Problem-Solving Through Problems / Edition 1by Loren C. Larson
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… See more details below
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.
Table of Contents
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.- 1.11. Consider Extreme Cases.- 1.12. Generalize.- 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.- 3. Arithmetic.- 3.1. Greatest Common Divisor.- 3.2. Modular Arithmetic.- 3.3. Unique Factorization.- 3.4. Positional Notation.- 3.5. Arithmetic of Complex Numbers.- 4. Algebra.- 4.1. Algebraic Identities.- 4.2. Unique Factorization of Polynomials.- 4.3. The Identity Theorem.- 4.4. Abstract Algebra.- 5. Summation of Series.- 5.1. Binomial Coefficients.- 5.2. Geometric Series.- 5.3. Telescoping Series.- 5.4. Power Series.- 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’Hôpital’s Rule.- 6.8. The Integral.- 6.9. The Fundamental Theorem.- 7. Inequalities.- 7.1. Basic Inequality Properties.- 7.2. Arithmetic-Mean-Geometric-Mean Inequality.- 7.3. Cauchy-Schwarz Inequality.- 7.4. Functional Considerations.- 7.5. Inequalities by Series.- 7.6. The Squeeze Principle.- 8. Geometry.- 8.1. Classical Plane Geometry.- 8.2. Analytic Geometry.- 8.3. Vector Geometry.- 8.4. Complex Numbers in Geometry.- Glossary of Symbols and Definitions.- Sources.
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