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Full Description
This book contains enrichment material for courses in first and second year calculus, differential equations, modeling, and introductory real analysis. It targets talented students who seek a deeper understanding of calculus and its applications. The book can be used in honors courses, undergraduate seminars, independent study, capstone courses taking a fresh look at calculus, and summer enrichment programs. The book develops topics from novel and/or unifying perspectives. Hence, it is also a valuable resource for graduate teaching assistants developing their academic and pedagogical skills and for seasoned veterans who appreciate fresh perspectives.
The explorations, problems, and projects in the book impart a deeper understanding of and facility with the mathematical reasoning that lies at the heart of calculus and conveys something of its beauty and depth. A high level of rigor is maintained. However, with few exceptions, proofs depend only on tools from calculus and earlier. Analytical arguments are carefully structured to avoid epsilons and deltas. Geometric and/or physical reasoning motivates challenging analytical discussions. Consequently, the presentation is friendly and accessible to students at various levels of mathematical maturity. Logical reasoning skills at the level of proof in Euclidean geometry suffice for a productive use of the book.
Contents
Preface
The Foundation On Which Calculus Stands
1. Critical Points and Graphing
2. Inverse Functions
3. Exponential and Logarithmic Functions
4. Linear Approximation and Newton's Method
5. Taylor Polynomial Approximation
6. Global Extreme Values
7. Angular Velocity and Curvature
8. π and e are Irrational
9. Hanging Cables
10. The Buffon Needle Problem
11. Optimal Location
12. Energy
13. Springs and Pendulums
14. Kepler's Laws of Planetary Motion
15. Newton's Law of Universal Gravitation
16. From Newton to Kepler and Beyond
Index
