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| Spivak M. — Calculus |
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| Предметный указатель |
Factorial 23
Factorials, table of 402
Factorization into primes 31
fibonacci 32
Fibonacci Association 32
Fibonacci Quarterly, The 32
Fibonacci sequence 32 486 537—538
Field 547
Field, characteristic of 552
Field, complete ordered 550 569
Field, ordered 549
First coordinate 57
First Fundamental Theorem of Calculus 268
Fixed point of a function 432
Four leaf clover 76
Fourier series 300 303
Fraction, continued 429
Function 39 47
Function, absolute value 506
Function, argument 507 511
Function, complex-valued 506
Function, composition of 44
Function, concave 205
Function, conjugate 506
Function, continuous 101 ff.
Function, convex 204
Function, critical point of 175
Function, critical value of 175
Function, decreasing 180
Function, derivative of 135 ff.
Function, differentiable 137 515
Function, elementary 336
Function, entire 533
Function, even 51 184
Function, exponential 317—318
Function, extension of 101—102
Function, fixed point of 432
Function, from A to B 567
Function, graphs of 56—65 183 507
Function, hyperbolic 326
Function, identity 43
Function, imaginary part 506
Function, increasing 180
Function, integrable 238
Function, integral of 238
Function, inverse 216 ff.
Function, linear 58
Function, local maximum point of 174
Function, local minimum point of 174
Function, local strict maximum point of 203
Function, logarithm 315 319
Function, maximum point of 173
Function, maximum value of 173
Function, minimum value of 173
Function, most general definition of 567
Function, negative part of 51
Function, nondecreasing 228
Function, nonincreasing 228
Function, nonnegative 51
Function, notation for 45
Function, odd 51 184
Function, onstant 43
Function, periodic 71 150 282
Function, polynomial 42
Function, positive part of 51
Function, power 60
Function, product of 43
Function, quotient of 43
Function, rational 42
Function, real part 506
Function, real-valued 506
Function, regulated 489
Function, step 258
Function, strict maximum point of 203
Function, sum of 43
Function, trigonometric 286 ff.
Function, value of at x 40
Function, “reasonable” 68 104 135 166
Fundamental theorem of algebra 351 503 513 533
Fundamental Theorem of Calculus, First 268
Fundamental Theorem of Calculus, Second 272
Gabriel 366
Galileo 134 150
Gamma function 369 411
Geometric mean 33
Geometric series 440
Global Properties 109
Goes to, “x goes to  ” 45
Graph sketching 181—186
graphs 56—65 75—76 78—79 184 507
Greatest lower bound 120
Grin and bear it 358—360
Gronwall’s inequality 330
Grow, at the same rate as 335
Grow, faster than 335
hadamard 534
Half-life (of radioactive substance) 329
hermite 410
High-school student’s real numbers 565
Higher-order derivatives 147
hilbert 410
Horizontal axis 57
Hyperbola 67
Hyperbolic functions 326
Hyperbolic spiral 299
Identity function 43
Identity operator 539
Identity, additive 9
Identity, multiplicative 9
Imaginary axis 499
Imaginary part function 506
Imaginary part of complex number 496
Implicit differentiation 226
Implicitly defined 226
Improper integral 283—285 371—373
Incantation for derivative of quotient 157
Increasing at a 202
Increasing function 180
Increasing sequence 424
Indefinite integral 338
Indefinite integrals, short table of 338—339
Induction, mathematical 21
Induction, mathematical, complete 23
Inductive set of real numbers 34
Inequalities 9
Inequalities, in an ordered field 550
Inequality, Bernoulli’s 32
Inequality, Cauchy — Schwarz 261
Inequality, geometric-arithmetic mean 33
Inequality, Gronwall’s 330
Inequality, Jensen’s 213
Inequality, Liouville’s 416
Inequality, Schwarz 18 33 261
Inequality, triangle 71
Inequality, Young’s 257
Infimum 120
Infinite intervals 57
Infinite products 312 370 463
Infinite sequence 419 520
Infinite series 439
Infinite series, multiplication of 453—455
Infinite sum 400 438
Infinite trumpet 366
Infinitely many primes 32
infinity 57
Infinity, minus 57
Inflection point 210
Initial conditions in differential equations 407
Instantaneous speed 138
Instantaneous velocity 138
| INTEGER 25
Integrable 238
integral 238
Integral form of the remainder 391 392
Integral sign 238
Integral test 445
Integral, classical notation for 244 245
Integral, definite 338
Integral, improper 283—285 371—373
Integral, indefinite 338
Integral, indefinite, short table of 338—339
Integral, Leibnizian notation for see “Integral classical
Integral, lower 277
Integral, Mean Value Theorem for 257
Integral, Second Mean Value Theorem for 367
Integral, upper 277
Integration, by parts 339 ff.
Integration, by substitution 342 ff.
Integration, limits of 238
Integration, of rational functions 350 ff.
Interest (finance) 328
Intermediate Value Theorem 110 117 121 282
Interpolation, Lagrange 49
Intersection of sets 43
interval 56
Interval, closed 57
Interval, infinite 57
Interval, open 56 (see also “Nested Intervals Theorem”)
Inverse of a function 216 ff.
Inverse, additive 9
Inverse, multiplicative 9
Inverses of trigonometric functions see “Trigonometric functions”
Irrational numbers 25
Isomorphic fields 568
Isomorphism 569
Jensen’s Inequality 213
Johnson, Samuel 573
Jump 60
Klingenstein, K., trials and tribulations of 637—647
Lagrange form of the remainder 391 392
Lagrange interpolation formula 49
Large negative 64
Least upper bound 119 ff. 550
Least upper bound property 121
Lebesgue see “Riemann — Lebesgue Lemma”
Left-hand derivative 142
Leibniz 134 141 244
Leibnizian notation for derivatives 141 152—253 172 227
Leibnizian notation for derivatives, for higher-order derivatives 148
Leibniz’s formula 170
Leibniz’s theorem 448
Lemma 88
Lemniscate 77
Length 258 264
LIMIT 78 ff. 84 507
Limit point 436 517
Limit superior 129 435
Limit, at infinity 93
Limit, from above 92
Limit, from below 92
Limit, of a sequence 420
Limit, of integration 238
Limit, uniqueness of 86
Limit, “does not exist” 87
Lindemann 414
Line, real 56
Line, tangent see “Tangent line”
Linear functions 58
Liouville 415
Liouville’s inequality 416
Liouville’s theorem 533
Lipschitz of order  195
Local maximum point of function 174
Local maximum point of function, higher-order derivative test for 385
Local maximum point of function, second derivative test for 186
Local minimum point of function 174 (see also “Local maximum point”)
Local property 95 109 152
Local strict maximum point 203
LOG 315
Log, Taylor polynomials for 381
Log, Taylor polynomials for, remainder term for 397
Logarithm, classical approach to 331
Logarithm, Naperian 331
Logarithm, of a complex number 536
Logarithm, to the base 10 313
Logarithmic derivative 325
Lower bound 120
Lower bound, almost 128
Lower bound, greatest 120
Lower integral 277
Lower limit of integration 238
Lower sum 234
Lowest terms 73
L’Hopital, Marquis de 134
L’Hopital’s Rule 189 198—199
Maclaurin 542
Mass, rate of change of 138
Mathematical induction 21
Maximum of two numbers 16
Maximum point of a function 173
Maximum point of a function, local 174 (see also “Local maximum point”)
Maximum point of a function, local strict 203
Maximum point of a function, strict 203
Maximum value of function 173
Mean value theorem 178 179
Mean value theorem, Cauchy 188
Mean Value Theorem, for integrals 257
Mean Value Theorem, for integrals, Second 367
Mean, arithmetic 33
Mean, arithmetic, geometric 33
Method of exhaustion 127
Minimum of function 173
Minimum of two numbers 16
Minimum point of a function, local 174 (see also “Local minimum point”)
Minus infinity 57
Mirifici logarithmonum canonis description 332
Modulus of a complex number 499
Mollerup, Johannes 370
Multiplication 5
Multiplication, associate law for 9
Multiplication, closure under 9
Multiplication, commutative law of 9
Multiplication, of arabic numerals 8
Multiplication, of complex numbers 496
Multiplication, of complex numbers, geometric interpretation 500—501
Multiplication, of infinite series 453—455
Multiplicative identity, existence of 9
Multiplicative inverses, existence of 9
Multiplicity (of a root) 116
Napier 332
Napierian logarithm 331
Natural numbers 21 34
Negative number 9
Negative numbers, product of two 7
Negative part of a function 51
Negative, large 64
Nested Intervals Theorem 128
Newton 141 256
Newton’s Law of Cooling 329
Newton’s law of motion 147
Newton’s method 431
Nondecreasing function 228
Nondecreasing sequence 424
Nondifferentiable complex functions 516
Nonincreasing function 228
Nonnegative function 51
Nonnegative sequence 441
Notational nonsense 539
Nowhere-differentiable continuous function 475
nth root 71 501
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