Details

<p>Preface ix</p> <p>Introduction xi</p> <p>Preliminary notation xv</p> <p><b>1 The real numbers 1</b></p> <p>1.1 Intuitive picture of R as points on the number line 2</p> <p>1.2 The field axioms 6</p> <p>1.3 Order axioms 8</p> <p>1.4 The Least Upper Bound Property of R 9</p> <p>1.5 Rational powers of real numbers 20</p> <p>1.6 Intervals 21</p> <p>1.7 Absolute value |·|and distance in R 23</p> <p>1.8 (∗) Remark on the construction of R 26</p> <p>1.9 Functions 28</p> <p>1.10 (∗) Cardinality 40</p> <p>Notes 43</p> <p><b>2 Sequences 44</b></p> <p>2.1 Limit of a convergent sequence 46</p> <p>2.2 Bounded and monotone sequences 54</p> <p>2.3 Algebra of limits 59</p> <p>2.4 Sandwich theorem 64</p> <p>2.5 Subsequences 68</p> <p>2.6 Cauchy sequences and completeness of R 74</p> <p>2.7 (∗) Pointwise versus uniform convergence 78</p> <p>Notes 85</p> <p><b>3 Continuity 86</b></p> <p>3.1 Definition of continuity 86</p> <p>3.2 Continuous functions preserve convergence 91</p> <p>3.3 Intermediate Value Theorem 99</p> <p>3.4 Extreme Value Theorem 106</p> <p>3.5 Uniform convergence and continuity 111</p> <p>3.6 Uniform continuity 111</p> <p>3.7 Limits 115</p> <p>Notes 124</p> <p><b>4 Differentiation 125</b></p> <p>4.1 Differentiable Inverse Theorem 136</p> <p>4.2 The Chain Rule 140</p> <p>4.3 Higher order derivatives and derivatives at boundary points 144</p> <p>4.4 Equations of tangent and normal lines to a curve 148</p> <p>4.5 Local minimisers and derivatives 157</p> <p>4.6 Mean Value, Rolle’s, Cauchy’s Theorem 159</p> <p>4.7 Taylor’s Formula 167</p> <p>4.8 Convexity 172</p> <p>4.9 0/0 form of l’Hôpital’s Rule 180</p> <p>Notes 182</p> <p><b>5 Integration 183</b></p> <p>5.1 Towards a definition of the integral 183</p> <p>5.2 Properties of the Riemann integral 198</p> <p>5.3 Fundamental Theorem of Calculus 210</p> <p>5.4 Riemann sums 226</p> <p>5.5 Improper integrals 232</p> <p>5.6 Elementary transcendental functions 245</p> <p>5.7 Applications of Riemann Integration 278</p> <p>Notes 296</p> <p><b>6 Series 297</b></p> <p>6.1 Series 297</p> <p>6.2 Absolute convergence 305</p> <p>6.3 Power series 320</p> <p>Appendix 335</p> <p>Notes 337</p> <p>Solutions 338</p> <p>Solutions to the exercises from Chapter 1 338</p> <p>Solutions to the exercises from Chapter 2 353</p> <p>Solutions to the exercises from Chapter 3 369</p> <p>Solutions to the exercises from Chapter 4 388</p> <p>Solutions to the exercises from Chapter 5 422</p> <p>Solutions to the exercises from Chapter 6 475</p> <p>Bibliography 493</p> <p>Index 495</p>

<strong>Amol Sasane</strong>, Mathematics Department, London School of Economics, UK.

<p><i>The How and Why of One Variable Calculus</i></p> <p><b>Amol Sasane</b>, Mathematics Department, London School of Economics</p> <p>First course calculus texts have traditionally been either engineering/ science-oriented with too little rigour, or have thrown students in the deep end with a rigorous analysis text. <i>The How and Why of One Variable Calculus </i>closes this gap in providing a rigorous treatment that takes an original and valuable approach between calculus and analysis.Logically organized and user-friendly, it covers 6 main topics; real numbers, sequences, continuity, differentiation, integration, and series. Primarily concerned with developing an understanding of the tools of calculus, it features numerous examples and exercises that illustrate how the techniques of calculus have universal application.</p> <p> <i>Key features:</i></p> <ul> <li>Provides a user-friendly text with more attention to rigour than is usually found in traditional calculus texts.</li> <li>Can be used both as a first course and as a text for a bridge course between calculus and upper level mathematics.</li> <li>Presents numerous illustrations, examples, exercises and detailed solutions to aid the reader’s understanding.</li> </ul> <p> <i>The How and Why of One Variable Calculus </i>presents an excellent text for a first course in calculus for students in the mathematical sciences, statistics and business analytics. It can also be used as a text for a bridge course between single and multi-variable calculus, as well as between single variable calculus and upper level theory courses for math majors.</p> <p> </p>

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