Approximately Calculus

by Shahriari, Shahriar

• ISBN13: 9780821837504
• ISBN10: 0821837508
• Format: Hardcover
• Copyright: 2006-12-01
• Publisher: Amer Mathematical Society

Summary

Is there always a prime number between $n$ and $2n$? Where, approximately, is the millionth prime? And just what does calculus have to do with answering either of these questions? It turns out that calculus has a lot to do with both questions, as this book can show you. The theme of the book is approximations. Calculus is a powerful tool because it allows us to approximate complicated functions with simpler ones. Indeed, replacing a function locally with a linear--or higher order--approximation is at the heart of calculus. The real star of the book, though, is the task of approximating the number of primes up to a number $x$. This leads to the famous Prime Number Theorem--and to the answers to the two questions about primes. While emphasizing the role of approximations in calculus, most major topics are addressed, such as derivatives, integrals, the Fundamental Theorem of Calculus, sequences, series, and so on. However, our particular point of view also leads us to many unusual topics: curvature, Pade approximations, public key cryptography, and an analysis of the logistic equation, to name a few. The reader takes an active role in developing the material by solving problems. Most topics are broken down into a series of manageable problems, which guide you to an understanding of the important ideas. There is also ample exposition to fill in background material and to get you thinking appropriately about the concepts. Approximately Calculus is intended for the reader who has already had an introduction to calculus, but wants to engage the concepts and ideas at a deeper level. It is suitable as a text for an honors or alternative second semester calculus course.

Table of Contents

Patterns and induction

Divisibility Primes

Derivatives and approximations of functions

Antiderivatives and integration

Distribution of primes Log, exponential, and the inverse trigonometric functions

The mean value theorem and approximations

Linearization topics Defining integrals, areas, and arclengths

Improper integrals and techniques of integration

The prime number theorem

Local approximation of functions and integral estimations

Sequences and series

Power series and Taylor series

More on series

Limits of functions

Differential equations

Logical arguments

Hints for selected problems

Bibliography

Index

Table of Contents provided by Publisher. All Rights Reserved.