Problem solving through problems问题解决通过问题
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作者： Loren C. Larson著
出 版 社：
出版时间： 1992-8-1字数：版次： 1页数： 332印刷时间： 1992/08/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387961712包装： 平装内容简介
This is a practical anthology of some of the best elementary problems in different branches of mathematics. They are selected for their aesthetic appeal as well as their instructional value, and are organized to highlight the most common problem-solving techniques encountered in undergraduate mathematics. Readers learn important principles and broad strategies for coping with the experience of solving problems, while tackling specific cases on their own. The material is classroom tested and has been found particularly helpful for students preparing for the Putnam exam. For easy reference, the problems are arranged by subject.
目录
Chapter 1. Heuristics
1.1. Search for a Pattern
1.2. Draw a Figure
1.3. Formulate an Equivalent Problem
1.4. Modify the Problem
1.5. Choose Effective Notation
1.6. Exploit Symmetry
1.7. Divide into Cases
1.8. Work Backward
1.9. Argue by Contradiction
1.10. Pursue Parity
l.ll. Consider Extreme Cases
1.12. Generalize
Chapter 2. Two Important Principles: Induction and Pigeonhole
2.1. Induction: Build on P(k)
2.2. Induction: Set Up P(k + 1)
2.3. Strong Induction
2.4. Induction and Generalization
2.5. Recursion
2.6. Pigeonhole Principle
Chapter 3. Arithmetic
3.1. Greatest Common Divisor
3.2. Modular Arithmetic
3.3. Unique Fac'torization
3.4. Positional Notation
3.5. Arithmetic of Complex Numbers
Chapter 4. Algebra
4.1. Algebraic Identities
4.2. Unique Factorization of Polynomials
4.3. The Identity Theorem
4.4. Abstract Algebra
Chapter 5. Summation of Series
5.1. Binomial Coefficients
5.2. Geometric Series
5.3. Telescoping Series
5.4. Power Series
Chapter 6. Intermediate Real Analysis
6.1. Continuous Functions
6.2. The Intermediate-Value Theorem
6.3. The Derivative
6.4. The Extreme-Value Theorem
6.5. Rolle's Theorem
6.6. The Mean Value Theorem
6.7. L'Hopital's Rule
6.8. The Integral
6.9. The Fundamental Theorem
Chapter 7. Inequalities
7.1. Basic Inequality Properties
7.2. Arithmetic-Mean-Geometric-Mean Inequa
7.3. Cauchy-Schwarz Inequality
7.4. Functional Considerations
7.5. Inequalities by Series
7.6. The Squeeze Principle
Chapter 8. Geometry
8.1. Classical Plane Geometry
8.2. Analytic Geometry
8.3. Vector Geometry
8.4. Complex Numbers in Geometry
Contents
Glosary of Symbols and Definitions
Sources
Index
