 sequence, 131, 136, 144 uniform Cauchy sequence, 325 Cauchy Criterion for Convergence, 147 Chain Rule, 264 chemical equilibrium, 46, 166, 201, 205 choose, 510 closed interval, 48, 142 set of numbers, 16, 21, 40 coefficients, 61 of a linear combination, 65, 74 with respect to a basis, 544 common denominator, 39 commutative rule for addition, 16 for multiplication, 16 competition, 45, 560 complete set of numbers, 147 complex analysis, 538 composition, 80 of continuous functions, 443 of differentiable functions, 263 of Lipschitz continuous functions, 96 compound interest, 370, 380 computing limits, 116, 117, 148 roots, 171 constant interpolant, 317 polynomial, 62 relative rate of change, 370, 374 constructive solution procedure, 12 constructivists, 19 continuity on an interval, 443 continuous function, 83, 442 model, 371 contraction map, 203, 573

7 Index 611 Contraction Mapping Principle, 205, 573 convergence factor, 207, 416 global, 408 linear, 207 local, 410 of a function, 158, 160 of a sequence, 102, 103, 144 of a series, 110 order of, 415 pointwise, 464 quadratic, 208, 415 second order rate of, 415 uniform, 323, 464 convergent sequence, 101 coordinate plane axis of the, 56 integer, 56 origin of the, 56 rational, 58 coordinates, 57 cost, 173 d Alembert, 3 Darboux, 491 integral, 491 Decasection Algorithm, 173 decimal expansion finite, 41 infinite, 99 infinite nonperiodic, 128 infinite periodic, 42 periodic, 42 truncated, 44 Dedekind, 149, 437 cut, 149 degenerate differential equation, 289 degree, 62 degrees of freedom, 544 denominator, 37 dense set of numbers, 148 Denseness of the Rational Numbers, 147 dependent variable, 53 derivative, 223, 237, 245 left, 255 right, 254 second, 267 difference of polynomials, 65 differentiability and strong differentiability, 242 differentiable at a point, 238 strongly, 223 differential, 307 differential equation, 285 degenerate, 289 energy argument for a, 395 general solution of a, 310 global existence of a solution of a, 579 initial value problem for a, 316, 559 integral form of a, 572 linear, 289 local solution of a, 586 nonlinear, 289 nonseparable, 290 order of a, 288 particular solution of a, 310 separable, 290 short time existence of a solution of a, 564 solution of a, uniqueness of a solution of a, 292, 598 weak form of a, 572 differential operator, 570 digit, 23, 41 Dinner Soup model, 8 10, 54 Dirichlet, 51, 436, 451 discontinuous, 84 function, 159 discrete model, 370 distance, 24 distributive rule, 16 divergence of a sequence, 107, 324

8 612 Index to infinity, 107 division of functions, 76 with remainder, 23 domain, 52 double precision, 119 dummy variable, 62, 101, 303, 308 Einstein, 288 Law of Motion, 288 Theory of Special Relativity, 288 electric current, 69 element, 100 endpoint, 48, 142 energy argument, 395 equality of polynomials, 67 equation, 8 differential, 285 equicontinuous family of functions, 588 equilibrium point, 233 stable, 233 unstable, 233 error bound for polynomial interpolation, 550 of a Bernstein polynomial, 518 of a quadratic approximation, 525 of a Taylor polynomial, 528 of a truncated decimal expansion, 100 of an interpolating polynomial, 548 of integration, 333, 334, 481, 482, 487 of Newton s method, 422 of the Bisection Algorithm, 170 of the Decasection Algorithm, 173 of the Fixed Point Iteration, 206, , of the linear approximation, 222 round-off, 119 tolerance, 424 Error Bound for Interpolation, 550 Error Formula for Interpolation, 549 Error Formulas for Taylor Polynomials, 533 estimate, 100 Euler, 52, 62, 436, 599 Existence of the Polynomial Interpolant, 548 Existence Result of Cauchy, Lipschitz, and Peano, 599 exponent, 118 exponential function, 372 extension of a function, 154 of a set of numbers, 22, 23 Extremum Principle for a Continuous Function, 456 factor, 127 factorial, 379 factorization, 23 family of functions, 586 Field Axioms, 149 finite decimal expansion, 41 Fixed Point Iteration, 198, 200, 573 Newton s method as a, 418 fixed point problem, 191 for a differential equation, 572 for an operator, 573 floating point number, 118 round-off error, 119 force, 286 Forward Euler Method, 584 Fourier, 311, 422 fraction, 37 Free Time model, 194, 200, 205 function, 51 antiderivative of a, 301

9 Index 613 average rate of change of a, 241 average value of a, 351 bounded, 93 bounded below, 95 constant interpolant of a, 317 continuity on an interval, 443 continuous, 83, 442 contraction map, 203 converges at, 494 converges at, 493 converges to, 495 converges to, 495 derivative of a, 223, 237, 245 differentiable at a point, 238 discontinuous, 84, 159 domain of a, 52 exponential, 372 extension of a, 154 fixed point problem, 191 global behavior of a, 269 graph of a, 55, 83, 156 Hölder continuous, 446 Hölder exponent of a, 447 instantaneous rate of change of a, 241 integrable, 302 integral of a, 302 inverse of a, 183 invertible, 183 left derivative of a, 255 left linearization of a, 255 limit of a, 158, 160 linear approximation of a, 223, 420 linear model for a, 420 linearization of a, 223 Lipschitz continuous, 85, 217 Lipschitz continuous inverse of a, 281 local behavior of a, 269 locally uniformly Lipschitz continuous, 564 maximum value of a, 456 minimum value of a, 456 modulus of continuity of a, 516 monotone, 182, 276 nth derivative of a, 267 one-sided limit of a, 253 piecewise constant interpolant of a, 320, 321 piecewise linear interpolant of a, 553 plot of a, 55 polynomial, 61 polynomial interpolation problem for a, 543 power, 171, 187, quadratic approximation of a, 525 range of a, 52 rational, 78 restriction of a, 183 right derivative of a, 254 right linearization of a, 254 second derivative of a, 267 strictly monotone, 182 strongly differentiable, 223 strongly differentiable on an interval, 245 strongly differentiable on the left, 254 strongly differentiable on the right, 254 sum, 73 tangent to a, 222, 238 Taylor polynomial of a, 528 uniformly continuous, 446 uniformly differentiable, 450 uniformly strongly differentiable, 251 uniformly strongly differentiable inverse, 283 weighted average of a, 352 functional relation, 360 functions composition of, 80 division of, 76 equicontinuous family of, 588

10 614 Index linear combination of, 74 linearlydependent,75 linearly independent, 75 Lipschitz continuous, 153 product of, 76 quotient of, 76 sequence of, 323, 463 spaces of, 570 fundamental sequence, 136 Fundamental Theorem of Calculus, 333, 334, 343, 481, 487 Galileo, 286 Galileo s law, 286, , 312 Gauss, 29, 127 General Order Rolle s Theorem, 548 general solution of a differential equation, 310 Generalized Integral Mean Value Theorem, 531 Generalized Mean Value Theorem, 499 geometric series, 109, 145 sum, 32, 62, 108 global behavior, 269 convergence, 408 existence, 579 globally convergent method, 424 graph of a function, 55, 83 a function of a real variable, 156 a polynomial, 68 an inverse function, 180 greatest lower bound, 454 Greeting Card Sales model, 192, 205 Gronwall, 598 Gronwall s Lemma, 598 half-life, 375 half-open interval, 48 harmonic series, 62, 119 Heine, 446 Hilbert, 149 Hölder, 446 continuity, 446 exponent, 447 Hooke, 387 law for a spring, 387 Horizontal Line Test, 181 hybrid-newton method, 425 identify, 137 ill-conditioned root, 423 image, 179 Implicit Function Theorem, 581 indefinite integral, 302 independent variable, 53 indeterminate form, 497 index, 34, 62, 100 induction, inductive step, 31 infimum, 454 infinite decimal expansion, 99 decimal expansion, uniqueness of, 111 interval, 143 nonperiodic decimal expansion, 128 number, 42 periodic decimal expansion, 42 series, 109 infinitesimal change, 240 infinity, 18, 48 initial condition, 296 population, 34 value, 559 value problem, 316, 559 input, 52 instantaneous rate of change, 241 INTEGER, 25

11 Index 615 integer coordinate plane, 56 number line, 22 integers, 22 integrable function, 302 integral, 302 equation, 571 form of a differential equation, 572 form of the remainder of a Taylor polynomial, 531 lower limit of a, 312 operator, 572 upper limit of a, 312 integrand, 303 integrating factor, 374, 375 integration, 302 by parts, 309 linearity of, 339 monotonicity of, 340 power rule, 303 variable, 303 Intermediate Value Theorem, 170, 444 interpolate, 221, 317, 543 interpolation nodes, 543 interval closed, 48, 142 endpoint, 142 half-open, 48 infinite, 143 open, 48, 142 into, 54 intrinsic growth rate, 371 intuitionists, 19 inverse function, 180, 183 graph of a, 180 Lipschitz continuous, 281 uniformly strongly differentiable, 282 Inverse Function Theorem, 183, 186, 444 inverse operation, 21, 23 invertible function, 183 irrational number, 128 Kovalevskaya, 107 Kronecker, 20 Lagrange, 246, 422, 436, 539 basis for polynomials, 545 form of the remainder of a Taylor polynomial, 532 Landau, 504 Law of Conservation of Momentum, 355 Law of Large Numbers, 515 least upper bound, 454 Least Upper Bound Principle, 455 Lebesgue, 473 integral, 473 left derivative, 255 linearization, 255 Leibniz, 51, 106, 302, 307, 435, 436, 539 length of a curve, 489 L Hôpital, 497 L Hôpital s rule, 498 limit computing, 116, 117, 148 indeterminate form, 497 lower, 62, 312 of a function, 158, 160 of a sequence, of a sequence of functions, 463 of a series, 110 one-sided, 253 point of a sequence, 452 uniqueness of a, 111 upper, 62, 312 linear approximation of a function, 223, 420 convergence, 207 differential equation, 289 model of a function, 420 operator, 570 polynomial, 62 linear combination, 65

12 616 Index of Lipschitz continuous functions, 92 coefficients of a, 65, 74 of differentiable functions, 259 of functions, 74 of integrable functions, 303 unique, 75 Linearity of Differentiation, 260 Linearity of Integration, 304 linearity of integration, 339 linearization left, 255 of a function, 223 right, 254 linearly dependent, 75 independent, 75 Liouville, 575 Lipschitz, 88, 599 Lipschitz constant, 85 Lipschitz continuous at a point, 229 function, 85, 153, 217 uniformly, 326 Lipschitz continuous functions composition of, 96 linear combination of, 92 product of, 94 quotient of, 95 little o, 505 local behavior, 269, 410 convergence, 410 solution, 586 Local Convergence for Newton s method, 418 Local Convergence for the Fixed Point Iteration, 410 locally convergent method, 424 uniformly Lipschitz continuous, 564 logarithm, 358 logarithmic differentiation, 378 logistic equation, 560, 562, 578, 584 long division, 23 of polynomials, 79 LONG INTEGER, 25 lower bound, 454 Darboux integral, 491 limit, 62, 312 machine number, 124 Maclaurin, 539 Malthus, 371 population growth model, 371 mantissa, 118 map, 53 contraction, 203 mass-spring system, 394 mathematical induction, model, 8 modeling, 11 maximum, 455 value of a function, 456 Mean Value Theorem, 269, 451, 456 mesh, 318, 347, 351, 478, 482, 584 nested, 319 nodes, 482 refinement, 485 size, 319, 482 method of exhaustion, 345 of successive approximations, 575 Method of Substitution, 312 method of substitution, 305 minimum, 455 value of a function, 456 model chemical equilibrium, 46 47, 166, 201, 205 compound interest, 370, 380 constant relative rate of change, 370, 374

13 Index 617 continuous, 371 Dinner Soup, 8 10, 54 discrete, 370 Einstein s Law of Motion, 288 Einstein s Theory of Special Relativity, 288 Free Time, 194, 200, 205 Galileo s law, 286, , 312 Greeting Card, 205 Greeting Card Sale, 192 Hooke s law for a spring, 387 insect population, 33 34, 231 Law of Conservation of Momentum, 355 logistic equation, 560, 562, 578, 584 Malthus population growth, 371 mass-spring system, 394 Muddy Yard, 10 11, 54, 114, 125 Newton s Law of Cooling, 370, 384 Newton s Law of Motion, 286 population growth, 370 radioactive decay, 370, 375 rocket propulsion, 357, spruce budworm, 560, 585 Verhulst population, 45 46, 114, 115 modulus of continuity, 516 monomials, 65, 68, 89, 248 monotone function, 182, 276 Monotonicity of Integration, 340 Muddy Yard model, 10 11, 54, 114, 125 Napier, 355 natural logarithm, 358 number line, 18 numbers, 15, 19 negative integers, 22 nested meshes, 319 net area underneath a curve, 349 Newton, 51, 106, 435, 436 Law of Cooling, 370, 384 Law of Motion, 286 Newton s method, 418 hybrid, 425 quasi, 427 stopping criterion for, 424 nodes, 482, 543 nonlinear differential equation, 289 nonseparable differential equation, 290 normalized weight function, 352 nth derivative, 267 number line integer, 22 natural, 18 rational, 47 numerator, 37 one-sided limit, 253 one-to-one, 185 onto, 53 open interval, 48, 142 operator, 570 linear, 570 order of a differential equation, 288 of convergence, 415 Order Axioms, 149 Order Properties of the Real Numbers, 142 ordered field, 149 pair of numbers, 38, 57 ordering numbers, 17 origin, 22, 56 output, 52 partial sums, 110 particular solution of a differential equation, 310 Peano, 40, 599

14 618 Index periodic decimal expansion, 42 Picard, 575 Picard Existence Theorem, 577 Picard s iteration, 575 piecewise constant interpolant, 320, 321 linear interpolant, 553 polynomial, 68 plot of a function, 55 Poincaré, 20 pointwise convergence, 464 polar coordinate system, 396 polynomial, 61 Bernstein, 517 binomial, 513 coefficients, 61 constant, 62 degree of a, 62 interpolation problem, 543 linear, 62 quadratic, 62 Taylor s representation of a, 528 zero, 62 polynomials basis for, 544 equality of, 67 Lagrange basis for, 545 linear combination of, 65 long division of, 79 monomials, 65, 68, 89, 248 piecewise, 68 product of, 66 sum of, 63 population growth, 370 model, 33, 231 positive integers, 22 power function, 171, 187, powers natural numbers, 17 predation, 560 prime number, 127 principle of induction, Principle of Uniform Continuity, 451 product of differentiable functions, 261 of functions, 76 of Lipschitz continuous functions, 94 of polynomials, 66 of polynomials and numbers, 64 Product Rule, 262 Pythagoreans, 128 quadratic approximation of a function, 525 convergence, 208, 415 polynomial, 62 quasi-newton method, 427 quotient function, 76 of differentiable functions, 265 of Lipschitz continuous functions, 95 Quotient Rule, 266 radioactive decay, 370, 375 range, 52 Raphson, 422 rate of change average, 241 instantaneous, 241 ratio, 37 rational coordinate plane, 58 function, 78 number line, 47 numbers, 23, 37, 39, 45 Real Number Theorem, 148 real numbers, 143 sequences of, 144 refinement of a mesh, 485 Regula Falsi Method, 213 relative change, 369

15 Index 619 rate of change, 370 remainder, 24 of a Taylor polynomial, 528 rescale, 166 residual, 422 restriction of a function, 183 Riemann, 334 sum, 335 right derivative, 254 linearization, 254 rocket propulsion model of, 357, Rolle, 272, 456 Rolle s Theorem, 271, 456 roots computing, 165, 171 ill-conditioned, 423 round-off error, 119 secant line, 238, 269 method, 428 second derivative, 267 order rate of convergence, 415 separable, 524 differential equation, 290 separation of variables, 563 sequence, 100 arithmetic, 111 bounded, 112 Cauchy, 144 convergence of a, , 144 divergence of a, 107, 324 divergence to infinity, 107 element of a, 100 extracting a subsequence from a, 452 fundamental, 136 index of a, 100 limit of a, limit point of a, 452 of real numbers, 144 subsequence of a, 452 uniform Cauchy, 325 sequence of functions, 323, 463 limit of a, 463 pointwise convergence of a, 464 uniform convergence of a, 323, 464 series convergence of a, 110 geometric, 109, 145 harmonic, 62, 119 infinite, 109 limit of a, 110 partial sum of a, 110 set, 53 greatest lower bound of a, 454 infimum of a, 454 least upper bound of a, 454 lower bound of a, 454 maximum of a, 455 minimum of a, 455 of natural numbers, 19 of real numbers, 143 supremum of a, 454 upper bound of a, 454 short time existence, 564 sigma notation, 62 Simpson, 422 single precision, 118, 120 size of a set of numbers, 89, 454 solubility, 46, 166, 201, 205 spaces of functions, 570 spring constant, 389 spruce budworm, 560, 585 stable equilibrium point, 233 step function, 69, 84 stopping criterion for Newton s method, 424 strictly monotone function, 182 strong differentiability and differentiability, 242 strongly differentiable, 223 on an interval, 245 on the left, 254 on the right, 254

16 620 Index subscript, 34 subsequence, 452 substitution, 305, 312 subtraction, 21 sum of polynomials, 63 summation notation, 62 supremum, 454 tangent, 222 line, 222, 223, 238, 269 Taylor, 539 Taylor polynomial, 528 integral form of the remainder, 531 Lagrange s form of the remainder, 532 remainder, 528 Taylor series, 538 Taylor s representation of a polynomial, 528 theorem Arithmetic Properties of Numbers, 140 Arzela s, 591 Bernstein Approximation, 518 Binomial Expansion, 511 Bolzano s, 169, 444 Cauchy Criterion for Convergence, 147 Chain Rule, 264 Contraction Mapping Principle, 205, 573 Denseness of the Rational Numbers, 147 Error Bound for Interpolation, 550 Error Formula for Interpolation, 549 Error Formulas for Taylor Polynomials, 533 Existence of the Polynomial Interpolant, 548 Existence Result of Cauchy, Lipschitz, and Peano, 599 Extremum Principle for a Continuous Function, 456 Fundamental Theorem of Calculus, 333, 334, 343, 481, 487 General Order Rolle s, 548 Generalized Integral Mean Value, 531 Generalized Mean Value, 499 Gronwall s Lemma, 598 Implicit Function, 581 Intermediate Value, 170, 444 Inverse Function, 183, 186, 444 L Hôpital s rule, 498 Law of Large Numbers, 515 Least Upper Bound Principle, 455 Linearity of Differentiation, 260 Linearity of Integration, 304 Local Convergence for Newton s method, 418 Local Convergence for the Fixed Point Iteration, 410 Mean Value, 269, 451, 456 Monotonicity of Integration, 340 Order Properties of Numbers, 142 Picard Existence, 577 Principle of Uniform Continuity, 451 Product Rule, 262 Quotient Rule, 266 Real Number, 148 Rolle s, 271, 456 Uniform Cauchy Criterion for Sequences of Functions, 326 Uniqueness for an Ordinary Differential Equation, 598 Weierstrass Approximation Theorem, 510 Weierstrass Principle, 453

17 Index 621 Weierstrass s Characterization of a Limit of a Function, 161 three components of mathematical modeling, 11 transformation, 53, 158 trial and error strategy, 129 trial-and-error strategy, 8, 9, 11, 12 triangle inequality, 25 truncated decimal expansion, 44 error, 100 uniform Cauchy sequence, 325 continuity, 446 convergence, 323, 464 Uniform Cauchy Criterion for Sequences of Functions, 326 uniformly differentiable, 450 Lipschitz continuous, 326 strongly differentiable, 251 unique linear combination, 75 uniqueness for first order, separable, linear differential equations, 295 of a solution of a differential equation, 292 of an infinite decimal expansion, 111 Uniqueness for an Ordinary Differential Equation, 598 unknown, 8 unstable equilibrium point, 233 upper bound, 454 Darboux integral, 491 estimate, 87 limit, 62, 312 independent, 53 velocity, 241 Verhulst, 46, 560 population model, 45 46, 114, 115 Vertical Line Test, 181 Viète, 421 Wallis, 44, 436 weak form of a differential equation, 572 Weierstrass, 106, 437, 442, 453, 456 Weierstrass Approximation Theorem, 510 Weierstrass Principle, 453 Weierstrass s Characterization of a Limit of a Function, 161 weight, 287 weight function, 352 weighted average of a function, 352 of numbers, 352 weights, 352 Weyl, 20 Zeno, 122 zero polynomial, 62 variable, 52 dependent, 53 dummy, 62, 101, 303, 308