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E-raamat: Course in Calculus and Real Analysis

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Course in Calculus and Real AnalysisSuurem pilt
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Offering a unified exposition of calculus and classical real analysis, this textbook presents a meticulous introduction to singlevariable calculus. Throughout, the exposition makes a distinction between the intrinsic geometric definition of a notion and its analytic characterization, establishing firm foundations for topics often encountered earlier without proof. Each chapter contains numerous examples and a large selection of exercises, as well as a Notes and Comments section, which highlights distinctive features of the exposition and provides additional references to relevant literature.

This second edition contains substantial revisions and additions, including several simplified proofs, new sections, and new and revised figures and exercises. A new chapter discusses sequences and series of realvalued functions of a real variable, and their continuous counterpart: improper integrals depending on a parameter. Two new appendices cover a construction of the real numbers using Cauchy sequences, and a selfcontained proof of the Fundamental Theorem of Algebra.

In addition to the usual prerequisites for a first course in singlevariable calculus, the reader should possess some mathematical maturity and an ability to understand and appreciate proofs. This textbook can be used for a rigorous undergraduate course in calculus, or as a supplement to a later course in real analysis. The authors A Course in Multivariable Calculus is an ideal companion volume, offering a natural extension of the approach developed here to the multivariable setting.



From reviews: [ The first edition is] a rigorous, well-presented and original introduction to the core of undergraduate mathematics first-year calculus. It develops this subject carefully from a foundation of high-school algebra, with interesting improvements and insights rarely found in other books. [ ] This book is a tour de force, and a necessary addition to the library of anyone involved in teaching calculus, or studying it seriously. N.J. Wildberger, Aust. Math. Soc. Gaz.

Arvustused

This book would be a valuable asset to a university library and that many instructors would do well to have a copy of this book in their personal libraries. In addition, I believe that some students would benefit if they possessed a copy of this book to use for reference purposes. (Jonathan Lewin, MAA Reviews, April 15, 2019)

1 Numbers and Functions
1(40)
1.1 Properties of Real Numbers
2(8)
1.2 Inequalities
10(3)
1.3 Functions and Their Geometric Properties
13(19)
Exercises
32(9)
2 Sequences
41(26)
2.1 Convergence of Sequences
41(13)
2.2 Subsequences and Cauchy Sequences
54(5)
2.3 Cluster Points of Sequences
59(3)
Exercises
62(5)
3 Continuity and Limits
67(38)
3.1 Continuity of Functions
67(6)
3.2 Basic Properties of Continuous Functions
73(10)
3.3 Limits of Functions of a Real Variable
83(15)
Exercises
98(7)
4 Differentiation
105(44)
4.1 Derivative and Its Basic Properties
106(13)
4.2 Mean Value Theorem and Taylor Theorem
119(8)
4.3 Monotonicity, Convexity, and Concavity
127(6)
4.4 L'Hopital's Rule
133(8)
Exercises
141(8)
5 Applications of Differentiation
149(32)
5.1 Absolute Minimum and Maximum
149(3)
5.2 Local Extrema and Points of Inflection
152(7)
5.3 Linear and Quadratic Approximations
159(4)
5.4 Picard and Newton Methods
163(12)
Exercises
175(6)
6 Integration
181(52)
6.1 Riemann Integral
181(10)
6.2 Integrable Functions
191(11)
6.3 Fundamental Theorem of Calculus
202(8)
6.4 Riemann Sums
210(6)
6.5 Riemann Integrals over Bounded Sets
216(9)
Exercises
225(8)
7 Elementary Transcendental Functions
233(62)
7.1 Logarithmic and Exponential Functions
234(12)
7.2 Trigonometric Functions
246(12)
7.3 Sine of the Reciprocal
258(7)
7.4 Polar Coordinates
265(9)
7.5 Transcendence
274(5)
Exercises
279(10)
Revision Exercises
289(6)
8 Applications and Approximations of Riemann Integrals
295(70)
8.1 Area of a Region Between Curves
295(7)
8.2 Volume of a Solid
302(12)
8.3 Arc Length of a Curve
314(8)
8.4 Area of a Surface of Revolution
322(6)
8.5 Centroids
328(12)
8.6 Quadrature Rules
340(17)
Exercises
357(8)
9 Infinite Series and Improper Integrals
365(60)
9.1 Convergence of Series
365(7)
9.2 Convergence Tests for Series
372(9)
9.3 Power Series
381(10)
9.4 Convergence of Improper Integrals
391(8)
9.5 Convergence Tests for Improper Integrals
399(7)
9.6 Related Improper Integrals
406(11)
Exercises
417(8)
10 Sequences and Series of Functions, Integrals Depending on a Parameter
425(78)
10.1 Pointwise Convergence of Sequences
426(3)
10.2 Uniform Convergence of Sequences
429(9)
10.3 Uniform Convergence of Series
438(10)
10.4 Weierstrass Approximation Theorems
448(10)
10.5 Bounded Convergence
458(8)
10.6 Riemann Integrals Depending on a Parameter
466(5)
10.7 Improper Integrals Depending on a Parameter
471(21)
Exercises
492(11)
A Construction of the Real Numbers 503(14)
A.1 Equivalence Relations
503(2)
A.2 Cauchy Sequences of Rational Numbers
505(7)
A.3 Uniqueness of a Complete Ordered Field
512(5)
B Fundamental Theorem of Algebra 517(6)
B.1 Complex Numbers and Complex Functions
517(2)
B.2 Polynomials and Their Roots
519(4)
References 523(4)
List of Symbols and Abbreviations 527(6)
Index 533
Sudhir R. Ghorpade is Institute Chair Professor in the Department of Mathematics at the Indian Institute of Technology (IIT) Bombay. He has received several awards, including the All India Council for Technical Education (AICTE) Career Award for Young Teachers and the Prof. S.C. Bhattacharya Award for Excellence in Pure Sciences. His research interests lie in algebraic geometry, combinatorics, coding theory, and commutative algebra.





Balmohan V. Limaye is Professor Emeritus in the Department of Mathematics at the Indian Institute of Technology (IIT) Bombay. He is the author of several research monographs and textbooks, including Linear Functional Analysis for Scientists and Engineers (Springer, 2016). He worked at IIT Bombay for more than 40 years and has twice received the Award for Excellence in Teaching from IIT Bombay. His research interests include Banach algebras, approximation theory, numerical functional analysis, and linear algebra.





The authors companion volume A Course in Multivariable Calculus and Analysis (2010) is also in the UTM series.
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