\documentclass[11pt,notitlepage]{article} \def\bare{n} \usepackage[all]{xy} \usepackage[english,greek]{babel} \usepackage{amsmath, graphicx, amssymb, datetime, multicol, stmaryrd, pifont, amscd, colortbl, mathtools, wasysym, needspace, import, longtable, overpic, enumitem, bbm, pdfpages, ../picins, array, setspace} \usepackage{tensor} \usepackage{txfonts} % for the likes of \coloneqq. \usepackage[usenames,dvipsnames]{xcolor} \usepackage[textwidth=8.5in,textheight=11in,centering]{geometry} \parindent 0in \usepackage[makeroom]{cancel} % Following http://tex.stackexchange.com/a/847/22475: \usepackage[setpagesize=false]{hyperref} \hypersetup{colorlinks, linkcolor={blue!50!black}, citecolor={blue!50!black}, urlcolor={blue!50!black} } % Following http://tex.stackexchange.com/questions/59340/how-to-highlight-an-entire-paragraph \usepackage[framemethod=tikz]{mdframed} \usepackage[T1]{fontenc} \def\myurl{http://www.math.toronto.edu/~drorbn} \def\thistalk{MAASeaway-1810} \def\title{My Favourite First-Year Analysis Theorem} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\thanks{{Thanks for inviting me to the fall 2018 MAA Seaway Section meeting!}} \def\webdef{{Handout, video, links at {\greektext web}$\coloneqq$\href{http://drorbn.net/maa18}{http://drorbn.net/maa18/}}} \def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}} \def\titleA{{\title}} \def\titleB{{\title}} \def\titleC{{\title}} \definecolor{mgray}{HTML}{B0B0B0} \definecolor{morange}{HTML}{FFA50A} \def\blue{\color{blue}} \def\gray{\color{gray}} \def\mgray{\color{mgray}} \def\morange{\color{morange}} \def\red{\color{red}} \def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}} \def\cbox#1#2{{\setlength{\fboxsep}{0pt}\colorbox{#1}{#2}}} \def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}} \def\qed{{\linebreak[1]\null\hfill\text{$\Box$}}} \def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}} \def\ad{\operatorname{ad}} \def\Ad{\operatorname{Ad}} \def\aft{$\overrightarrow{\text{4T}}$} \def\AS{\mathit{AS}} \def\bbZZ{{\mathbb Z\mathbb Z}} \def\CW{\text{\it CW}} \def\diag{\operatorname{diag}} \def\FL{\text{\it FL}} \def\Hom{\operatorname{Hom}} \def\IHX{\mathit{IHX}} \def\mor{\operatorname{mor}} \def\PvT{{\mathit P\!v\!T}} \def\remove{\!\setminus\!} \def\STU{\mathit{STU}} \def\SW{\text{\it SW}} \def\TC{\mathit{TC}} \def\tr{\operatorname{tr}} \def\vT{{\mathit v\!T}} \def\barT{{\bar T}} \def\bbe{\mathbbm{e}} \def\bbE{{\mathbb E}} \def\bbO{{\mathbb O}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\bcA{{\bar{\mathcal A}}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calF{{\mathcal F}} \def\calG{{\mathcal G}} \def\calI{{\mathcal I}} \def\calK{{\mathcal K}} \def\calO{{\mathcal O}} \def\calP{{\mathcal P}} \def\calR{{\mathcal R}} \def\calS{{\mathcal S}} \def\calT{{\mathcal T}} \def\calU{{\mathcal U}} \def\fraka{{\mathfrak a}} \def\frakb{{\mathfrak b}} \def\frakg{{\mathfrak g}} \def\frakh{{\mathfrak h}} \def\tilE{\tilde{E}} \def\tilq{\tilde{q}} \def\tDelta{\tensor[^t]{\Delta}{}} \def\tf{\tensor[^t]{f}{}} \def\tF{\tensor[^t]{F}{}} \def\tg{\tensor[^t]{g}{}} \def\tI{\tensor[^t]{I}{}} \def\tm{\tensor[^t]{m}{}} \def\tR{\tensor[^t]{R}{}} \def\tsigma{\tensor[^t]{\sigma}{}} \def\tS{\tensor[^t]{S}{}} \def\tSW{\tensor[^t]{\SW}{}} % From http://tex.stackexchange.com/questions/154672/how-to-get-a-medium-sized-otimes \DeclareMathOperator*{\midotimes}{\text{\raisebox{0.25ex}{\scalebox{0.8}{$\bigotimes$}}}} %%% \def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf Abstract.} Whatever it may be, it should say something useful and exciting and it should not be *about* rigour, yet it should *demand* rigour. You can't guess. You probably think it the dreariest. You are wrong. }}}} \def\S{{$\circledS$}} \def\SpivakContents{{\raisebox{-1mm}{\parbox[t]{3.95in}{ {\large\bf Contents}\qquad\S \par{\bf Prologue} \newline\phantom{$\ast$1}1\quad Basic Properties of Numbers\quad 3 \newline\phantom{$\ast$1}2\quad Numbers of Various Sorts\quad 21 \par{\bf Foundations} \newline\phantom{$\ast$1}3\quad Functions\quad 39 \newline\phantom{$\ast$1}4\quad Graphs\quad 56 \newline\phantom{$\ast$1}5\quad Limits\quad 90 \newline{\red\phantom{$\ast$1}6\quad Continuous Functions\quad 113} \newline{\red\phantom{$\ast$1}7\quad Three Hard Theorems\quad 120} \newline\phantom{$\ast$1}8\quad Least Upper Bounds\quad 142 \par{\bf Derivatives and Integrals} \newline\phantom{$\ast$1}9\quad Derivatives\quad 147 \newline\phantom{$\ast$}10\quad Differentiation\quad 166 \newline{\red\phantom{$\ast$}11\quad Significance of the Derivative\quad 185} \newline\phantom{$\ast$}12\quad Inverse Functions\quad 227 \newline\phantom{$\ast$}13\quad Integrals\quad 250 \newline{\red\phantom{$\ast$}14\quad The Fundamental Theorem of Calculus\quad 282} \newline\phantom{$\ast$}15\quad The Trigonometric Functions\quad 300 \newline{\red$\ast$16\quad$\pi$ is Irrational\quad 321} \newline$\ast$17\quad Planetary Motion\quad 327 \newline\phantom{$\ast$}18\quad The Logarithm and Exponential Functions\quad 336 \newline\phantom{$\ast$}19\quad Integration in Elementary Terms\quad 359 \par{\bf Infinite Sequences and Infinite Series} \newline{\red\phantom{$\ast$}20\quad Approximation by Polynomial Functions\quad 405} \quad\includegraphics[width=1.5pt]{Atom2.jpg} \ \includegraphics[width=1.5pt]{RedBloodCell.jpg} \ \includegraphics[width=1.5pt]{Saturn_.jpg} }}}} \def\SpivakFair{{\raisebox{0mm}{\scalebox{0.85}{\parbox[t]{1.65in}{\footnotesize Several excerpts here are from \text{Spivak's} ``Calculus'' \S. I \text{believe} they fall under ``fair use''. }}}}} %\def\SpivakFair{{\raisebox{0mm}{\parbox[t]{1.65in}{\footnotesize %Several excerpts here are from Spivak's ``Calculus'' \S. I \text{believe} they fall under %``fair use''. %}}}} \def\SectionSix{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf 6\quad Continuous Functions.} }}}} \def\SectionSeven{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf 7\quad Three Hard Theorems.} }}}} \def\SectionEleven{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf 11\quad Significance of the Derivative.} }}}} \def\SectionFourteen{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf 14\quad The Fundamental Theorem of Calculus.} }}}} \def\SectionSixteen{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf$\ast$16\quad $\pi$ is Irrational.} }}}} \def\SectionTwenty{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf 20\quad Approximation by Polynomial Functions.} \par\parshape 1 2.125in 1.825in For example for $f(x)=\sin(x)$ at $a=0$, $f^{(k)}=$ $\sin$, $\cos$, $-\sin$, $-\cos$, $\sin$, \ldots, so \[ a_k = \begin{cases} \frac{(-1)^{(k-1)/2}}{k!} & k\text{ odd} \\ 0 & k\text{ even} \end{cases} \] }}}} \def\Sizes{{\raisebox{2mm}{\parbox[t]{2.2in}{\footnotesize {\red\bf Some sizes} (in multiples of the diameter of a Hydrogen atom: \vspace{-1mm} \begin{center}\begin{tabular}{|l|r|} \hline A red blood cell & $1.56\times 10^5$ \\ The CN Tower & $1.11\times 10^{13}$ \\ The rings of Saturn & $5.6\times 10^{18}$ \\ The Milky Way galaxy & $1.89\times 10^{31}$ \\ The observable universe & $1.76\times 10^{37}$ \\ \hline \end{tabular}\end{center} }}}} \def\RemainderFormula{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 9 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 3.95in {\red\bf The Taylor Remainder Formulas.} Let $f$ be a smooth function, let $P_{n,a}(x)$ be the $n$th order Taylor polynomial of $f$ around $a$ and evaluated at $x$, so with $a_k=f^{(k)}(a)/k!$, \[ P_{n,a}(x) \coloneqq \sum_{k=0}^n a_k(x-a)^k, \] and let $R_{n,a}(x) \coloneqq f(x)-P_{n,a}(x)$ be the ``mistake'' or ``remainder term''. Then \begin{equation}\label{eq:Integral} R_{n,a}(x) = \int_a^xdt\,\frac{f^{(n+1)}(t)}{n!}(x-t)^n, \end{equation} or alternatively, for some $t$ between $a$ and $x$, \begin{equation}\label{eq:MVT} R_{n,a}(x) = \frac{f^{(n+1)}(t)}{(n+1)!}(x-a)^{n+1}. \end{equation} (In particular, the Taylor expansions of $\sin$, $\cos$, $\exp$, and of several other lovely functions converges to these functions {\em everywhere}, no matter the odds.) \parshape 7 0in 1.88in 0in 1.88in 0in 1.88in 0in 1.88in 0in 1.88in 0in 1.88in 0in 3.95in {\red\bf Proof of \eqref{eq:Integral}} (for adults; I learned it from my son Itai). The fundamental theorem of calculus says that if $g(a)=0$ then ${g(x)=\int_a^xdx_1g(x_1)}$. By design, $R_{n,a}^{(k)}(a) = 0$ for $0\leq k\leq n$. Therefore \begin{multline*} R_{n,a}(x) = \int_a^xdx_1R'_{n,a}(x_1) \\ = \int_a^xdx_1\int_a^{x_1}dx_2R''_{n,a}(x_2) \\ =\ldots= \int_a^xdx_1\int_a^{x_1}dx_2 \ldots \int_a^{x_n}dx_n\int_a^tdt\,R_{n,a}^{(n+1)}(t) \\ = \int_a^xdx_1\int_a^{x_1}dx_2 \ldots \int_a^{x_n}dx_n\int_a^tdt\,f^{(n+1)}(t), \end{multline*} when $x>a$, and with similar logic when $x