E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 6: CalculusAbout the exposition of CalculusIn lecture, I have tried to summarize (and derive!) the most important points of calculus. Thehandout does this on two pages. I have tried in class to present the core of calculus in a quarterof an hour a nd we have seen Ed Burger presenting calculus in 20 minutes (18 and 1/2 minutesactually).My efforts to reduce things to the core is a reaction to an ”focus free” attitude towards calculuswhich ma nif ests in the insane inflation of text book sizes. Eli Maor expresses it well in hisbook ”The facts on file: Calculus Handbook, 2003”:”Over the pas t 25 years or so , the typical college calculus textbook has grown from a modest350-page book to a huge volume of some 1,200 pages, with thousands of exercises, special topics,interviews with career mathematicians, 10 or more appendixes, and much, much more. But as theold adag e goes, m ore is not always better. The enormous si ze and sheer volume of these monsters(not to men tion their weight!) have made their use a daunting task. Both student and instructorare lost in a sea of information, not know i ng which materia l is important and which can be skipped.As if the study of calculus is not a challenge already, these huge texts make the task even moredifficult.”I personally find tha t there truth in that. Textbooks like Snyder’s Elementary Textbook onthe Calculus was 20 0 pages long. Also Maor’s book is a short dictionary or glossary and doesnot substitute a textbook. But it can help to focus.The text book issue will change anyway. The scandalous prizes of textboo ks which I can onlyexplained by priz e fixing will change f ast. The reason is the appearance of electronic textbookreaders which forces publishers to become reasonable with prizes. If buying two textbooks costsmore than a new computer, things are out of proportion. If this is not going to change, it willforce students to get electronic versions of a text book (I myself can get a 1200 page book intoelectronic OCR’ed PDF form with a time effort of maybe half an hour. Most of my time for thatis spent with cutting off the spine of the book and afterwards flipping through the pa ges to checkafter scanning to see that all pages are there. OCR and PDF optimization which can take sometime is done by the computer in the background.The fundamental t heorem o f calculusIn the handout, I tried to give a self contained exposition of trigonometric functions thatdoes not make use of anything we know before. If you read it you will find that it contains a proofof Euler’s formula eix= cos(x) + i sin(x) without Taylor series. In reality, one can compute wellwith the polynomial exponentials and their inverses. In order to verify that expn(x) expn(y) =expn(x + y) plus something small, we verify((1 + x/n)n∗ ( 1 + y/n)n)/(1 + (x + y)/n)n= (1 + xy/(n(n + x + y)))n∼ expn(xy/n) .The two page exposition contained in that handout is the heart of calculus. Without taking limits,it is simple algebra and before taking limits, there are absolutely no restric tions on functions.It is the concept of limit which gives students the impression that the subject is hard. Here again,history leads the way and shows tha t the struggle to understand limits is o ne of the hardest partof calculus. While it is reasonable to ask whether it is worth the trouble, and almost everybodyI have talked to in the last couple of years about the importance of limit consider t he questionheretic. But developments like nonstandard calculus 30 years ago or quantum calculus 1 0 yearsago a nd entire business calculus sequences not mentioning limits with one word have shown thatit is possible to avoid limits. While I still think that limits have to be discussed, the question isvalid, whether one has to read 300 pages before coming to the essentials of calculus.About SeriesHenri Poincar´e mentions in his ”New Methods of Celestial Mechanics” t he two series Sn=Pn1000n/n! and Sn=Pnn!/1000n. In this lecture, we had a worksheet on this. We knowthat the first series has a limit and the second not. An astronomer however would rule t he first tobe divergent and the second to be convergent. Also the harmonic seriesP∞n=11/n which is knownto diverge has a limit when we do experiments (it would take times of the age of the universeto reach the value 50 even if we could do trillions of additions every second). The fact t ha t theconcept of limits needed virtually 1000 years to develop is no accident. The struggle started withthe Z eno paradoxa and continues until today.Divergent series likeP∞n=11/nsfor negative s are more important today than convergent series.Both in number theory, where zeta functions are central as well as in physics, where the descriptionof nature uses divergent series in many ways. Series like 1 + 3 + 9 + 27 + 81 + ... make sense if wenote that it is a geometric series 1 + a + a2+ a3+ ... with limit 1/(1 − a) which becomes −1/2for a = 3. Again, also here, the concept of limit is bypassed on a fundamental level and thatthe limit of 1 + 3 + 9 + 27 + 81 + ... does not exist is irrelevant. There is much cultural baggagewhich makes us believe t hat smo othness is nice, similarly as astrono mers before Kepler thoughtthe circle is nice and modeled everything with epicycles.Are limits at all necessary? As calculus sequences for business students has shown, it can be donewithout doing big damage. Even nature tries to avoid limits. Our calculus for smo oth functions isan idealization which was originally invented t o make classical mechanics work. In reality, on themicroscopic and f undamental level, nature behaves differently and ”calculus without limits” workswell. We still do not know how space looks like on a very small scale but we have indications likethe Planck constant ¯h telling us that the continuum is an idealization and simplification. Becausewe like smooth objects, we have to deal with limits. If we relax the class of functions, mathemat-ics becomes easier. Quantum mechanics deals with Hilbert spaces of f unctions in which smoothfunctions form a tiny slice only. Nature likes simplicity and quantum mechanics with a linearevolution is on a fundamental level much easier than classical nonlinear mechanics with chaoticmotion. This is a well known principle in analysis: solutions to equations a r e often modeled b etterwith non smooth functions. Basic properties of primes