From 1de9584c135e74ee2ed0c8a516bd40917dcab98f Mon Sep 17 00:00:00 2001 From: Hextical Date: Wed, 20 Nov 2024 14:19:17 -0500 Subject: [PATCH] 12/12 137 --- README.md | 3 +- year-1/semester-1/MATH 137/math137.pdf | Bin 816047 -> 833809 bytes year-1/semester-1/MATH 137/math137.tex | 1 + year-1/semester-1/MATH 137/wk_11_2.tex | 240 ++++++++++++++++++++++++- 4 files changed, 242 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index cb93d40..dc7cd6d 100755 --- a/README.md +++ b/README.md @@ -82,4 +82,5 @@ Incomplete notes and graduate notes can be found [here](https://github.com/Hexti ### Semester 1 | Course | Completion | Online | | :-------------------------------------------------------------------------------------------------------------------------------------: | :--------: | :----: | -| [MATH 137 *(Calculus 1 for Honours Mathematics)*](https://hextical.github.io/university-notes/year-1/semester-1/MATH%20137/math137.pdf) | 11/12 | :x: | +| [MATH 137 *(Calculus 1 for Honours Mathematics)*](https://hextical.github.io/university-notes/year-1/semester-1/MATH%20137/math137.pdf) | Complete\* | :x: | +\* Needs revision. \ No newline at end of file diff --git a/year-1/semester-1/MATH 137/math137.pdf b/year-1/semester-1/MATH 137/math137.pdf index 7e5d62524cb499afd8d288e24d8f1a208c588228..ebdcb173d97820e33390fc69f7bc151b1a42eb91 100644 GIT binary patch delta 88853 zcmV)DK*7JS=P{A;GO)1*0x%$xfe0v*+Xf&7)HEnf_>u=SW4dF=>8jneOw$8Pgs~K( z*)Sn!RmaPMp`=wf_6*bDaoW~B;4W|8i!yJv3>94+*h{vI3J{#uqEo;P7)cV8;FO61 z?xsu{nl1WYr8xo}xKyM8mssGb^SF*G;^xda8pZD-la~iWf9i|KYUL@b@R?o&AI4M` zb>I1}#0RE0t*Q@yMWlJX-G|3wyjlk-H$U5cvp+O-8>YBwnDP@I6`?I?mkuJ9O!zbGI@1G6!gVew~;~xU=Ej|Tm%yXGc`4n5pXJh zTw8P7HWq&OuVDL9YBoY~CBSU6vu@+`mU}$2+e!PN*o>{#k{wCjxclq(;7Wi7B`Q{I zcV-`KQ6LB$92}hUoePDFlL8mt?|pl?_w^G(Xi=lOqE&JDzL2z#OcjU6#oO|`ANF4! zj>e=a4_=SPRH{;-(YV62JRA{1%KZ_4r{ynCUyL{|U5A(E%hOTK%U_<4h$>$^`*Hs% z+*9Qn+hhOvV|$SsgURwM*YEI$(U_C+V8ly%^WfR&x5J;lej-UkE*BN5DTUP-iY3>5 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magnitude less than (or equal to) $ g $ +near $ x=a $. +\begin{Remark}{}{} + We can always insist $ 0<\varepsilon\le 1 $ + since once we find an $ \varepsilon $ that works, + so will any smaller $ \varepsilon $ value. +\end{Remark} +\begin{Example}{}{} + Suppose $ f(x)=\bigo{x^n} $ for some $ n\in \N $ + as $ x\to 0 $. Find $ \lim\limits_{{x} \to {0}}f(x) $. + \tcblower{} + \textbf{Solution}. + Since $ \lim\limits_{{x} \to {0}}-M\abs{x^n}= + \lim\limits_{{x} \to {0}}M\abs{x^n} $, by the + Squeeze Theorem we get + \[ \lim\limits_{{x} \to {0}}f(x)=0. \] + So, if $ f(x)=\bigo{x^n} $, we get + $ \lim\limits_{{x} \to {0}}f(x)=0 $. Denote this as + \[ \lim\limits_{{x} \to {0}}\bigo{x^n}=0. \] +\end{Example} +The idea is to compare functions though, so let's extend our +definition. +\begin{Definition}{}{} + Suppose $ f $ and $ g $ are defined on an open interval + containing $ x=a $, except possibly at $ x=a $, + write + \[ f(x)=g(x)+\bigo{h(x)}\text{ as $x\to a$} \] + if + \[ f(x)-g(x)=\bigo{h(x)}\text{ as $x\to a$}. \] + This means $ f(x)\approx g(x) $ near $ x=a $, + with error that has magnitude at most $ h(x) $. +\end{Definition} +\begin{Example}{}{} + We saw earlier than for $ f(x)=\sqrt{1+x} $, + if we use $ T_{2,0}(x) $ to approximate it, then + \[ \abs{f(x)-T_{2,0}(x)}\le \frac{3}{48}x^3 \] + and $ f(x)\ge T_{2,0}(x) $. So + \[ \sqrt{1+x}-T_{2,0}(x)=\bigo{x^3} \] + or + \[ \sqrt{1+x}=T_{2,0}(x)+\bigo{x^3}. \] +\end{Example} +We can collect this result as a theorem: +\begin{Theorem}{Taylor's Approximation Theorem II (TAT II)}{} + Let $ r>0 $. If $ f $ is $ (n+1) $-times differentiable + on $ [-r,r] $ and $ f^{(n+1)} $ is continuous on $ [-r,r] $, + then + \[ f(x)=T_{n,0}(x)+\bigo{x^{n+1}}\text{ as $x\to 0$}. \] + \tcblower{} + \textbf{Proof}: Since $ f^{(n+1)} $ + is continuous on $ [-r,r] $, the EVT + says that it attains its maximum. Let + $ M $ be chosen so that + \[ \abs{f^{(n+1)}(x)}\le M\text{ for $x\in[-r,r]$}. \] + Taylor's Theorem says there exists $ c $ + between $ x $ and $ 0 $ so that + \[ \abs{f(x)-T_{n,0}(x)}=\abs*{\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}} + \le \frac{M}{(n+1)!}\abs{x^{n+1}}. \] + So, $ f(x)-T_{n,0}(x)=\bigo{x^{n+1}} $ as + $ x\to 0 $, as desired. +\end{Theorem} +Q\@: If $ f(x)=\bigo{x^m} $ and +$ g(x)=\bigo{x^n} $ as $ x\to 0 $, what +can we say about $ f+g $? Well, there exists +$ M_1,M_2>0 $ so that +\[ \abs{f(x)}\le M_1\abs{x^m}\text{ and } + \abs{g(x)}\le M_2\abs{x^n} \] +for $ x $ near zero. Using the triangle equality, we obtain +\[ \abs{f(x)+g(x)}\le M_1x^n+M_2x^m\text{ as $x\to 0$}. \] +Let $ k=\min(n,m) $, then for $ x $ near +zero ($ x\in[-1,1] $), we get +\[ x^m\le x^k,x^n\le x^k, \] +so +\[ \abs{f(x)+g(x)}\le (M_1+M_2)x^k \] +or $ f(x)+g(x)=\bigo{x^k} $ for $ k=\min(n,m) $. +We denote this as $ \bigo{x^n}+\bigo{x^m}=\bigo{x^k} $, +where $ k=\min(n,m) $. +Let's collect all the arithmetic properties +of Big-O. +\begin{Theorem}{Arithmetic of Big-O}{} + Let $ f(x)=\bigo{x^n} $ and $ g(x)=\bigo{x^m} $ + as $ x\to 0 $ for $ m,n\in\R $. + \begin{enumerate}[(1)] + \item $ c(\bigo{x^n})=\bigo{x^n} $, i.e., + $ c f(x)=\bigo{x^n} $. + \item $ \bigo{x^n}+\bigo{x^m}=\bigo{x^k} $ + where $ k=\min(n,m) $. + \item $ \bigo{x^n}\bigo{x^m}=\bigo{x^{n+m}} $, i.e., + $ f(x)g(x)=\bigo{x^{n+m}} $. + \item If $ k\le n $, then $ f(x)=\bigo{x^k} $. + \item If $ k\le n $, then $ \frac{1}{x^k}\bigo{x^n}=\bigo{x^{n-k}} $, + i.e., $ \frac{f(x)}{x^k}=\bigo{x^{n-k}} $. + \item $ f(u^k)=\bigo{u^{kn}} $, i.e., we can + sub in $ x=u^k $. + \end{enumerate} + \tcblower{} + \textbf{Proof}: Exercises. +\end{Theorem} +\begin{Example}{}{} + We know $ \sqrt{1+x}-T_{2,0}(x)=\bigo{x^3} $, + but what about + \[ \underbrace{\bigl(\sqrt{1+x}-T_{2,0}(x)\bigr)x^5}_{\bigo{x^{3+5}}=\bigo{x^{8}}} + +\underbrace{x^{10}}_{\bigo{x^{10}}}? \] + Clearly, we get $ \bigo{x^{8}} $ by property 2. +\end{Example} +We can also use Big-O notation to help us evaluate limits, +let's look at some examples. +\begin{Example}{}{} + Evaluate $ \displaystyle \lim\limits_{{x} \to {0}}\frac{e^{x^2}-1-x^2}{3} $ + using Big-O\@. + \tcblower{} + \textbf{Solution}. Note that + $ T_{1,0}(x)=1+x $ for $ e^x $, so + \[ e^x-(1+x)=\bigo{x^2} \] + by TAT II\@. So, by arithmetic rules of Big-O (sub $ x=u^2 $), + \[ e^{u^2}-(1+u^2)=\bigo{(u^2)^2}=\bigo{u^4}. \] + Therefore, + \[ \frac{e^{x^2}-1-x^2}{x^3}=\frac{\bigo{x^4}}{x^3}=\bigo{x}, \] + and so + \[ \lim\limits_{{x} \to {0}}\frac{e^{x^2}-1-x^2}{x^3}= + \lim\limits_{{x} \to {0}}\bigo{x}=0. \] +\end{Example} +\begin{Example}{}{} + Evaluate $ \displaystyle \frac{(e^x-1)[\cos(x)-1]x^3}{(e^{x^2}-1)\sin(x^2)\sin^2(x)} $ + using Big-O\@. + \tcblower{} + \textbf{Solution}. As an exercise, first find/show: + \begin{align*} + e^x & =1+x+\bigo{x^2}, \\ + \cos x & =1-\frac{x^2}{2}+\bigo{x^4}, \\ + \sin x & =x+\bigo{x^3}. + \end{align*} + Then, we get + \[ \sin^2(x)=(x+\bigo{x^3})^2=x^2+2x\bigo{x^3}+\bigo{x^3}^2 + =x^2+\bigo{x^4}+\bigo{x^6}=x^2+\bigo{x^4}. \] + \begin{align*} + (e^x-1)[\cos(x)-1]x^3 + & =(x+\bigo{x^2})(-\tfrac{x^2}{2}+\bigo{x^4})x^3 \\ + & =(-\tfrac{x^3}{2}+\bigo{x^4}+\bigo{x^5}+\bigo{x^6})x^3 \\ + & =-\tfrac{x^6}{2}+\bigo{x^7}+\bigo{x^8}+\bigo{x^9} \\ + & =-\tfrac{x^6}{2}+\bigo{x^7}. + \end{align*} + Next, + \begin{align*} + (e^{x^2}-1)\sin(x^2)\sin^2(x) \\ + & =(x^2+\bigo{x^4})(x^2+\bigo{x^6})(x^2+\bigo{x^4}) \\ + & =(x^4+\bigo{x^6}+\bigo{x^8}+\bigo{x^{10}})(x^2+\bigo{x^4}) \\ + & =(x^4+\bigo{x^6})(x^2+\bigo{x^4}) \\ + & =x^6+\bigo{x^8}+\bigo{x^8}+\bigo{x^{10}} \\ + & =x^6+\bigo{x^8}. + \end{align*} + Putting it together, + \begin{align*} + \lim\limits_{{x} \to {0}} + \frac{(e^x-1)[\cos(x)-1]x^3}{(e^{x^2}-1)\sin(x^2)\sin^2(x)} + & =\lim\limits_{{x} \to {0}}\frac{-\tfrac{x^6}{2}+\bigo{x^7}}{x^6+\bigo{x^8}} \\ + & =\lim\limits_{{x} \to {0}}\frac{-\tfrac{1}{2}+\bigo{x}}{1+\bigo{x^2}} \\ + & =\frac{-\tfrac{1}{2}}{1} \\ + & =-\frac{1}{2}. + \end{align*} +\end{Example} +\subsection*{Characterization of Taylor Polynomials} +Consider $ \cos(x^2)-1 $. We know +\[ \cos(x)-1=-\frac{x^2}{2}+\bigo{x^4} + \implies \cos(x^2)-1=-\frac{x^4}{2}+\bigo{x^8}. \] +However, we know +\[ \cos(x^2)-1=T_{7,0}(x)+\bigo{x^8}. \] +So, is $ T_{7,0}(x)=-\frac{x^4}{2} $? Yes! +Let's examine the theorem. +\begin{Theorem}{Characterization of Taylor Polynomials}{} + Let $ r>0 $, $f$ be $ (n+1) $-times differentiable + on $ [-r,r] $, and $ f^{(n+1)} $ be continuous + on $ [-r,r] $. If $ p $ + is a polynomial of degree at most $ n $ + such that $ f(x)=p(x)+\bigo{x^{n+1}} $, then + \[ p(x)=T_{n,0}(x). \] + \tcblower{} + \textbf{Proof}: First, we need the following fact: + \begin{itemize} + \item If $ p $ is a polynomial of degree + at most $ n $ and $ p(x)=\bigo{x^{n+1}} $, + then $ p(x)=0 $ for all $ x $. + + \emph{The proof is an exercise and uses induction}. + \end{itemize} + By assumption, + \[ f(x)-p(x)=\bigo{x^{n+1}}. \] + Using TAT II, we have + \[ f(x)-T_{n,0}(x)=\bigo{x^{n+1}}. \] + So, + \[ p(x)-T_{n,0}(x)=[f(x)-T_{n,0}(x)]-[f(x)-p(x)] + =\bigo{x^{n+1}}+\bigo{x^{n+1}}=\bigo{x^{n+1}}. \] + But $ p(x)-T_{n,0}(x) $ + is a polynomial of degree at most $ n $, + so by the fact above, for all $ x $ we have + \[ p(x)-T_{n,0}(x)=0. \] + Therefore, $ p(x)=T_{n,0}(x) $. +\end{Theorem} +\begin{Example}{}{} + Previously, we calculated + \[ (e^x-1)[\cos(x)-1]x^3=-\frac{x^6}{2}+\bigo{x^7}, \] + so for this function we have + \[ T_{6,0}(x)=-\frac{x^6}{2}. \] + The derivatives would be terrible to take in practice. + \begin{itemize} + \item Q\@: What is $ f^{(4)}(0) $, $ f^{(5)}(0) $ + and $ f^{(6)}(0) $? + \item A\@: We know that + \begin{align*} + T_{6,0}(x)= + f(0)+f'(0)+ + \frac{f''(0)}{2!}x^2+ + \frac{f^{(3)}(0)}{3!}x^3+ + \frac{f^{(4)}(0)}{4!}x^4+ + \frac{f^{(5)}(0)}{5!}x^5+ + \frac{f^{(6)}(0)}{6!}x^6. + \end{align*} + Matching coefficients to + $ T_{6,0}(x)=-\frac{x^6}{2} $, we get + \[ f^{(4)}(0)=0,\quad f^{(5)}(0)=0,\quad + \frac{f^{(6)}(0)}{6!}=-\frac{1}{2}\implies + f^{(6)}(0)=-\frac{6!}{2}=-360. \] + \end{itemize} +\end{Example} \ No newline at end of file