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eBookISBN:  9781470411138 
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HardcoverISBN:  9780821837504 
eBookISBN:  9781470411138 
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Hardcover ISBN:  9780821837504 
Product Code:  ACALC 
List Price:  $57.00 
MAA Member Price:  $51.30 
AMS Member Price:  $45.60 
eBook ISBN:  9781470411138 
Product Code:  ACALC.E 
List Price:  $53.00 
MAA Member Price:  $47.70 
AMS Member Price:  $42.40 
Hardcover ISBN:  9780821837504 
eBookISBN:  9781470411138 
Product Code:  ACALC.B 
List Price:  $110.00$83.50 
MAA Member Price:  $99.00$75.15 
AMS Member Price:  $88.00$66.80 

Book Details2006; 292 ppMSC: Primary 26; Secondary 11; 41;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.ReadershipUndergraduate students interested in calculus and number theory.

Table of Contents

Cover

Title page

Copyright

Dedication

Contents

Preface

Chapter 1: Patterns and induction

Chapter 2: Divisibility

Chapter 3: Primes

Chapter 4: Derivatives and approximations of functions

Chapter 5: Antiderivatives and integration

Chapter 6: Distribution of primes

Chapter 7: Log, exponential, and the inverse trigonometric functions

Chapter 8: The Mean Value Theorem and approximations

Chapter 9: Linearization topics

Chapter 10: Defining integrals, areas, and arclengths

Chapter 11: Improper integrals and techniques of integration

Chapter 12: The Prime Number Theorem

Chapter 13: Local approximation of functions and integral estimations

Chapter 14: Sequences and series

Chapter 15: Power series and Taylor series

Chapter 16: More on series

Chapter 17: Limits of functions

Chapter 18: Differential equations

Chapter 19: Logical arguments

Hints for selected problems

Bibliography

Index

Back Cover


Additional Material

Reviews

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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 Book Details
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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.

Cover

Title page

Copyright

Dedication

Contents

Preface

Chapter 1: Patterns and induction

Chapter 2: Divisibility

Chapter 3: Primes

Chapter 4: Derivatives and approximations of functions

Chapter 5: Antiderivatives and integration

Chapter 6: Distribution of primes

Chapter 7: Log, exponential, and the inverse trigonometric functions

Chapter 8: The Mean Value Theorem and approximations

Chapter 9: Linearization topics

Chapter 10: Defining integrals, areas, and arclengths

Chapter 11: Improper integrals and techniques of integration

Chapter 12: The Prime Number Theorem

Chapter 13: Local approximation of functions and integral estimations

Chapter 14: Sequences and series

Chapter 15: Power series and Taylor series

Chapter 16: More on series

Chapter 17: Limits of functions

Chapter 18: Differential equations

Chapter 19: Logical arguments

Hints for selected problems

Bibliography

Index

Back Cover

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