Calculus of one variable

Description

Preface to the Second Edition Preface to the First Edition Why another book? Key features. Acknowledgments Note to the Reader Contents About the Author 1 Sequence and Series of Real Numbers 1.1 Sequence of Real Numbers 1.1.1 Convergence and Divergence 1.1.2 Some Tests for Convergence and Divergence 1.1.3 Monotonic Sequences 1.1.4 Subsequences 1.1.5 Further Examples 1.1.6 Cauchy Criterion 1.2 Series of Real Numbers 1.2.1 Convergence and Divergence of Series 1.2.2 Some Tests for Convergence 1.2.3 Alternating Series 1.2.4 Madhava-Nilakantha Series 1.2.5 Absolute Convergence 1.3 Additional Exercises 1.3.1 Sequences 1.3.2 Series 2 Limit, Continuity and Differentiability of Functions 2.1 Limit of a Function 2.1.1 Limit Point of a Set 2.1.2 Limit of a Function at a Point 2.1.3 Limit of a Function in Terms of Sequences 2.1.4 Some Properties 2.1.5 Left Limit and Right Limit 2.1.6 Limit at pminfty and Limit pminfty 2.2 Continuity of a Function 2.2.1 Definition and Some Basic Results 2.2.2 Some More Examples 2.2.3 Some Properties of Continuous Functions 2.2.4 Exponential and Logarithm Functions 2.3 Differentiability of a Function 2.3.1 Definition and Examples 2.3.2 Left and Right Derivatives 2.3.3 Some Properties of Differentiable Functions 2.3.4 Local Maxima and Local Minima 2.3.5 Rolle's Theorem and Mean Value Theorems 2.3.6 A Sufficient Condition for a Local Extremum Point 2.3.7 L'Hospital's Rules 2.3.8 Higher Derivatives and Taylor's Formula 2.3.9 Determination of Shapes of a Curves 2.4 Additional Exercises 2.4.1 Limit 2.4.2 Continuity 2.4.3 Differentiation 3 Definite Integral 3.1 Integrability and Integral 3.1.1 Introduction 3.1.2 Lower and Upper Sums 3.1.3 The Integral and Its Characterizations 3.1.4 Some Basic Properties of Integral 3.1.5 Integral of Continuous Functions 3.2 Mean Value Theorems 3.3 Fundamental Theorems 3.4 Some Consequences 3.5 Some Applications 3.5.1 Computing Area Under the Graph of a Function 3.5.2 Computing the Arc Length 3.5.3 Computing Volume of a Solid 3.5.4 Computing the Volume of Solid of Revolution 3.5.5 Computing the Area of Surface of Revolution 3.5.6 Centre of Gravity 3.6 Appendix 3.7 Additional Exercises 4 Improper Integrals 4.1 Definitions 4.1.1 Integrals over Unbounded Intervals 4.1.2 Improper Integrals over Bounded Intervals 4.1.3 Typical Examples 4.2 Tests for Integrability 4.2.1 Integrability by Comparison 4.2.2 Integral Test for Series of Numbers 4.2.3 Integrability Using Limits 4.3 Gamma and Beta Functions 4.3.1 Gamma Function 4.3.2 Beta Function 4.4 Additional Exercises 5 Sequence and Series of Functions 5.1 Sequence of Functions 5.1.1 Pointwise Convergence and Uniform Convergence 5.1.2 Uniform Convergence and Continuity 5.1.3 Uniform Convergence and Integration 5.1.4 Uniform Convergence and Differentiation 5.2 Series of Functions 5.2.1 Dominated Convergence 5.3 Power Series 5.3.1 Convergence and Absolute Convergence 5.3.2 Term by Term Differentiation and Integration 5.4 Additional Exercises 6 Fourier Series 6.1 Fourier Series of 2π-Periodic Functions 6.2 Best Approximation Property 6.3 Fourier Series for Even and Odd Functions 6.4 Sine and Cosine Series Expansions 6.5 Fourier Series of 2ell-Periodic Functions 6.6 Fourier Series on Arbitrary Intervals 6.7 Additional Exercises Appendix References Index