Elementary Calculus - mecmath

1 ElementaryCalculus0v20gv202gMichael CorralElementary CalculusMichael CorralSchoolcraft CollegeAbout the author:Michael Corral is an Adjunct Faculty member of the Department of Mathematics at School-craft College. He received a in Mathematics from the University of California, Berkeley,and received an in Mathematics and an in Industrial& Operations Engineeringfrom the University of text was typeset in LATEX with theKOMA-Scriptbundle, using the GNU Emacstext editor on a Fedora Linux system. The graphics were created using TikZ and 2020 Michael is granted to copy, distribute and/or modify this document under the terms of theGNU Free Documentation License, Version or any later version published by the FreeSoftware Foundation; with no Invariant Sections, no Front-Cover Texts, and no book covers Calculus of a single variable.

2 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 highschool or college algebra, geometry and trigonometry. The book is designed for students inengineering, physics, mathematics, chemistry and other 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, whichI feel are bloated and keep moving further away from the subject s roots in physics.

3 In addi-tion, many of the intuitive approaches and techniques from the early days of Calculus whichI think often yield more insights for students seem to have been agree with the views of the late Russian mathematician on teaching mathe-matics, in particular the idea that Mathematics is the partof physics where experiments arecheap. 1 The ties to physics are especially important in Calculus , sothis book tries to introducenew 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 inphysics and controversially, the book uses infinitesimals, making it a bit of a throwback or retro Calculus text.

4 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 seecalculus 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 theusual limit-based approach, so an instructor could simply ignore the parts involving infinites-imals and teach the material as he or she normally would.

5 I didnot want to be dogmatic, so Iused infinitesimals where I thought it made sense, and used limits where appropriate ( indiscussing continuity, series). Again, pedagogy was my exercises at the end of each section are divided into three categories: A, B and C. TheA exercises are mostly of a routine computational nature, the B exercises are slightly moreinvolved, and the C exercises usually require some effort orinsight to solve. A crude way of1 ARNOLD, , On Teaching Mathematics ,Russian Math. Surveys53 (1998), No.

6 1, 229-236. An HTML versionis ~ book covers some of the types of problems and techniques for solving them that such students will likelyencounter. Facility with using named constants ( ,h,T) is also 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 aredifficult. Appendix A providesanswers and hints to many of the odd-numbered and some of the even-numbered few exercises require the student to write a computer program to solve numerical approx-imation problems ( numerical methods for approximating definite integrals).

7 Algorithmsare presented in pseudocode, with code implementations in various languages (primarily Java,but also python , Octave, Sage). I hope the code comments willhelp the reader figure out whatis being done, regardless of familiarity with those languages. Students are free to implementsolutions using the language of their choice. There are no dedicated calculator exercises, asthose have been rendered pointless by modern computing (with which students need to becomeacquainted).Stylistically I made a conscious effort to break from an unfortunate but all too commonmode of writing in mathematics texts, lamented in the preface of a physics book: Nothingis more repellent to normal human beings than the clinical succession of definitions, axioms,and theorems generated by the labours of pure mathematicians.

8 3I have been guilty of thatsin myself, but I have changed my ways and banished all tracesof that sort of thing from thisbook. So you won t find Definition , Theorem , , Lemma , Axiom 1B,etc. Instead, I tried to borrow the best of the styles from thephysics and foreign languagestextbooks I enjoyed so much in college. I also deliberately avoided what the author Gore Vidalcalled the we-ness that prevails in academic writing. There is no good reason for the royalwe in a textbook, and it comes off as a bit pompous, sowewon t use book is released under the GNU Free Documentation License (GFDL), which allowsothers to not only copy and distribute the book but also to modify it.

9 For more details, seethe included copy of the GFDL. So that there is no ambiguity onthis matter, anyone canmake as many copies of this book as desired and distribute it as desired, without needing mypermission. The PDF version will always be freely availableto the public at no cost (go ). Feel free to contact me questions on this or any other matter regarding the welcome your CollegeMICHAELCORRALD ecember 20203 ZIMAN, ,Elements of Advanced Quantum Theory, Cambridge, : Cambridge University Press, The Introduction.

10 The Derivative: Limit Approach .. The Derivative: Infinitesimal Approach .. Derivatives of Sums, Products and Quotients .. The Chain Rule .. Higher Order Derivatives .. 322 Derivatives of Common Inverse Functions .. Trigonometric Functions and Their Inverses .. The Exponential and Natural Logarithm Functions .. General Exponential and Logarithmic Functions .. 533 Topics in Differential Tangent Lines .. Limits: Formal Definition .. Continuity .. Implicit Differentiation.