Maxima and minima without calculus
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Washington, DC
Math. Assoc. of America
1981
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| Schriftenreihe: | The Dolciani mathematical expositions
6 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
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| 245 | 1 | 0 | |a Maxima and minima without calculus |c by Ivan Niven |
| 264 | 1 | |a Washington, DC |b Math. Assoc. of America |c 1981 | |
| 300 | |a XV, 303 S. |b graph.Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Datensatz im Suchindex
| _version_ | 1819701308434677760 |
|---|---|
| adam_text | CONTENTS
PAGE
PREFACE ix
CHAPTER ONE. BACKGROUND MATERIAL 1
1.1. Language and Notation 1
1.2. Geometry and Trigonometry 4
1.3. Areas and Volumes 7
1.4. Inequalities 11
1.5. The Sigma Notation 13
CHAPTER TWO. SIMPLE ALGEBRAIC RESULTS 15
2.1. Sums and Products 15
2.2. Any Square Is Positive or Zero 16
2.3. The Inequality of the Arithmetic Geometric Means 20
2.4. An Alternative Approach 23
2.5. Cauchy s Proof 24
2.6. Techniques for Finding Extrema 26
2.7. The Inequality of the Arithmetic Harmonic Means 36
2.8. TheNumbere 38
2.9. Cauchy s Inequality 41
CHAPTER THREE. ELEMENTARY GEOMETRIC QUESTIONS 47
3.1. Introduction 47
3.2. Triangles 47
3.3. Quadrilaterals 50
3.4. Miscellaneous Results in Geometry 55
3.5. The Reflection Principle 63
3.6. Equivalent Results 67
3.7. Auxiliary Circles 70
CHAPTER FOUR. ISOPERIMETRIC RESULTS 77
4.1. Some Definitions 77
4.2. Polygons 79
4.3. The Isoperimetric Theorem 81
4.4. The Isoperimetric Quotient 85
4.5. Existence and Uniqueness 87
xiii
xiv MAXIMA AND MINIMA WITHOUT CALCULUS
CHAPTER FIVE. BASIC TRIGONOMETRIC INEQUALITIES 92
5.1. A New Direction 92
5.2. Some Trigonometric Inequalities 92
5.3. The Jensen Inequalities 97
5.4. Other Trigonometric Functions 100
5.5. Extreme Values of a sinS + ft cos 0 103
5.6. Tacking Against a Headwind 106
CHAPTER SIX. POLYGONS INSCRIBED AND CIRCUMSCRIBED Ill
6.1. Introduction Ill
6.2. Regular Polygons 113
6.3. Inscribed and Circumscribed Polygons 115
6.4. A Definition of v 120
6.5. Circles Versus Regular Polygons 123
CHAPTER SEVEN. ELLIPSES 126
7.1. A Basic Mapping 126
7.2. Parametric Equations 129
7.3. Polygons Inscribed in an Ellipse 130
7.4. Circumscribed Polygons 132
7.5. Tangent Lines and Extreme Values 133
7.6. Shortest Distance from a Point to a Curve 137
7.7. Extreme Points on an Ellipse 141
CHAPTER EIGHT. THE BEES AND THEIR HEXAGONS 144
8.1. The Two Problems 144
8.2. Tiling by Regular Polygons 147
8.3. Nonconvex Polygons 148
8.4. Tiling by Convex Polygons 149
8.5. The Summing up 153
CHAPTER NINE. FURTHER GEOMETRIC RESULTS 156
9.1. Introduction 156
9.2. A Problem of Fermat 156
9.3. The Inscribed Triangle 162
9.4. A Theorem of Erdds and Mordell 168
9.5. Lines of Division 171
9.6. Enclosing a Convex Region in a Rectangle 175
CHAPTER TEN. APPLIED AND MISCELLANEOUS PROBLEMS 179
10.1. Lines of Best Fit 179
10.2. The Least Squares Line in General 181
10.3. The Most Likely Number of Occurrences 183
10.4. Experimental Solutions of Minimal Problems 188
10.5. Ptolemy s Theorem 193
10.6. The Refraction of Light 195
10.7. Time and Distance Problems 198
CONTENTS XV
10.8. Minimax Problems 203
10.9. The Jeep Crossing the Desert 204
CHAPTER ELEVEN. EUCLIDEAN THREE SPACE 211
11.1. Preliminary Results 211
11.2. The Isoperimetric Theorem for Tetrahedra 213
11.3. Inscribed and Circumscribed Spheres of a Tetrahedron 217
11.4. Shortest Paths on a Sphere 218
CHAPTER TWELVE. ISOPERIMETRIC RESULTS NOT ASSUMING EXISTENCE . . 226
12.1. The Need for a Closer Study 226
12.2. The Inner Parallel Polygon 227
12.3. The Isoperimetric Theorem 231
12.4. The Isoperimetric Theorem for Polygons 234
12.5. Polygons with Prescribed Sides 236
POSTSCRIPT ON CALCULUS 239
SOLUTIONS OF PROBLEMS 244
REFERENCES 293
INDEX 299
|
| any_adam_object | 1 |
| author | Niven, Ivan 1915-1999 |
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| dewey-full | 511/.66 511.66 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511/.66 511.66 |
| dewey-search | 511/.66 511.66 |
| dewey-sort | 3511 266 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV006004599 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T11:58:47Z |
| institution | BVB |
| isbn | 088385306X 0883853000 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003773934 |
| oclc_num | 154176174 |
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| physical | XV, 303 S. graph.Darst. |
| publishDate | 1981 |
| publishDateSearch | 1981 |
| publishDateSort | 1981 |
| publisher | Math. Assoc. of America |
| record_format | marc |
| series | The Dolciani mathematical expositions |
| series2 | The Dolciani mathematical expositions |
| spellingShingle | Niven, Ivan 1915-1999 Maxima and minima without calculus The Dolciani mathematical expositions Maxima and minima Extremwert (DE-588)4137272-4 gnd |
| subject_GND | (DE-588)4137272-4 |
| title | Maxima and minima without calculus |
| title_auth | Maxima and minima without calculus |
| title_exact_search | Maxima and minima without calculus |
| title_full | Maxima and minima without calculus by Ivan Niven |
| title_fullStr | Maxima and minima without calculus by Ivan Niven |
| title_full_unstemmed | Maxima and minima without calculus by Ivan Niven |
| title_short | Maxima and minima without calculus |
| title_sort | maxima and minima without calculus |
| topic | Maxima and minima Extremwert (DE-588)4137272-4 gnd |
| topic_facet | Maxima and minima Extremwert |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003773934&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001900740 |
| work_keys_str_mv | AT nivenivan maximaandminimawithoutcalculus |