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ОбложкаMercer P.R. More calculus of a single variable. - New York: Springer Science+Business Media, 2014. - xvi, 411 p.: ill. - (Undergraduate texts in mathematics). - Bibliogr. at the end of the chapters. - Ind.: p.407-411. - ISBN 978-1-4939-1925-3; ISSN 0172-6056
Шифр: (И/В16-M57) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
1  The Real Numbers ............................................ 1
   1.1  Intervals and Absolute Value ........................... 1
   1.2  Rational and Irrational Numbers ........................ 3
   1.3  Sequences .............................................. 5
   1.4  Increasing Sequences .................................. 11
   1.5  The Increasing Bounded Sequence Property .............. 13
   1.6  The Nested Interval Property .......................... 17
   Exercises .................................................. 18
   References ................................................. 24
2  Famous Inequalities ........................................ 25
   2.1  Bernoulli's Inequality and Euler's Number e ........... 25
   2.2  The AGM Inequality .................................... 28
   2.3  The Cauchy-Schwarz Inequality ......................... 34
   Exercises .................................................. 37
   References ................................................. 50
3  Continuous Functions ....................................... 53
   3.1  Basic Properties ...................................... 53
   3.2  Bolzano's Theorem ..................................... 56
   3.3  The Universal Chord Theorem ........................... 59
   3.4  The Intermediate Value Theorem ........................ 61
   3.5  The Extreme Value Theorem ............................. 65
   Exercises .................................................. 66
   References ................................................. 71
4  Differentiable Functions ................................... 73
   4.1  Basic Properties ...................................... 73
   4.2  Differentiation Rules ................................. 77
   4.3  Derivatives of Transcendental Functions ............... 80
   4.4  Fermat's Theorem and Applications ..................... 83
   Exercises .................................................. 87
   References ................................................. 94
5  The Mean Value Theorem ..................................... 97
   5.1  The Mean Value Theorem ................................ 97
   5.2  Applications ......................................... 100
   5.3  Cauchy's Mean Value Theorem .......................... 104
   Exercises ................................................. 107
   References ................................................ 116
6  The Exponential Function .................................. 119
   6.1  The Exponential Function, Quickly .................... 119
   6.2  The Exponential Function, Carefully .................. 122
   6.3  The Natural Logarithmic Function ..................... 126
   6.4  Real Exponents ....................................... 130
   6.5  The AGM Inequality Again ............................. 132
   6.6  The Logarithmic Mean ................................. 135
   6.7  The Harmonic Series and Some Relatives ............... 137
   Exercises ................................................. 142
   References ................................................ 156
7  Other Mean Value Theorems ................................. 159
   7.1  Darboux's Theorem .................................... 159
   7.2  Flett's Mean Value Theorem ........................... 162
   7.3  Pompeiu's Mean Value Theorem ......................... 163
   7.4  A Related Result ..................................... 165
   Exercises ................................................. 165
   References ................................................ 169
8  Convex Functions and Taylor's Theorem ..................... 171
   8.1  Higher Derivatives ................................... 171
   8.2  Convex Functions ..................................... 176
   8.3  Jensen's Inequality .................................. 179
   8.4  Taylor's Theorem: e Is Irrational .................... 183
   8.5  Taylor Series ........................................ 187
   Exercises ................................................. 189
   References ................................................ 207
9  Integration of Continuous Functions ....................... 209
   9.1  The Average Value of a Continuous Function ........... 209
   9.2  The Definite Integral ................................ 215
   9.3  The Definite Integral as Area ........................ 220
   9.4  Some Applications .................................... 223
   9.5  Famous Inequalities for the Definite Integral ........ 228
   9.6  Epilogue ............................................. 233
   Exercises ................................................. 234
   References ................................................ 247
10 The Fundamental Theorem of Calculus ....................... 249
   10.1 The Fundamental Theorem .............................. 249
   10.2 The Natural Logarithmic and Exponential Functions 
        Again ................................................ 260
   Exercises ................................................. 264
   References ................................................ 279
11 Techniques of Integration ................................. 283
   11.1 Integration by u-Substitution ........................ 283
   11.2 Integration by Parts ................................. 287
   11.3 Two Consequences ..................................... 290
   11.4 Taylor's Theorem Again ............................... 294
   Exercises ................................................. 296
   References ................................................ 309
12 Classic Examples .......................................... 311
   12.1 Wallis's Product ..................................... 311
   12.2 π Is Irrational ...................................... 314
   12.3 More Irrational Numbers .............................. 315
   12.4 Euler's Sum Σ l/n2 = π2/6 ............................ 318
   12.5 The Sum Σ l/p of the Reciprocals of the Primes 
        Diverges ............................................. 321
   Exercises ................................................. 323
   References ................................................ 328
13 Simple Quadrature Rules ................................... 331
   13.1 The Rectangle Rules .................................. 331
   13.2 The Trapezoid and Midpoint Rules ..................... 333
   13.3 Stirling's Formula ................................... 337
   13.4 Trapezoid Rule or Midpoint Rule: Which Is Better? .... 341
   Exercises ................................................. 342
   References ................................................ 354

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