Table Of Content

Contents Page: vii Preface Page: xi Chapter 1. Accumulation Page: 1 1.1. Archimedes and the Volume of the Sphere Page: 1 1.2. The Area of the Circle and the Archimedean Principle Page: 7 1.3. Islamic Contributions Page: 11 1.4. The Binomial Theorem Page: 17 1.5. Western Europe Page: 19 1.6. Cavalieri and the Integral Formula Page: 21 1.7. Fermat’s Integral and Torricelli’s Impossible Solid Page: 25 1.8. Velocity and Distance Page: 29 1.9. Isaac Beeckman Page: 32 1.10. Galileo Galilei and the Problem of Celestial Motion Page: 35 1.11. Solving the Problem of Celestial Motion Page: 38 1.12. Kepler’s Second Law Page: 42 1.13. Newton’s Principia Page: 44 Chapter 2. Ratios of Change Page: 49 2.1. Interpolation Page: 50 2.2. Napier and the Natural Logarithm Page: 57 2.3. The Emergence of Algebra Page: 64 2.4. Cartesian Geometry Page: 70 2.5. Pierre de Fermat Page: 75 2.6. Wallis’s Arithmetic of Infinitesimals Page: 81 2.7. Newton and the Fundamental Theorem Page: 87 2.8. Leibniz and the Bernoullis Page: 90 2.9. Functions and Differential Equations Page: 93 2.10. The Vibrating String Page: 99 2.11. The Power of Potentials Page: 103 2.12. The Mathematics of Electricity and Magnetism Page: 104 Chapter 3. Sequences of Partial Sums Page: 108 3.1. Series in the Seventeenth Century Page: 110 3.2. Taylor Series Page: 114 3.3. Euler’s Influence Page: 120 3.4. D’Alembert and the Problem of Convergence Page: 125 3.5. Lagrange Remainder Theorem Page: 128 3.6. Fourier’s Series Page: 134 Chapter 4. The Algebra of Inequalities Page: 141 4.1. Limits and Inequalities Page: 142 4.2. Cauchy and the Language of є and δ Page: 144 4.3. Completeness Page: 149 4.4. Continuity Page: 151 4.5. Uniform Convergence Page: 154 4.6. Integration Page: 157 Chapter 5. Analysis Page: 163 5.1. The Riemann Integral Page: 163 5.2. Counterexamples to the Fundamental Theorem of Integral Calculus Page: 166 5.3. Weierstrass and Elliptic Functions Page: 173 5.4. Subsets of the Real Numbers Page: 178 5.5. Twentieth-Century Postscript Page: 183 Appendix. Reflections on the Teaching of Calculus Page: 186 Teaching Integration as Accumulation Page: 186 Teaching Differentiation as Ratios of Change Page: 189 Teaching Series as Sequences of Partial Sums Page: 191 Teaching Limits as the Algebra of Inequalities Page: 193 The Last Word Page: 196 Notes Page: 199 Bibliography Page: 209 Index Page: 215 Image Credits Page: 223

Description:

How our understanding of calculus has evolved over more than three centuries, how this has shaped the way it is taught in the classroom, and why calculus pedagogy needs to change Calculus Reordered takes readers on a remarkable journey through hundreds of years to tell the story of how calculus evolved into the subject we know today. David Bressoud explains why calculus is credited to seventeenth-century figures Isaac Newton and Gottfried Leibniz, and how its current structure is based on developments that arose in the nineteenth century. Bressoud argues that a pedagogy informed by the historical development of calculus represents a sounder way for students to learn this fascinating area of mathematics. Delving into calculus’s birth in the Hellenistic Eastern Mediterranean—particularly in Syracuse, Sicily and Alexandria, Egypt—as well as India and the Islamic Middle East, Bressoud considers how calculus developed in response to essential questions emerging from engineering and astronomy. He looks at how Newton and Leibniz built their work on a flurry of activity that occurred throughout Europe, and how Italian philosophers such as Galileo Galilei played a particularly important role. In describing calculus’s evolution, Bressoud reveals problems with the standard ordering of its curriculum: limits, differentiation, integration, and series. He contends that the historical order—integration as accumulation, then differentiation as ratios of change, series as sequences of partial sums, and finally limits as they arise from the algebra of inequalities—makes more sense in the classroom environment. Exploring the motivations behind calculus’s discovery, Calculus Reordered highlights how this essential tool of mathematics came to be.