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Cấu trúc
Introductory Analysis: A Deeper View of Calculus
Copyright Page
Contents
Acknowledgments
Preface
Chapter 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
Chapter 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
Chapter III. Limits
1. Sequences and Limits
2. Limits and Removing Discontinuities
3. Limits Involving infinity
Chapter 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
Chapter 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
Chapter 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
Chapter 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
Chapter 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
Chapter IX. Additional Computational Methods
1. L’Hôpital’s Rule
2. Newton’s Method
3. Simpson’s Rule
4. The Substitution Rule for Integrals
References
Index
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