Aspects of Calculus

ISBN-13: 9781461395638 / Angielski / Miękka / 2011 / 516 str.

Gabriel Klambauer

Opis

This book is intended for students familiar with a beginner's version of differential and integral calculus stressing only manipulation offormulas and who are now looking for a closer study of basic concepts combined with a more creative use of information. The work is primarily aimed at students in mathematics, engineering, and science who find themselves in transition from elementary calculus to rigorous courses in analysis. In addition, this book may also be of interest to those preparing to teach a course in calculus. Instead of exposing the reader to an excess of premature abstractions that so easily can degenerate into pedantry, I felt it more useful to stress instruc- tive and stimulating examples. The book contains numerous worked out examples and many of the exercises are provided with helpful hints or a solution in outline. For further exercises the interested reader may want to consult a problem book by the author entitled Problems and Propositions in Analysis (New York: Marcel Dekker, 1979). For the history of calculus I recommend the book by C. B. Boyer, The Concepts of the Calculus (New York: Dover, 1949).

Spis treści

1 The Logarithmic and Exponential Functions.- 1. An Area Problem.- 2. The Natural Logarithm.- 3. The Exponential Function.- 4. The Hyperbolic Functions.- 5. Miscellaneous Examples.- Exercises to Chapter 1.- 2 Limits and Continuity.- 1. Limits.- 2. Continuity.- 3. Monotonic Functions.- 4. Miscellaneous Examples.- Exercises to Chapter 2.- 3 Differentiation.- 1. Basic Rules of Differentiation.- 2. Derivatives of Basic Functions.- 3. Mechanics of Differentiation.- 4. Asymptotes.- 5. Tangent to a Conic Section.- Exercises to Chapter 3.- 4 Applications of Differentiation.- 1. Mean Value Theorems.- 2. Taylor’s Theorem.- 3. Concave Functions.- 4. Newton’s Method for Approximating Real Roots of Functions.- 5. Arithmetic and Geometric Means.- 6. Miscellaneous Examples.- Exercises to Chapter 4.- 5 Integration.- 1. Examples of Area Calculation.- 2. Area of a Planar Region.- 3. The Riemann Integral.- 4. Basic Propositions of Integral Calculus.- 5. Numerical Integration.- Exercises to Chapter 5.- 6 Additional Topics in Integration.- 1. The Indefinite Integral.- 2. Some Techniques of Integration.- 3. Integration of Rational Functions.- 4. Integration by Rationalization.- 5. Some Applications of Integration.- Exercises to Chapter 6.- 7 Infinite Series.- 1. Numerical Sequences.- 2. The Formulas of Wallis and Stirling.- 3. Numerical Series.- 4. Groupings and Rearrangements.- 5. Uniform Convergence.- 6. Power Series.- 7. Some Important Power Series.- 8. Miscellaneous Examples.- Exercises to Chapter 7.