Problem-Solving Through Problems / Edition 1

Problem-Solving Through Problems / Edition 1

by Loren C. Larson
     
 

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ISBN-10: 0387961712

ISBN-13: 9780387961712

Pub. Date: 09/03/1992

Publisher: Springer New York

This is a practical anthology of some of the best elementary problems in different branches of mathematics. Arranged by subject, the problems highlight the most common problem-solving techniques encountered in undergraduate mathematics. This book teaches the important principles and broad strategies for coping with the experience of solving problems. It has been

Overview

This is a practical anthology of some of the best elementary problems in different branches of mathematics. Arranged by subject, the problems highlight the most common problem-solving techniques encountered in undergraduate mathematics. This book teaches the important principles and broad strategies for coping with the experience of solving problems. It has been found very helpful for students preparing for the Putnam exam.

Product Details

ISBN-13:
9780387961712
Publisher:
Springer New York
Publication date:
09/03/1992
Series:
Problem Books in Mathematics Series
Edition description:
1st ed. 1983. Corr. 3rd printing 1992
Pages:
352
Product dimensions:
6.00(w) x 9.20(h) x 0.90(d)

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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