Author | Larson, Loren C. author |
---|---|
Title | Problem-Solving Through Problems [electronic resource] / by Loren C. Larson |
Imprint | New York, NY : Springer New York, 1983 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-5498-0 |
Descript | XI, 352 p. 11 illus. online resource |
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. Rolleโs Theorem -- 6.6. The Mean Value Theorem -- 6.7. LโHรดpitalโs 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