This textbook provides a gentle overview of fundamental concepts related to one-variable calculus. The original approach is a result of the author’s forty years of experience in teaching the subject at universities around the world. In this book, Dr. Zalduendo makes use of the history of mathematics and a friendly, conversational approach to attract the attention of the student, emphasizing what is more conceptually relevant and putting key notions in a historical perspective. Such an approach was conceived to help them to overcome potential difficulties in teaching and learning of this subject — caused, in many cases, by an excess of technicalities and computations.
Besides covering the core of the discipline — real number, sequences and series, functions, derivatives, integrals, convexity and inequalities — the book is enriched by “side trips” to relevant subjects not usually seen in traditional calculus textbooks, touching on topics like curvature, the isoperimetric inequality, Riemann’s rearrangement theorem, Snell’s law, Buffon’s needle problem, Gregory’s series, random walk and the Gauss curve, and more. An insightful collection of exercises and applications completes this book, making it ideal as a supplementary textbook for a calculus course or the main textbook for an honors course on the subject.
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1 | (20) |
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1 | (1) |
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2 | (1) |
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3 | (1) |
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4 | (1) |
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5 | (3) |
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8 | (3) |
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Dyadic Series---A Construction of R |
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11 | (2) |
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13 | (1) |
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14 | (2) |
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16 | (3) |
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19 | (2) |
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21 | (18) |
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21 | (1) |
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22 | (2) |
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Cantor's Nested Intervals Theorem |
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24 | (1) |
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25 | (3) |
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28 | (1) |
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29 | (1) |
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30 | (2) |
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Series with Positive and Negative Terms |
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32 | (2) |
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The Riemann Series Theorem |
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34 | (2) |
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Absolute and Unconditional Convergence |
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36 | (1) |
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36 | (3) |
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39 | (32) |
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39 | (1) |
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40 | (1) |
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40 | (3) |
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The Exponential Function: Bernoulli's Inequality |
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43 | (6) |
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49 | (1) |
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Convergence of Π∞k=1(1 + ak) and of Σ∞k=1 ak |
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50 | (1) |
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51 | (1) |
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Injectivity and Inverse Functions |
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52 | (3) |
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Curves in the Plane: Parametrized Curves |
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55 | (1) |
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56 | (1) |
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57 | (2) |
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59 | (1) |
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60 | (2) |
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62 | (1) |
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Limits in Ancient Greece: The Area of a Circle |
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62 | (3) |
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65 | (2) |
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67 | (4) |
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71 | (28) |
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71 | (1) |
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72 | (3) |
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75 | (2) |
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Derivatives of the Elementary Functions |
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77 | (2) |
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79 | (1) |
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Derivative of the Inverse Function |
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80 | (2) |
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The Derivative of a Parametrized Curve |
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82 | (2) |
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First Derivative, Tangent Line, and Growth |
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84 | (1) |
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84 | (3) |
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87 | (2) |
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89 | (3) |
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92 | (2) |
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94 | (5) |
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99 | (38) |
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99 | (6) |
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The Fundamental Theorem of Calculus |
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105 | (3) |
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108 | (3) |
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111 | (2) |
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113 | (1) |
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114 | (4) |
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Integration and Sums: Linearity of the Integral |
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118 | (1) |
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Uniform Convergence---The Weierstrass M-Test |
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118 | (5) |
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123 | (1) |
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Integration and Products: Integration by Parts |
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124 | (1) |
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125 | (1) |
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Integration and Composition: Integration by Substitution |
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126 | (1) |
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127 | (1) |
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Length of Curves. The Catenary |
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128 | (3) |
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Area Enclosed by a Simple Closed Curve |
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131 | (2) |
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133 | (4) |
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137 | (18) |
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Second Derivative, Best-Fitting Parabola, and Curvature |
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137 | (1) |
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The Taylor Polynomial of Order Two |
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138 | (9) |
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147 | (6) |
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153 | (2) |
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7 Convexity and the Isoperimetric Inequality |
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155 | (16) |
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The Arithmetic-Geometric Inequality |
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155 | (1) |
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156 | (3) |
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Young, Holder, Jensen, Cauchy--Schwarz |
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159 | (5) |
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The Isoperimetric Inequality |
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164 | (4) |
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168 | (3) |
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171 | (30) |
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171 | (3) |
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174 | (1) |
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175 | (3) |
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178 | (5) |
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Density Functions, Barycenter, and Expectation |
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183 | (1) |
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Center of Mass or Barycenter |
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184 | (4) |
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188 | (2) |
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190 | (2) |
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192 | (2) |
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Normal Distribution. Gauss, Laplace, and Stirling |
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194 | (4) |
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198 | (3) |
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201 | (6) |
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201 | (1) |
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202 | (3) |
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Growth of the Harmonic Series, Again |
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205 | (1) |
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206 | (1) |
| Bibliography |
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207 | (2) |
| Index |
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209 | |
Ignacio Zalduendo holds a PhD in Mathematical Sciences (1983) from the University of Buenos Aires, Argentina. He is currently a Full Professor at the Torcuato di Tella University, where he also served as vice-rector (2010-2013). His previous activities include positions as a Visiting Professor at the University of California (UCLA), Complutense University of Madrid, Spain, and Kent State University, in the USA. In 2004, he received a scholarship from the Fulbright Program. Dr. Zalduendo has published over 40 articles and has served as a reviewer for journals as the Journal of Mathematical Analysis and Applications, Annals of Mathematics, Mathematical Reviews, and zbMath. He also authored the book Matemática para Iñaki (in Spanish).
Lisainfo e-raamatute kohta