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Elementary Calculus Elementary Calculus 2 v0 2g 2 0 v0 g Michael Corral Elementary Calculus Michael Corral Schoolcraft College About the author: Michael Corral is an Adjunct Faculty member of the Department of Mathematics at School- craft College. He received a B.A. in Mathematics from the University of California, Berkeley, and received an M.A. in Mathematics and an M.S. in Industrial & Operations Engineering from the University of Michigan. This text was typeset in LATEXwith the KOMA-Script bundle, using the GNU Emacs text editor on a Fedora Linux system. The graphics were created using TikZ and Gnuplot. Copyright © 2020 Michael Corral. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. Preface This book covers calculus of a single variable. It is suitable for a year-long (or two-semester) course, normally known as Calculus I and II in the United States. The prerequisites are high school or college algebra, geometry and trigonometry. The book is designed for students in engineering, physics, mathematics, chemistry and other sciences. One reason for writing this text was because I had already written its sequel, Vector Cal- culus. More importantly, I was dissatisfied with the current crop of calculus textbooks, which I feel are bloated and keep moving further away from the subject’s roots in physics. In addi- tion, many of the intuitive approaches and techniques from the early days of calculus—which I think often yield more insights for students—seem to have been lost. I agree with the views of the late Russian mathematician V.I. Arnold on teaching mathe- matics, in particular the idea that “Mathematics is the part of physics where experiments are cheap.”1 The ties to physics are especially important in calculus, so this book tries to introduce new concepts with physical motivations (what other motivations can there be?). The book con- tains exercises and examples that I hope will adequately prepare students who continue on in physics and engineering.2 Perhaps controversially, the book uses infinitesimals, making it a bit of a “throwback” or “retro” calculus text. My justification for this heretical act was purely pedagogical: infinitesi- mals make learning calculus easier, and their use aligns more with the way students will see calculus in their physics, chemistry and other science classes and textbooks (where infinitesi- mals are employed liberally). This might ruffle some feathers among mathematical “purists,” but they are not the main audience for this book. That said, the book is still compatible with the usual limit-based approach, so an instructor could simply ignore the parts involving infinites- imals and teach the material as he or she normally would. I did not want to be dogmatic, so I used infinitesimals where I thought it made sense, and used limits where appropriate (e.g. in discussing continuity, series). Again, pedagogy was my priority. The exercises at the end of each section are divided into three categories: A, B and C. The A exercises are mostly of a routine computational nature, the B exercises are slightly more involved, and the C exercises usually require some effort or insight to solve. A crude way of 1ARNOLD, V.I., “On Teaching Mathematics”, Russian Math. Surveys 53 (1998), No. 1, 229-236. An HTML version is at https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html 2The book covers some of the types of problems and techniques for solving them that such students will likely encounter. Facility with using named constants (e.g. c, h, T) is also emphasized. iii iv PREFACE describing A, B and C would be “Easy”, “Moderate” and “Challenging”, respectively. However, many of the B exercises are easy and not all the C exercises are difficult. Appendix A provides answers and hints to many of the odd-numbered and some of the even-numbered exercises. A few exercises require the student to write a computer program to solve numerical approx- imation problems (e.g. numerical methods for approximating definite integrals). Algorithms are presented in pseudocode, with code implementations in various languages (primarily Java, but also Python, Octave, Sage). I hope the code comments will help the reader figure out what is being done, regardless of familiarity with those languages. Students are free to implement solutions using the language of their choice. There are no dedicated “calculator exercises,” as those have been rendered pointless by modern computing (with which students need to become acquainted). Stylistically I made a conscious effort to break from an unfortunate but all too common mode of writing in mathematics texts, lamented in the preface of a physics book: “Nothing is more repellent to normal human beings than the clinical succession of definitions, axioms, and theorems generated by the labours of pure mathematicians.”3 I have been guilty of that sin myself, but I have changed my ways and banished all traces of that sort of thing from this book. So you won’t find Definition 1.2, Theorem 3.3, Corollary 4.6, Lemma 5.7, Axiom 1B, etc. Instead, I tried to borrow the best of the styles from the physics and foreign languages textbooks I enjoyed so much in college. I also deliberately avoided what the author Gore Vidal called the “we-ness” that prevails in academic writing. There is no good reason for the “royal we” in a textbook, and it comes off as a bit pompous, so we won’t use it. This book is released under the GNU Free Documentation License (GFDL), which allows others to not only copy and distribute the book but also to modify it. For more details, see the included copy of the GFDL. So that there is no ambiguity on this matter, anyone can make as many copies of this book as desired and distribute it as desired, without needing my permission. The PDF version will always be freely available to the public at no cost (go to http://www.mecmath.net/calculus). Feel free to contact me at [email protected] for any questions on this or any other matter regarding the book. I welcome your feedback. Schoolcraft College MICHAEL CORRAL December 2020 3 ZIMAN, J.M., Elements of Advanced Quantum Theory, Cambridge, U.K.: Cambridge University Press, 1969. Contents Preface iii 1 The Derivative 1 1.1 Introduction.................................... .... 1 1.2 TheDerivative:LimitApproach . ....... 8 1.3 TheDerivative: InfinitesimalApproach . ......... 15 1.4 DerivativesofSums,ProductsandQuotients . .......... 21 1.5 TheChainRule.................................... 27 1.6 HigherOrderDerivatives. ...... 32 2 DerivativesofCommonFunctions 37 2.1 InverseFunctions ................................ 37 2.2 Trigonometric Functions and Their Inverses . .......... 41 2.3 The Exponential and NaturalLogarithm Functions . .......... 45 2.4 General Exponential and Logarithmic Functions . ........... 53 3 Topics in Differential Calculus 56 3.1 TangentLines.................................... 56 3.2 Limits:FormalDefinition. ...... 62 3.3 Continuity...................................... 73 3.4 ImplicitDifferentiation . ........ 79 3.5 RelatedRates .................................... 82 3.6 Differentials................................... ..... 84 4 ApplicationsofDerivatives 90 4.1 Optimization .................................... 90 4.2 CurveSketching .................................. 102 4.3 Numerical Approximation of Roots of Functions . .......... 109 4.4 TheMeanValueTheorem............................. 118 5TheIntegral 124 5.1 TheIndefiniteIntegral ............................ 124 v vi CONTENTS 5.2 TheDefiniteIntegral.............................. 132 5.3 TheFundamentalTheoremofCalculus . 140 5.4 IntegrationbySubstitution. ........ 146 5.5 ImproperIntegrals............................... 152 6 MethodsofIntegration 159 6.1 IntegrationbyParts .............................. 159 6.2 TrigonometricIntegrals . ....... 167 6.3 TrigonometricSubstitutions . ........ 172 6.4 PartialFractions ................................ 178 6.5 MiscellaneousIntegrationMethods . ......... 185 6.6 NumericalIntegrationMethods . ....... 195 7 AnalyticGeometryandPlaneCurves 202 7.1 Ellipses........................................ 202 7.2 Parabolas ....................................... 210 7.3 Hyperbolas...................................... 216 7.4 TranslationsandRotations. ....... 224 7.5 HyperbolicFunctions ............................. 230 7.6 ParametricEquations ............................. 239 7.7 PolarCoordinates................................ 245 8 ApplicationsofIntegrals 252 8.1 AreaBetweenCurves ............................... 252 8.2 AverageValueofaFunction .. .. .. .. .. .. .. .. .. .. .. .. 258 8.3 ArcLengthandCurvature ........................... 263 8.4 SurfacesandSolidsofRevolution . ........ 271 8.5 ApplicationsinPhysicsandStatistics . .......... 277 9 InfiniteSequencesandSeries 286 9.1 SequencesandSeries .............................. 286 9.2 TestsforConvergence ............................. 293 9.3 AlternatingSeries ............................... 300 9.4 PowerSeries..................................... 303 9.5 Taylor’sSeries.................................. 308 AppendixA AnswersandHintstoSelectedExercises 315 GNUFreeDocumentationLicense 321 History 330 Index 332 The Greek Alphabet Letters Name Letters Name Letters Name A α alpha I ι iota P ρ rho B β beta K κ kappa Σ σ sigma Γ γ gamma Λ λ lambda T τ tau ∆ δ delta M µ mu Υ υ upsilon E ǫ epsilon
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