Approximately calculus:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, R.I.
AMS, American Mathematical Society
2006
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (p. 283-286) and index |
| Beschreibung: | XVII, 292 S. graph. Darst. |
| ISBN: | 9780821837504 0821837508 |
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| 100 | 1 | |a Shahriari, Shahriar |d 1956- |e Verfasser |0 (DE-588)141856734 |4 aut | |
| 245 | 1 | 0 | |a Approximately calculus |c Shahriar Shahriari |
| 264 | 1 | |a Providence, R.I. |b AMS, American Mathematical Society |c 2006 | |
| 300 | |a XVII, 292 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references (p. 283-286) and index | ||
| 650 | 4 | |a Approximation theory |v Problems, exercises, etc | |
| 650 | 4 | |a Calculus |v Problems, exercises, etc | |
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| adam_text | Contents
Preface xiii
Chapter 1. Patterns and Induction 1
Goals 1
1.1. Mathematics, Patterns, and Computers 1
Problems 4
1.2. Writing Mathematical Sentences and Proof by Induction 5
Problems 9
1.3. Additional Problems 11
Problems 11
Chapter 2. Divisibility 17
Goals 17
2.1. The Division Algorithm and Strange Properties of Positive Integers 17
Problems 19
2.2. Writing Mathematics in Paragraphs, Proof by Contradiction, and
Irrational Numbers 22
Problems 24
2.3. Introducing the planet mod n 25
Problems 26
2.4. Additional Problems 27
Problems 27
Chapter 3. Primes 33
Goals 33
3.1. Prime Numbers 33
Problems 34
3.2. Formulas for Primes 38
Problems 41
3.3. Fermat s Theorem, Pseudo Primes, and Carmichael Numbers 42
Problems 42
3.4. Dynamical Systems and a Proof of Fermat s Theorem 45
Problems 50
3.5. Public Key Cryptography 53
Problems 55
3.6. Open Conjectures about Primes 57
Chapter 4. Derivatives and Approximations of Functions 59
Goals 59
4.1. A Quick Review of Derivatives 59
vii
viii CONTENTS
Problems 63
4.2. Continuous Functions and Differentiability 66
Problems 70
4.3. Linearization and Approximation of Functions 73
Problems 81
4.4. Taylor Polynomials 84
Problems 87
4.5. Additional Problems 89
Problems 89
Chapter 5. Antiderivatives and Integration 91
Goals 91
5.1. Why Can Areas Be Found Using Antiderivatives? A Quick Review of
Integration 91
Problems 98
5.2. Approximating Integrals: Inscribed and Circumscribed Rectangles 100
Problems 103
5.3. Functions Defined by Integrals 105
Problems 106
5.4. What if F (x) = G (x)l 108
Problems 109
Chapter 6. Distribution of Primes 111
Goals 111
6.1. Approximating x/n(x) 111
Problems 112
6.2. The Sieve of Eratosthenes 113
Problems 113
Chapter 7. Log, Exponential, and the Inverse Trigonometric Functions 115
Goals 115
7.1. The Natural Log Function and the Distribution of Primes 115
Problems 116
7.2. Properties of the Log Function 117
Problems 118
7.3. The Exponential Function 120
Problems 122
7.4. Inverse Trigonometric Functions 127
Problems 128
7.5. Additional Problems 128
Problems 128
Chapter 8. The Mean Value Theorem and Approximations 131
Goals 131
8.1. Real Numbers and Properties of Continuous Functions 131
Problems 133
8.2. The Mean Value Theorem for Integrals 134
Problems 134
8.3. The Mean Value Theorem 135
Problems 137
CONTENTS ix
8.4. The Error in a Taylor Polynomial Approximation 139
Problems 142
Chapter 9. Linearization Topics 147
Goals 147
9.1. L Hospital s Rule 147
Problems 148
9.2. Newton s Method 151
Problems 152
Chapter 10. Defining Integrals, Areas, and Arclengths 155
Goals 155
10.1. Going Beyond an Intuitive Notion of Area 155
Problems 162
10.2. Arc Length 164
Problems 165
Chapter 11. Improper Integrals and Techniques of Integration 167
Goals 167
11.1. Improper Integrals 167
Problems 168
11.2. Integration Methods 170
Problems 171
Chapter 12. The Prime Number Theorem 175
Goals 175
12.1. The Prime Number Theorem 175
Problems 176
12.2. Primes between n and 2n 176
Problems 177
12.3. Logarithmic integral 177
Problems 178
12.4. Where approximately is the nth prime? 179
Problems 179
12.5. Primes and the Riemann Hypothesis 180
Chapter 13. Local Approximation of Functions and Integral Estimations 183
Goals 183
13.1. Taylor Polynomials and Approximations of Integrals. Are they
related? 183
Problems 184
13.2. Approximating Integrals: Rectangles, Trapezoids, and Parabolas 187
Problems 188
13.3. Curvature 189
Problems 190
13.4. Pade Approximants 191
Problems 192
Chapter 14. Sequences and Series 195
Goals 195
x CONTENTS
14.1. Sequences, Convergence, and Mathematical Rigor 195
Problems 199
14.2. Series 203
Problems 206
14.3. Monotone Bounded Sequences and Limit Properties 208
Problems 210
14.4. The nth Term Test and the Comparison Test 213
Problems 214
14.5. Euler s Constant and the Alternating Harmonic Series 217
Problems 217
14.6. The Integral Test and p series 218
Problems 219
14.7. Additional Problems 221
Problems 221
Chapter 15. Power Series and Taylor Series 223
Goals 223
15.1. Taylor Polynomials and Series 223
Problems 225
15.2. Power Series and the Ratio Test 225
Problems 226
15.3. Analytic Functions and Convergence of Taylor Series 231
Problems 232
15.4. The Interval of Convergence of a Power Series 233
Problems 235
15.5. New Power Series from Old 236
Problems 238
Chapter 16. More On Series 243
Goals 243
16.1. The Limit Comparison Test 243
Problems 243
16.2. Leibniz s Alternating Series Test 247
Problems 248
16.3. Additional Problems 249
Problems 249
Chapter 17. Limits of Functions 251
Goals 251
Introduction 251
17.1. The Precise Definition of Limits 251
Problems 255
Chapter 18. Differential Equations 259
Goals 259
18.1. Differential Equations and Modeling 259
Problems 260
18.2. Qualitative Analysis of Differential Equations 262
Problems 263
18.3. Additional Problems 266
CONTENTS xi
Problems 266
Chapter 19. Logical Arguments 271
Goals 271
19.1. Logical Reasoning through Puzzles 271
Problems 271
Hints for Selected Problems 277
Bibliography 283
Index 287
|
| adam_txt |
Contents
Preface xiii
Chapter 1. Patterns and Induction 1
Goals 1
1.1. Mathematics, Patterns, and Computers 1
Problems 4
1.2. Writing Mathematical Sentences and Proof by Induction 5
Problems 9
1.3. Additional Problems 11
Problems 11
Chapter 2. Divisibility 17
Goals 17
2.1. The Division Algorithm and Strange Properties of Positive Integers 17
Problems 19
2.2. Writing Mathematics in Paragraphs, Proof by Contradiction, and
Irrational Numbers 22
Problems 24
2.3. Introducing the planet mod n 25
Problems 26
2.4. Additional Problems 27
Problems 27
Chapter 3. Primes 33
Goals 33
3.1. Prime Numbers 33
Problems 34
3.2. Formulas for Primes 38
Problems 41
3.3. Fermat's Theorem, Pseudo Primes, and Carmichael Numbers 42
Problems 42
3.4. Dynamical Systems and a Proof of Fermat's Theorem 45
Problems 50
3.5. Public Key Cryptography 53
Problems 55
3.6. Open Conjectures about Primes 57
Chapter 4. Derivatives and Approximations of Functions 59
Goals 59
4.1. A Quick Review of Derivatives 59
vii
viii CONTENTS
Problems 63
4.2. Continuous Functions and Differentiability 66
Problems 70
4.3. Linearization and Approximation of Functions 73
Problems 81
4.4. Taylor Polynomials 84
Problems 87
4.5. Additional Problems 89
Problems 89
Chapter 5. Antiderivatives and Integration 91
Goals 91
5.1. Why Can Areas Be Found Using Antiderivatives? A Quick Review of
Integration 91
Problems 98
5.2. Approximating Integrals: Inscribed and Circumscribed Rectangles 100
Problems 103
5.3. Functions Defined by Integrals 105
Problems 106
5.4. What if F'(x) = G'(x)l 108
Problems 109
Chapter 6. Distribution of Primes 111
Goals 111
6.1. Approximating x/n(x) 111
Problems 112
6.2. The Sieve of Eratosthenes 113
Problems 113
Chapter 7. Log, Exponential, and the Inverse Trigonometric Functions 115
Goals 115
7.1. The Natural Log Function and the Distribution of Primes 115
Problems 116
7.2. Properties of the Log Function 117
Problems 118
7.3. The Exponential Function 120
Problems 122
7.4. Inverse Trigonometric Functions 127
Problems 128
7.5. Additional Problems 128
Problems 128
Chapter 8. The Mean Value Theorem and Approximations 131
Goals 131
8.1. Real Numbers and Properties of Continuous Functions 131
Problems 133
8.2. The Mean Value Theorem for Integrals 134
Problems 134
8.3. The Mean Value Theorem 135
Problems 137
CONTENTS ix
8.4. The Error in a Taylor Polynomial Approximation 139
Problems 142
Chapter 9. Linearization Topics 147
Goals 147
9.1. L'Hospital's Rule 147
Problems 148
9.2. Newton's Method 151
Problems 152
Chapter 10. Defining Integrals, Areas, and Arclengths 155
Goals 155
10.1. Going Beyond an Intuitive Notion of Area 155
Problems 162
10.2. Arc Length 164
Problems 165
Chapter 11. Improper Integrals and Techniques of Integration 167
Goals 167
11.1. Improper Integrals 167
Problems 168
11.2. Integration Methods 170
Problems 171
Chapter 12. The Prime Number Theorem 175
Goals 175
12.1. The Prime Number Theorem 175
Problems 176
12.2. Primes between n and 2n 176
Problems 177
12.3. Logarithmic integral 177
Problems 178
12.4. Where approximately is the nth prime? 179
Problems 179
12.5. Primes and the Riemann Hypothesis 180
Chapter 13. Local Approximation of Functions and Integral Estimations 183
Goals 183
13.1. Taylor Polynomials and Approximations of Integrals. Are they
related? 183
Problems 184
13.2. Approximating Integrals: Rectangles, Trapezoids, and Parabolas 187
Problems 188
13.3. Curvature 189
Problems 190
13.4. Pade Approximants 191
Problems 192
Chapter 14. Sequences and Series 195
Goals 195
x CONTENTS
14.1. Sequences, Convergence, and Mathematical Rigor 195
Problems 199
14.2. Series 203
Problems 206
14.3. Monotone Bounded Sequences and Limit Properties 208
Problems 210
14.4. The nth Term Test and the Comparison Test 213
Problems 214
14.5. Euler's Constant and the Alternating Harmonic Series 217
Problems 217
14.6. The Integral Test and p series 218
Problems 219
14.7. Additional Problems 221
Problems 221
Chapter 15. Power Series and Taylor Series 223
Goals 223
15.1. Taylor Polynomials and Series 223
Problems 225
15.2. Power Series and the Ratio Test 225
Problems 226
15.3. Analytic Functions and Convergence of Taylor Series 231
Problems 232
15.4. The Interval of Convergence of a Power Series 233
Problems 235
15.5. New Power Series from Old 236
Problems 238
Chapter 16. More On Series 243
Goals 243
16.1. The Limit Comparison Test 243
Problems 243
16.2. Leibniz's Alternating Series Test 247
Problems 248
16.3. Additional Problems 249
Problems 249
Chapter 17. Limits of Functions 251
Goals 251
Introduction 251
17.1. The Precise Definition of Limits 251
Problems 255
Chapter 18. Differential Equations 259
Goals 259
18.1. Differential Equations and Modeling 259
Problems 260
18.2. Qualitative Analysis of Differential Equations 262
Problems 263
18.3. Additional Problems 266
CONTENTS xi
Problems 266
Chapter 19. Logical Arguments 271
Goals 271
19.1. Logical Reasoning through Puzzles 271
Problems 271
Hints for Selected Problems 277
Bibliography 283
Index 287 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T18:52:04Z |
| indexdate | 2024-07-09T21:07:45Z |
| institution | BVB |
| isbn | 9780821837504 0821837508 |
| language | English |
| lccn | 2006048370 |
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| spelling | Shahriari, Shahriar 1956- Verfasser (DE-588)141856734 aut Approximately calculus Shahriar Shahriari Providence, R.I. AMS, American Mathematical Society 2006 XVII, 292 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (p. 283-286) and index Approximation theory Problems, exercises, etc Calculus Problems, exercises, etc Infinitesimalrechnung (DE-588)4072798-1 gnd rswk-swf Elementare Zahlentheorie (DE-588)4294368-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Infinitesimalrechnung (DE-588)4072798-1 s Elementare Zahlentheorie (DE-588)4294368-1 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091287&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Shahriari, Shahriar 1956- Approximately calculus Approximation theory Problems, exercises, etc Calculus Problems, exercises, etc Infinitesimalrechnung (DE-588)4072798-1 gnd Elementare Zahlentheorie (DE-588)4294368-1 gnd |
| subject_GND | (DE-588)4072798-1 (DE-588)4294368-1 (DE-588)4123623-3 |
| title | Approximately calculus |
| title_auth | Approximately calculus |
| title_exact_search | Approximately calculus |
| title_exact_search_txtP | Approximately calculus |
| title_full | Approximately calculus Shahriar Shahriari |
| title_fullStr | Approximately calculus Shahriar Shahriari |
| title_full_unstemmed | Approximately calculus Shahriar Shahriari |
| title_short | Approximately calculus |
| title_sort | approximately calculus |
| topic | Approximation theory Problems, exercises, etc Calculus Problems, exercises, etc Infinitesimalrechnung (DE-588)4072798-1 gnd Elementare Zahlentheorie (DE-588)4294368-1 gnd |
| topic_facet | Approximation theory Problems, exercises, etc Calculus Problems, exercises, etc Infinitesimalrechnung Elementare Zahlentheorie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091287&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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