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2006; 292 pp; hardcover
List Price: US$53
Member Price: US$42.40
Order Code: ACALC
2015 Recipient of the MAA University Teaching of Mathematics Award
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, Padé 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.
Undergraduate students interested in calculus and number theory.
"This fascinating book is a novel approach to undergraduate analysis, which combines most topics in single variable calculus with some elementary number theory. ... It is very well written and fully engages readers in its developments, often beginning with examples and leading them to develop generalizations and, ultimately, theorems and proofs. ... An attractive book, well worth consulting for ideas on presenting topics, or for examples."
-- J.H. Ellison, Choice
"The book is very well written and contains many references to articles in journals that are accessible to students ..."
-- MAA Reviews
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