How and Why of One Variable Calculus

Amol Sasane, Mathematics Department, London School of Economics, UK.

• Preface ixIntroduction xiPreliminary notation xv1 The real numbers 11.1 Intuitive picture of R as points on the number line 21.2 The field axioms 61.3 Order axioms 81.4 The Least Upper Bound Property of R 91.5 Rational powers of real numbers 201.6 Intervals 211.7 Absolute value |·|and distance in R 231.8 (∗) Remark on the construction of R 261.9 Functions 281.10 (∗) Cardinality 40Notes 432 Sequences 442.1 Limit of a convergent sequence 462.2 Bounded and monotone sequences 542.3 Algebra of limits 592.4 Sandwich theorem 642.5 Subsequences 682.6 Cauchy sequences and completeness of R 742.7 (∗) Pointwise versus uniform convergence 78Notes 853 Continuity 863.1 Definition of continuity 863.2 Continuous functions preserve convergence 913.3 Intermediate Value Theorem 993.4 Extreme Value Theorem 1063.5 Uniform convergence and continuity 1113.6 Uniform continuity 1113.7 Limits 115Notes 1244 Differentiation 1254.1 Differentiable Inverse Theorem 1364.2 The Chain Rule 1404.3 Higher order derivatives and derivatives at boundary points 1444.4 Equations of tangent and normal lines to a curve 1484.5 Local minimisers and derivatives 1574.6 Mean Value, Rolle’s, Cauchy’s Theorem 1594.7 Taylor’s Formula 1674.8 Convexity 1724.9 0/0 form of l’Hôpital’s Rule 180Notes 1825 Integration 1835.1 Towards a definition of the integral 1835.2 Properties of the Riemann integral 1985.3 Fundamental Theorem of Calculus 2105.4 Riemann sums 2265.5 Improper integrals 2325.6 Elementary transcendental functions 2455.7 Applications of Riemann Integration 278Notes 2966 Series 2976.1 Series 2976.2 Absolute convergence 3056.3 Power series 320Appendix 335Notes 337Solutions 338Solutions to the exercises from Chapter 1 338Solutions to the exercises from Chapter 2 353Solutions to the exercises from Chapter 3 369Solutions to the exercises from Chapter 4 388Solutions to the exercises from Chapter 5 422Solutions to the exercises from Chapter 6 475Bibliography 493Index 495