A

AalbmendoE, 276

Abel, Niels Henrik, 410, 522

Abel summable, 522

Abel's formula for summation by parts, 392

Abel's Lemma, 393

Abel's test, 496

Abel's Theorem, 522

Absolute value, 11

• of a complex number, 534

Absolutely convergent, 480, 556

Acceleration, 161

Acta Eruditorum, 148

Addition, 3

• associative law for, 9
• commutative law for, 9
• of complex numbers, 531
• geometric interpretation of, 535
• of vector-valued functions, 245
• of vectors, 75

Addition formula

• for arcsin, 317
• for arctan, 317
• for cos, 314
• for sin, 313, 314
• for tan, 317

Additive identity

• existence of, 9
• for vectors, 76

Additive inverses

• existence of, 9

Algebra, Fundamental Theorem of, 377, 538, 548, 567

Algebraic functions, 363

Algebraic number, 442

Algebraist's real numbers, 598

Almost lower bound, 142

Almost upper bound, 142

Analyst's real numbers, 598

Angle, 303

• directed, 303

Antidiagonal, 243

Arabic numerals, multiplication of, 8

Arc length, 278, 283

Arccos, 311

• derivative of, 311

Archimedes, 138, 141, 263

Archimedian property

• for the rational numbers, 584
• for the real numbers, 138

Archimedian spiral, 85, 249

Arcsec, 320, 383

Arcsin, 310

• addition formula for, 317
• derivative of, 311
• Taylor series for, 517

Arctan, 311

• addition formula for, 317
• derivative of, 311
• Taylor polynomials for, 413, 420
• remainder term for, 426

Area, 253, 258

Arg cosh, 353

Arg sinh, 353

Arg tanh, 353

Argument, 536

Argument function, 542

• discontinuities of, 546

Argument of the hyperbolic functions, 353

Arithmetic mean, 33

Arrow, 75, 76

Associative law

• for addition, 9
• of vectors, 76
• for multiplication, 9

Average velocity, 152

Axis

• horizontal, 57
• imaginary, 534
• real, 533
• vertical, 57

B

Bacon, Francis, vi

Basic properties of numbers, 3

Bernoulli, Jakob, 148, 574

Bernoulli numbers, 572

Bernoulli polynomials, 575

Bernoulli's inequality, 32

Binary operation, 581

Binomial coefficient, 27, 437

Binomial series, 495, 518

Binomial theorem, 28

Bisection argument, 142, 552

Bohr, Harold, 394

Bolzano-Weierstrass Theorem, 458, 469, 552

Bound

• almost lower, 142
• almost upper, 142
• greatest lower, 134
• least upper, 133, 584
• lower, 134
• upper, 133

Bounded above, 122, 133, 457, 584

Bounded below, 134, 457

Boundedness criterion, 474

Bourbaki, Nicholas, 148

C

Cantor, Georg, 448

Cardioid, 89, 250

Cartesian coordinates, 84

Cauchy

• Condensation Theorem, 496
• criterion, 473
• form of the remainder, 437
• Mean Value Theorem, 204
• product, 493, 513
• sequence, 459, 571

Cauchy-Hadamard formula, 569

Cauchy-Schwarz inequality, 281

Cavalieri, Bonaventura, 275

Cesaro summable, 493

Chain Rule, 174

• proof of, 178

Circle, 65

Circle of convergence, 560

Classical notation

• for derivatives, 154-156, 162, 167, 187, 241-242
• for integrals, 265

Closed interval, 57

Closed rectangle, 547

Closure under addition, 9

Closure under multiplication, 9

Commutative law

• for addition, 9
• of vectors, 76
• for multiplication, 9

Comparison test, 474

• limit, 475

Comparison Theorem, Sturm, 323

Complete induction, 23

Complete ordered field, 584, 603

Completing the square, 17, 379

Complex analysis, 565

Complex function

• continuous, 545, 546
• differentiable, 550
• graph of, 542
• limit of, 542
• nondifferentiable, 551
• Taylor series for, 563

Complex nth root, 536

Complex numbers, 526, 531

• absolute value of, 534
• addition of, 531
• geometric interpretation of, 535
• imaginary part of, 531
• infinite series of, 555-557
• logarithm of, 570
• multiplication of, 531
• real part of, 531

Complex plane, 533

Complex power series, 557

• circle of convergence of, 560
• radius of convergence of, 559

Complex-valued functions, 541

Composition of functions, 44

Concave function, 220

Conditionally convergent series, 481

Cone, 80

• generating line of, 80
• surface area of, 404

Conic sections, 80

Conjugate, 534, 539

Conjugate function, 541

Constant function, 43

Construction of the real numbers, 588 ff.

Continued fraction, 462

Continuous, uniformly, 144

Continuous at a, 115, 545

Continuous function, 115, 118, 546

• nowhere differentiable, 159, 509

Continuous on (a, b), 118

Continuous on [a, b], 118

Contraction, 466

Contraction lemma, 466

Converge

• pointwise, 502
• uniformly, 502, 506

Convergent sequence, 453, 555

Convergent series, 472, 556

• absolutely, 480, 556
• conditionally, 481

Convex function, 219

• strictly, 228
• weakly, 228

Convex subset of the plane, 229, 553

Cooling, Newton's law of, 356

Coordinate

• first, 57
• second, 57

Coordinate system, 57

• cartesian, 84
• origin of, 57

Coordinates, polar, 84

Cos, 303-304, 306, 321-322, 563

• addition formula for, 314
• derivative of, 172, 307
• Taylor polynomials for, 413

Cosh, 353

Cosine, hyperbolic, 353

Cot, 310

• derivative of, 310

Countable, 449

Counting numbers, 21

Critical point, 190

Critical value, 190

Csc, 310

• derivative of, 310

Cubic equation, general solution, 528-529

Curve

• parameterized, tangent line of, 246
• parametric representation of, 244
• reparameterization of, 247

Cycloid, 250

D

Darboux's Theorem, 214

De Moivre's Theorem, 536

Decimal expansion, 73, 492, 599

Decreasing function, 195

Decreasing sequence, 457

Dedekind, Richard, 38

Defined implicitly, 241

Definite integral, 365

Definition, recursive, 23

Degree (of a polynomial), 42

Degree measurement, 304-305

Delicate ratio test, 493

Delicate root test, 493

Dense, 140

Derivative, 149 ff., 151

• classical notation for, 154-156, 162, 167, 187, 241-242
• higher-order, 161
• "infinite", 158
• left-hand, 156
• logarithmic, 351
• of f, 151
• of f at a, 151
• of vector-valued function, 246
• right-hand, 156
• Schwarz, 439
• Schwarzian, 184
• second, 161

Derivative of quotient, incantation for, 171

Descartes, René, 84

Determinant, 79

• of vector-valued functions, 248

Diagonal, 233

Difference operator, 573

Differentiable, 151, 550

Differential equation, 292, 300, 321, 322-323, 356, 361, 438-440

• initial conditions for, 440

Differentiation, 168 ff.

• implicit, 241
• logarithmic, 351

Differentiation operator, 573

Dini's Theorem, 524

Directed angle, 303

Dirichlet's test, 495

Disc method, 402

Discontinuities of a nondecreasing function, 450

Discontinuity, removable, 121

Disraeli, Benjamin, 2

Distance, 58, 534

• shortest between two points, 278

Distributive law, 9

Diverge, 453, 556

Division, 6

• by zero, 6

Domain, 40, 41, 47, 601

Dot product

• of vector-valued functions, 246
• of vectors, 78

Double intersection, 165

Double root, 185

Durège, 38

E

e, 343

• irrationality of, 429
• relation with π, 448, 564
• transcendentality of, 444
• value of, 344, 426

Eccentricity of ellipse, 87

Elementary function, 363

Ellipse, 66, 82

• axes of, 87
• eccentricity of, 87
• equation in polar coordinates, 86-87
• focus point of, 66, 86
• major axis of, 87
• minor axis of, 87

Ellipsoid of revolution, 405

Empty collection, 23

Entire function, 567

Epsilon, 18

Equal up to order n, 418

Equality, order of, 418

Equivalent Cauchy sequences, 599

Etymology lesson, 82

Euler, Leonhard, 575

Euler's number, 463

Euler-Maclaurin Summation Formula, 576

Even function, 51, 199

Even number, 25

Eventually inside, 555

Exhaustion, method of, 141

Exp, 343 ff., 563

• classical approach to, 357
• elementary definition of, 468
• Taylor polynomials for, 413
• remainder term for, 426

Expansion, decimal, 73, 492, 599

Extension of a function, 115-116

F

Factorial, 23

Factorials, table of, 432

Factorization into primes, 31

Fibonacci, 32

Fibonacci Association, 32

Fibonacci Quarterly, The, 32

Fibonacci sequence, 32, 521, 572

Field, 581

• characteristic of, 586
• complete ordered, 584, 603
• ordered, 583

First coordinate, 57

First Fundamental Theorem of Calculus, 285

Fixed point of a function, 465

Focus point, 66, 86

Force, as vector, 76

Four leaf clover, 88

Fourier series, 318, 320, 323

Fraction, continued, 462

Function, 39, 47

• absolute value, 541
• argument, 542
• complex valued, 541
• composition of, 44
• concave, 220
• conjugate, 541
• constant, 43
• continuous, 115 ff.
• convex, 219
• critical point of, 190
• critical value of, 190
• decreasing, 195
• derivative of, 149
• differentiable, 151, 550
• elementary, 363
• entire, 567
• even, 51, 199
• exponential, 343-344
• extension of, 115-120
• fixed point of, 465
• from A to B, 601
• from real numbers to the plane, 244
• graphs of, 57-65, 198, 542
• hyperbolic, 353
• identity, 43
• imaginary part, 541
• implicitly defined, 241
• increasing, 195
• integrable, 258
• integral of, 258
• inverse, 231 ff.
• linear, 58
• local maximum point of, 189
• local minimum point of, 189
• local strict maximum point of, 218
• logarithm, 341, 346
• maximum point of, 188
• maximum value of, 188
• minimum value of, 188
• most general definition of, 601
• negative part of, 51
• nondecreasing, 243
• nonincreasing, 243
• nonnegative, 51
• notation for, 40, 45
• odd, 51, 199
• one-one, 230
• periodic, 71, 164, 298
• polynomial, 42
• positive part of, 51
• power, 60
• product of, 43
• quotient of, 43
• rational, 42
• real part, 541
• real-valued, 541
• "reasonable", 68, 118, 149, 180
• regulated, 524
• square root, 546-547
• step, 278
• strict maximum point of, 218
• sum of, 43
• trigonometric, 303 ff.
• value at x, 40
• vector-valued, 244

Fundamental Theorem of Algebra, 377, 538, 549, 567

Fundamental Theorem of Calculus

• First, 285
• Second, 289

G

Gabriel, 408

Galileo Galilei, 148, 164

Gamma function, 394, 444

Generating line, of a cone, 80

Geometric mean, 33

Geometric series, 473

Global property, 123

Graph of polynomial function, 197-198

Graph sketching, 196-201

Graphs, 57-65, 85 ff., 90-91, 197-198, 542

Gravitation, 330

Greatest lower bound, 134

Grin and bear it, 385-386

Gronwall's inequality, 356

Grow

• at the same rate as, 361
• faster than, 361

H

Half-life (of radioactive substance), 356

Hermite, Charles, 443

High-school student's real numbers, 599

Higher-order derivatives, 161

Hilbert, David, 443

Horizontal axis, 57

Hyperbola, 67, 82

• equation in polar coordinates, 88

Hyperbolic cosine, 353

Hyperbolic functions, 353

Hyperbolic sine, 353

Hyperbolic spiral, 316

Hyperbolic tangent, 353

I

Identity

• additive, 9
• multiplicative, 9

Identity function, 43

Identity operator, 574

Imaginary axis, 534

Imaginary part function, 541

Imaginary part of a complex number, 531

Implicit differentiation, 241

Implicitly defined, 241

Improper integral, 301-302, 397-399

Increasing at a, 217

Increasing function, 195

Increasing sequence, 457

Indefinite integral, 365

Induction, mathematical, 21

• complete, 23

Inductive set of real numbers, 34

Inequalities, 9

• in an ordered field, 584

Inequality

• Bernoulli's, 32
• Cauchy-Schwarz, 281
• geometric-arithmetic mean, 33
• Gronwall's, 356
• Jensen's, 228
• Liouville's, 448
• Schwarz, 17, 33, 281
• triangle, 71
• Young's, 276

Infimum, 134

"Infinite" derivative, 158

Infinite intervals, 57

Infinite product, 329, 395, 497

Infinite sequence, 452, 555

Infinite series, 471

• multiplication of, 486

Infinite sum, 430, 471

Infinite trumpet, 408

Infinitely many primes, 32

"Infinitely small", 155, 264

Infinity, 57

• minus, 57

Inflection point, 225

Initial conditions for differential equations, 440

Instantaneous speed, 152

Instantaneous velocity, 152

Integer, 25

Integrable, 258

Integral, 258

• classical notation for, 264-265
• definite, 365
• improper, 301-302, 397-399
• indefinite, 365
• lower, 295
• Mean Value Theorem for, 277
• Second Mean Value Theorem for, 391
• upper, 295

Integral form of the remainder, 423

Integral sign, 258

Integral test, 478

Integration

• by parts, 366 ff.
• by substitution, 369 ff.
• limits of, 258
• of rational functions, 377 ff.

Interest (finance), 355

Intermediate Value Theorem, 124, 131, 135, 299

Interpolation, Lagrange, 49

Intersection of sets, 43

Interval, 56

• closed, 57
• infinite, 57
• open, 56

Inverse

• additive, 9
• multiplicative, 9

Inverse of a function, 231 ff.

Inverse square law, 330

Irrational numbers, 25

Isomorphic fields, 602

Isomorphism, 602

J

Jensen's inequality, 228

Johnson, Samuel, 607

Jump, 60

K

Kepler, Johannes, 330

Kepler's laws of planetary motion, 330

L

Lagrange form of the remainder, 423, 436

Lagrange interpolation formula, 49

Large negative, 64

Least upper bound, 133 ff., 584

Least upper bound property, 135

Left-hand derivative, 156

Leibniz, Gottfried Wilhelm, 155, 264

Leibniz's formula, 184

Leibniz's Theorem, 481

Lemma, 102

Lemniscate, 89

Length, 278, 283

L'Hôpital, Marquis de, 148

L'Hôpital's Rule, 204, 213-214

Limit, 90 ff., 98, 542

• at infinity, 106
• "does not exist", 101
• from above, 106
• from below, 106
• of a sequence, 453
• of vector-valued function, 246, 252
• uniqueness of, 100

Limit comparison test, 475

Limit of integration, 258

Limit point, 469, 552

Limit superior, 143, 467

Lindemann, Ferdinand von, 447

Line, real, 56

Linear functions, 58

Liouville, Joseph, 448

Liouville's inequality, 448

Liouville's Theorem, 567

Lipschitz of order a, 210

Local maximum point of function, 189

Local minimum point of function, 189

Local property, 109, 123, 166

Local strict maximum point, 218

Log, 341, 346

• Taylor polynomials for, 413
• remainder term for, 427

Logarithm

• classical approach to, 357
• Napierian, 358
• of a complex number, 570
• to the base 10, 339

Logarithmic derivative, 351

Lower bound, 134

• almost, 142
• greatest, 134

Lower integral, 295

Lower limit of integration, 258

Lower sum, 254

Lowest terms, 73

M

Major axis of ellipse, 87

Mass, rate of change of, 152

Mathematical induction, 21

Maximum of two numbers, 16

Maximum point of a function, 188

• local, 189
• local strict, 218
• strict, 218

Maximum value of function, 188

Mean

• arithmetic, 33
• geometric, 33

Mean Value Theorem, 193, 194

• Cauchy, 204
• for integrals, 277
• Second, 391

Method of exhaustion, 141

Minimum of function, 188

Minimum of two numbers, 16

Minor axis of ellipse, 87

Minus infinity, 57

Modulus of a complex number, 534

Mollerup, Johannes, 394

Multiplication, 5

• associative law for, 9
• closure under, 9
• commutative law for, 9
• of complex numbers, 531
• of infinite series, 486
• of number and vector, 77
• of vectors, 77

Multiplicative identity, existence of, 9

Multiplicative inverses, existence of, 9

Multiplicity (of a root), 130, 185-186

N

Napier, John, 358

Napierian logarithm, 358

Natural numbers, 21, 34

Negative, large, 64

Negative number, 9

Negative part of a function, 51

Nested Interval Theorem, 142

Newton, Isaac, 155, 276, 330

Newton's law of cooling, 356

Newton's laws of motion, 161

Newton's method, 464

Nondecreasing function, 243

Nondecreasing sequence, 457

Nondifferentiable complex functions, 551

Nonincreasing function, 243

Nonincreasing sequence, 457

Nonnegative function, 51

Nonnegative sequence, 474

Norm, 78, 252

Nowhere differentiable continuous function, 509

nth root, 71, 536

• existence of, 125, 536, 553
• primitive, 540

Null set, 23

Number

• algebraic, 442
• complex, 526, 531
• counting, 21
• even, 25
• imaginary, 526
• irrational, 25
• natural, 21, 34
• odd, 25
• prime, 31
• rational, 25
• real, 25, 534, 589
• transcendental, 442

O

Odd function, 51, 199

Odd number, 25

One-one function, 230

Open interval, 56

Order of equality, 418

Ordered field, 583

• complete, 584

Ordered pair, 47 (footnote), 54

Origin (of a coordinate system), 57

P

π, 305

• Archimedes approximation of, 141
• irrationality of, 326
• relation to e, 448, 564
• transcendentality of, 447
• value of, 433
• Viète's product for 2/π, 329
• Wallis' product for π/2, 395

Pair, 46

Parabola, 60, 82

• area under, 263
• equation in polar coordinates, 88

Parallelogram, 76

Partial fraction decomposition, 378

Partial sums, 471

Partition, 254

Pascal's triangle, 27

Peak point, 458

Period of a function, 71, 164, 298

Periodic function, 71, 164, 298

Perpendicularity of lines, 70

Petard, H., 552

Plane, 58

• complex, 533

Point, 56

Point of contact, 220

Polar coordinates, 84 ff.

Polynomial function, 42

• graph of, 61, 197 ff.
• multiplicity of roots, 130, 185-186

Polynomials, Bernoulli, 575

Pope, Alexander, 330

Position, rate of change of, 152

Positive element of R, 593

Positive number, 9

Positive part of a function, 51

Power functions, 60

Power series, 510

• centered at a, 510, 564
• complex, 557

Prime number, 31

• characteristic of a field, 586
• infinitely many of, 32
• unique factorization into, 31

Primitive, 363

Primitive nth root, 540

Principia, 276

Product, 5

• Cauchy, 493, 513
• infinite, 329, 395, 497
• of functions, 43
• of number and vector, 77
• of two negative numbers, 7
• of vectors, 77

Pyramid

• surface area of, 403
• volume of, 407

Pythagorean theorem, 25, 58

Q

Quaternions, 587

Quotient of functions, 43

R

Rabbits

• growth of population, 32

Radian measure, 63, 304-305

Radioactive decay, 355-356

Ratio test, 476

• delicate, 493

Rational functions, 42

• integration of, 377 ff.

Rational numbers, 25

Real axis, 533

Real line, 56

Real number (formal definition), 589

Real numbers, 25

• algebraist's, 598
• analyst's, 598
• Archimedian property of, 138
• construction of, 588 ff.
• high-school student's, 599
• inductive set of, 34

Real part function, 541

Real part of a complex number, 531

Rearrangement of a sequence, 483

Rectangle, closed, 547

Recursive definition, 23

Reduction formulas, 377

Regulated function, 524

Removable discontinuity, 121

Reparameterization, 247

Revolution

• ellipsoid of, 405
• solid of, 402

Riemann sum, 282

Riemann-Lebesgue Lemma, 320, 391

Right-hand derivative, 156

Rising Sun Lemma, 143

Rolle, Michel, 186

Rolle's Theorem, 193

Root

• double, 185
• multiplicity of, 130, 185-186

Root of a polynomial function, 50

Root test, 493

• delicate, 493

S

Scalar, 78

Scalar product of vectors, 78

Schwarz, H. A., 217

Schwarz inequality, 17, 33, 281

Schwarz second derivative, 438

Schwarzian derivative, 184

Sec, 310

• derivative of, 310

Secant line, 150

Second coordinate, 57

Second derivative, 161

Second Fundamental Theorem of Calculus, 289

Sequence

• absolutely summable, 480
• Cauchy, 459
• complex numbers, 555
• convergent, 453
• decreasing, 457
• divergent, 453, 556
• Fibonacci, 32, 521, 572
• increasing, 457
• infinite, 452, 555
• limit of, 453
• nondecreasing, 457
• nonincreasing, 457
• nonnegative, 474
• rearrangement of, 483
• summable, 472

Series

• absolutely convergent, 480
• conditionally convergent, 481
• convergent, 472, 556
• Fourier, 318, 320, 323
• geometric, 473
• power, 510, 557
• Taylor, 511

Set, 22

• empty, 23

Sets

• intersection of, 43
• notation for, 43-44

Shell method, 403

Sigma, 24

Sign, 12

Simpson's rule, 401

Sin, 43, 303-304, 306, 321-322, 563

• addition formula for, 313, 314
• derivative of, 172, 307
• Taylor polynomials for, 412

Sine, hyperbolic, 353

Sinh, 353

Sketching graphs, 196-201

Skew field, 587

Slope of a straight line, 58

Solid of revolution, 402

Speed, instantaneous, 152

Spiral

• Archimedian, 85, 249
• hyperbolic, 316

Square root, 12, 527

• existence of, 124

Step function, 278

Stirling's Formula, 578

Straight line

• analytic definition, 58
• shortest distance between two points, 278
• slope of, 58

Strict maximum point, 218

Strictly convex, 228

Sturm Comparison Theorem, 323

Subsequence, 458

Substitution

• integration by, 369 ff.
• world's sneakiest, 386

Sum

• finite, 3-4
• infinite, 430, 471
• lower, 254
• of an infinite sequence, 472
• of functions, 43
• of vectors, 75
• partial, 471
• upper, 254

Summable, 472, 556

• Abel, 522
• absolutely, 480
• Cesaro, 493
• uniformly, 506

Supremum, 134

T

Tan, 310

• derivative of, 310
• Taylor series for, 573

Tangent, hyperbolic, 353

Tangent line, 149, 151

Tanh, 353

Taylor polynomial, 412 ff.

Taylor series, 511

Taylor's Theorem, 424

Torus, 405

Transcendental number, 442

Trapezoid rule, 400

Triangle inequality, 71

Trichotomy law, 9

Trigonometric functions, 303 ff.

U

Uniform limit, 502

Uniformly continuous function, 144

Uniformly convergent sequence, 502

Uniformly convergent series, 506

Uniformly distributed sequence, 469

Uniformly summable, 506

Uniqueness

• of factorization into primes, 31
• of limits, 100

Unit circle, 66

Upper bound, 133, 584

• almost, 142
• least, 133

Upper integral, 295

Upper limit of integration, 258

Upper sum, 254

V

Value of f at x, 40

Vanishing condition, 473

Vector-valued functions, 244

• derivative of, 246
• determinant of, 246
• dot product of, 246
• limit of, 246, 252
• sum of, 245

Vectors, 75

• addition of, 75
• as forces, 76
• dot product of, 78
• multiplication by numbers, 77
• multiplication of, 77
• scalar product of, 78

Velocity

• average, 152
• instantaneous, 152

Vertical axis, 57

Viète, François, 329

Volume, 402-403

W

Wallis' product, 395

Weakly convex, 228

Weierstrass M-test, 507

Well-ordering principle, 23

Wright, Edward, 388

Y

Young's inequality, 276

Z

Zahl, 25

Zero, division by, 6