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Introductory Analysis, A Deeper View of Calculus by Richard J. Bagby

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About this book :-
Introductory Analysis, A Deeper View of Calculus written by Richard J. Bagby
Introductory Analysis addresses the needs of students taking a course in analysis after completing a semester or two of calculus, and offers an alternative to texts that assume that math majors are their only audience. By using a conversational style that does not compromise mathematical precision, the author explains the material in terms that help the reader gain a firmer grasp of calculus concepts. * Written in an engaging, conversational tone and readable style while softening the rigor and theory * Takes a realistic approach to the necessary and accessible level of abstraction for the secondary education students * A thorough concentration of basic topics of calculus * Features a student-friendly introduction to delta-epsilon arguments * Includes a limited use of abstract generalizations for easy use * Covers natural logarithms and exponential functions * Provides the computational techniques often encountered in basic calculus.
Every aspect of this book was influenced by the desire to present calculus not merely as a prelude to but as the first real encounter with mathematics. Since the foundations of analysis provided the arena in which modern modes of mathematical thinking developed, calculus ought to be the place in which to expect, rather than avoid, the strengthening of insight with logic. In addition to developing the students' intuition about the beautiful concepts of analysis, it is surely equally important to persuade them that precision and rigor are neither deterrents to intuition, nor ends in themselves, but the natural medium in which to formulate and think about mathematical questions.
This goal implies a view of mathematics which, in a sense, the entire book attempts to defend. No matter how well particular topics may be developed, the goals of this book will be realized only if it succeeds as a whole. For this reason, it would be of little value merely to list the topics covered, or to mention pedagogical practices and other innovations. Even the cursory glance customarily bestowed on new calculus texts will probably tell more than any such extended advertisement, and teachers with strong feelingsabout particular aspects of calculus will knowjust where to look to see if this book fulfillstheir requirements.

Book Detail :-
Title: Introductory Analysis, A Deeper View of Calculus
Edition:
Author(s): Richard J. Bagby
Publisher: Harcourt/Academic Press
Series:
Year: 2001
Pages: 219
Type: PDF
Language: English
ISBN: 0120725509,9780120725502,9780080549422
Country: Maxico
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About Author :-
Author Richard J. Bagby is Professor at the Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico.
He does not deprecate rigour. But by presenting his narrative in a casual way, he gains the flexibility in recasting important concepts of proofs in a way understandable to more readers.
There are many ways to teach calculus. Typically, for maths majors, a rigorous epsilon-delta approach has been favoured. So we get the classic texts by Spivak, Apostol and Marsden, for example. But a problem is that this approach often turns off many students who need a firm understanding of calculus and classical analysis. They may not be maths majors. Instead, they could be majoring in engineering or the physical sciences. So Bagby offers them a more informal approach.


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Book Contents :-
Introductory Analysis, A Deeper View of Calculus written by Richard J. Bagby cover the following topics.
I THE REAL NUMBER SYSTEM
1. Familiar Number Systems
2. Intervals
3. Suprema and Infima
4. Exact Arithmetic in R
5. Topics for Further Study
II CONTINUOUS FUNCTIONS
1. Functions in Mathematics
2. Continuity of Numerical Functions
3. The Intermediate Value Theorem
4. More Ways to Form Continuous Functions
5. Extreme Values
III LIMITS
1. Sequences and Limits
2. Limits and Removing Discontinuities
3. Limits Involving
IV THE DERIVATIVE
1. Differentiability
2. Combining Differentiable Functions
3. Mean Values
4. Second Derivatives and Approximations
5. Higher Derivatives
6. Inverse Functions
7. Implicit Functions and Implicit Differentiation
V THE RIEMANN INTEGRAL
1. Areas and Riemann Sums
2. Simplifying the Conditions for Integrability
3. Recognizing Integrability
4. Functions Defined by Integrals
5. The Fundamental Theorem of Calculus
6. Topics for Further Study
VI EXPONENTIAL AND LOGARITHMIC FUNCTIONS
1. Exponents and Logarithms
2. Algebraic Laws as Definitions
3. The Natural Logarithm
4. The Natural Exponential Function
5. An Important Limit
VII CURVES AND ARC LENGTH
1. The Concept of Arc Length
2. Arc Length and Integration
3. Arc Length as a Parameter
4. The Arctangent and Arcsine Functions
5. The Fundamental Trigonometric Limit
VIII SEQUENCES AND SERIES OF FUNCTIONS
1. Functions Defined by Limits
2. Continuity and Uniform Convergence
3. Integrals and Derivatives
4. Taylor’s Theorem
5. Power Series
6. Topics for Further Study
IX ADDITIONAL COMPUTATIONAL METHODS
1. L’Hopital’s Rule
2. Newton’s Method
3. Simpson’s Rule
4. The Substitution Rule for Integrals
References
Index


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