Introductory Analysis, A Deeper View of Calculus by Richard J. Bagby
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Introductory Analysis, A Deeper View of Calculus by
Richard J. Bagby , Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico.
Introductory real analysis can be an exciting course; it is the gateway to an impressive panorama of higher mathematics. But for all too many students, the excitement takes the form of anxiety or even terror; they are overwhelmed. For many, their study of mathematics ends one course sooner than they expected, and for many others, the doorways that should have been opened now seem rigidly barred. It shouldn’t have to be that way, and this book is offered as a remedy.
The goals of first courses in real analysis are often too ambitious. Students are expected to solidify their understanding of calculus, adopt an abstract point of view that generalizes most of the concepts, recognize how explicit examples fit into the general theory and determine whether they satisfy appropriate hypotheses, and not only learn definitions, theorems, and proofs but also learn how to construct valid proofs and relevant examples to demonstrate the need for the hypotheses. Abstract properties such as countability, compactness and connectedness must be mastered. The students who are up to such a challenge emerge ready to take on the world of mathematics.
A large number of students in these courses have much more modest immediate needs. Many are only interested in learning enough mathematics to be a good high-school teacher instead of to prepare for high-level mathematics. Others seek an increased level of mathematical maturity, but something less than a quantum leap is desired. What they need is a new understanding of calculus as a mathematical theory — how to study it in terms of assumptions and consequences, and then check whether the needed assumptions are actually satisfied in specific cases. Without such an understanding, calculus and real analysis seem almost unrelated in spite of the vocabulary they share, and this is why so many good calculus students are overwhelmed by the demands of higher mathematics. Calculus students come to expect regularity but analysis students must learn to expect irregularity; real analysis sometimes shows that incomprehensible levels of pathology are not only possible but theoretically ubiquitous. In calculus courses, students spend most of their energy using finite procedures to find solutions, while analysis addresses questions of existence when there may not even be a finite algorithm for recognizing a solution, let alone for producing one. The obstacle to studying mathematics at the next level isn’t just the inherent difficulty of learning definitions, theorems, and proofs; it is often the lack of an adequate model for interpreting the abstract concepts involved. This is why most students need a different understanding of calculus before taking on the abstract ideas of real analysis. For some students, such as prospective high-school teachers, the next step in mathematical maturity may not even be necessary.
Introductory Analysis, A Deeper View of Calculus 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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