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zRQ~*0WoW-Z2Ag`;%-x!vzF4yIOdkFD;r@0RudbXB0RiFfB7Kep-~0C+dAk@0qp_M& z(ie+cuHAi$oMEh2)s2!KFrb zMgq?D7MHtAcaR#Ln8cwjS4@nQlz)v=r1k=aZ## z5P|Rif3Jl*Tj#@>sX0z!P9&9FTG&9&N%w~fJaM%Qy;Khb;8V;|#||p`>!oZ@&;Bw< znqpFFEi{_;f_DxW-eE(To+%!K4HP#pqNQzWOk8|K)PmPwx+TPUGSQ3%8`aZB@13J2K39!SXmmB;0=9b__mDKg!kIKZ)cX)pW z=<#L#(fT;YZ9PemA&d?Pkb%|-Ki##UI|Uy6;282OPDX!%>P`TV7FQ4}7c~g{KYyTr AF8}}l literal 0 HcmV?d00001 diff --git a/exponential.tex b/exponential.tex index dc137d2..853c104 100644 --- a/exponential.tex +++ b/exponential.tex @@ -92,14 +92,14 @@ $$ These formulas should be memorized, both in their long polynomial form and their more concise summation notation form. \subsection*{Example} -Use Euler's formula to show that $e^{i \pi}=-1$. Answer +Use Euler's formula to show that $e^{i \pi}=-1$. \hyperref[ExponentialsAnswer1]{Answer} \subsection*{Example} -Compute $1-\frac{\pi^2}{2!}+\frac{\pi^4}{4!}-\cdots$. Answer +Compute $1-\frac{\pi^2}{2!}+\frac{\pi^4}{4!}-\cdots$. \hyperref[ExponentialsAnswer2]{Answer} \subsection*{Example} -Check that taking the derivative of the long polynomial for $\sin x$ gives the long polynomial for $\cos x$ (hence, verify that $\frac{d}{d x} \sin x=\cos x$ ). Answer +Check that taking the derivative of the long polynomial for $\sin x$ gives the long polynomial for $\cos x$ (hence, verify that $\frac{d}{d x} \sin x=\cos x$ ). \hyperref[ExponentialsAnswer3]{Answer} \subsection*{Example} -Show that the long polynomial for $e^x$ satisfies the first property above, namely $e^{x+y}=e^x e^y$. Hint: start with the long polynomials for $e^x$ and $e^y$ and multiply these together, and carefully collect like terms to show it equals the long polynomial for $e^{x+y}$. Answer +Show that the long polynomial for $e^x$ satisfies the first property above, namely $e^{x+y}=e^x e^y$. Hint: start with the long polynomials for $e^x$ and $e^y$ and multiply these together, and carefully collect like terms to show it equals the long polynomial for $e^{x+y}$. \hyperref[ExponentialsAnswer4]{Answer} \section{More on the long polynomial} @@ -114,4 +114,27 @@ $$ \end{aligned} $$ -Each polynomial in the sequence is, in a sense, the best approximation possible of that degree. Put another way, taking the first several terms of the long polynomial gives a good polynomial approximation of the function. The more terms included, the better the approximation. This is how calculators compute the exponential function (without having to add up infinitely many things). \ No newline at end of file +Each polynomial in the sequence is, in a sense, the best approximation possible of that degree. Put another way, taking the first several terms of the long polynomial gives a good polynomial approximation of the function. The more terms included, the better the approximation. This is how calculators compute the exponential function (without having to add up infinitely many things). + +\begin{figure}[h] + \centering + \includegraphics[width=0.7\linewidth]{ExponentialApproximants} +% \caption{} + \label{fig:exponentialapproximants} +\end{figure} + +\section{EXERCISES} +\begin{itemize} + + \item[\textcolor{red}{\tikzcircle{3pt}}] So, how good of an approximation is a polynomial truncation of $e^{\text {? }}$. Use a calculator to compare how close $e$ is to the linear, quadratic, cubic, quartic, and quintic approximations. How many digits of accuracy do you seem to be gaining with each additional term in the series? +\end{itemize} +%- +%- Now, do the same thing with $1 / e$ by plugging in $x=-1$ into the series. Do you have the same results? Are you surprised? +%- Calculate the following sum using what you know: +%$$ +%\sum_{n=0}^{\infty}(-1)^n \frac{(\ln 3)^n}{n!} +%$$ +%- Write out the first four terms of the following series +%$$ +%\sum_{n=0}^{\infty}(-1)^n \frac{\pi^{2 n}}{2^n n!} +%$$ \ No newline at end of file diff --git a/solutions.tex b/solutions.tex index dc65399..bedcbae 100644 --- a/solutions.tex +++ b/solutions.tex @@ -91,9 +91,39 @@ \chapter*{EXPONENTIALS} \section*{Euler's formula} \subsection*{Answer 1} + \label{ExponentialsAnswer1} Setting $x=\pi$ in Euler's formula gives $e^{i \pi}=\cos \pi+i \sin \pi=-1$. -% + + \subsection*{Answer 2} + \label{ExponentialsAnswer2} + Note that this is the long polynomial for $\cos x$, evaluated at $x=\pi$. So the value is $\cos \pi=-1$. + + \subsection*{Answer 3} + \label{ExponentialsAnswer3} + Computing the derivative term by term gives + $$ + \begin{aligned} + \frac{d}{d x} \sin (x) & =\frac{d}{d x}\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots\right) \\ + & =1-3 \frac{x^2}{3!}+5 \frac{x^4}{5!}-\ldots \\ + & =1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots + \end{aligned} + $$ + which is the long polynomial for $\cos x$, as desired. + + \subsection*{Answer 4} + \label{ExponentialsAnswer4} + Beginning with $e^x \cdot e^y$, we find + $$ + \begin{aligned} + e^x \cdot e^y & =\left(1+x+\frac{x^2}{2!}+\cdots\right)\left(1+y+\frac{y^2}{2!}+\cdots\right) \\ + & =1+(x+y)+\left(\frac{x^2}{2!}+x y+\frac{y^2}{2!}\right)+\cdots \\ + & =1+(x+y)+\frac{x^2+2 x y+y^2}{2!}+\cdots \\ + & =1+(x+y)+\frac{(x+y)^2}{2!}+\cdots, + \end{aligned} + $$ + which is the long polynomial for $e^{x+y}$, as desired. + % \subsection{Euler's Formula} % % \subsection{Additional Examples} diff --git a/upenn.pdf b/upenn.pdf index 105b255b944017a7781044b57b9f6d678a377f27..95444e68c8de4040667368e29996581291fa2797 100644 GIT binary patch delta 58814 zcmX_nbyybP^ERM#cS?hFmvqBJcL>rbjf9lK(kUQ3G)Q-McY|~zT>{eG@LNCM_j>>1 zT6Xu$nLRUe-}lVZT!UK_fL|Fu0m~0PQ^u_+njlAc#mUE)2>TU|5inSDB<08PnK$kl zX!)8cKo%{KK93pJBAh%3Uz;lhF>b0!eX>A^zK^b-jc2dFCzx}MO?LX#q*^L>rG zwv(C#h)7Gr14Oqc-031w-f#M&Huj9Z`Hms;)SXXzO1V&j#6wNZ0f*|}qTm;alr70B z@s>fA{xbHOHfqrF5{ZpYkL*yd>0kxs0y^)A4gGPyCes+4vN+?jt?m07^M~#R7hYy? z3r$->9pSG~J>V;O`rACZD-p&h287QOfP%k}!~lw)uA0ni(JWcNLk@PVK=MxU(&p`K z8Z8H13+c%sgyIGbkBY4sjn$Grk;`ZlYFb0olyd47bZxltiW-sGe_5C;HGJR*H8p$| zFG6QpMSC=JWA7vdCAYE}=C!oGJ}DJbl%T^%Y{DpMC;-LYb1m5<9^VLO>6ig=bPPAA z%k3enZ`UV7vsZ_@7u#VHzzR>|>hra!O)IfjMn@|9n5Yt4!N6tAG|PGGighD7v>Q>9I(s$uX6y4w-_fD7r$`3xv|T&T#i>s)so?4Io81%a-eeam zTlojqdEf={cY*4tdaD+%`|FC)TT!Du(bToEnP7QOqCY>Da`)o`Q`bh>c6Bx1$jP#n zz7PsJKdubyxVl-6z3_UvbMuaVa@pFSp1i#>^YP^S;J%gEdEtMX*34&GcJcp=N?QBad6ocGLh7) ztt88Sk<#bEp%B))yk>=ccLMPYx^zqU8*$rcI)bn)x3gP_+_AiD3NwPI(2SXLp*l8j?uYZx^`r}0P zJLdCWEcNYN${J(y={}3LHKz-w0*1sK307_Gqo@`xO}=>LrS$En%%!9@7Kp-oBYq`F3tI zcrfJr?!(AP$}92HbGvu?Dx?;zoH=KxjrKuv;+4n02Mj6Qr%(I_ZjUav&e`lQ{V9j} zkj#WpUURN2W?Gau`aFoAQj#7jhPI{T)t2d_U{2UQj@t1V+Yk**g~sSyt1xFxDJ;|i z8m5Zb9~sWCj@W~7V;_Ho# z*Vt9W2pd0G&I{UM8(>HVESe@@IlDib0USN#FbFM$_cYj@ffrkx8rSfD0@vZDBg@zdckGbEs)X$`wbroYLuOSmYp=i+W71!-|;6-gU{fQ>^z)LceZ$2QVa-Xo&g3p_?dILE)8Hi$( z?Q<+R+>BCKY(h&;NOeXq;LOTBP{)TE-`{5uxDp?UFH2(9(x!8lSCnEx0zC5lxw4!+ zHkdS=b<7H>EGyhpd2>Z+H2Wmx#UQFK_UrThJbyJIA{@t^b?8?Fxl0?L5x4kSDUhtG z8zQYmq}|X8uLM#Ratju9?1^x8fIyOUn*3Q_HsWD%PZv8Z(llqWk*6Pum@{tFjeRwv zUan}0QHPgyxe4z9aV&C;AKG+C;yhGR>Qtmv^x^Uoez(4qrp%{TYG(FzM^mfD4i8qE z=cl<^ld`=<7UkLKg^^U>!hVz&Y5Y8O&@IxjoO%_C7s90J<>fAv{}g~S3;gPPyq0qD zznrEoPImS-|HkLjk~HEVx>xvX&1AdgdMiy=)|j{U+s;9WLZVxI$t#_PJ{SD5a$!0V z5+p$k+pjGFp>F4nZ;!lf;|cr)`?8_iz~e6a%Vei+vQgW-X8V~>r7y~@hYu1@s{zvE zCww@w4nkgAvDuvj*r!^4aloK!USGq9>!G~pYOl3)4}1o);h8{BDxJI5p9=X7-H{_Bd>ub(YIf$<`q9= z*bP>NCumleR|H>8a@pu#YuA$3n%Xu!=yA@l*&TV<00Lj-BT-vSamn%c zSeJqgJFfR9$n_pBC%8Uqo+qw~9H1e#Tg_;RW0i1IK&LM6! z_A6T*qVK!&f6t#O-|H2`I-(3Ymv0{|^tWHRV3vupGX=HxW;dcYVqh&b{tjsK`CjLA ztJXSnd``bZ^Xu@hAK-n_lBu0KSB9qVMup~9e(#E<_!~}Fqw?N759X%j+d370%f{dG zfk1d6C3=y-xJbnOSb5Eo!8abIzj!h!;ZqLY2~m_iBSjGV9Z+J`che$XuL`eA`DPY0 z7T&QYDF=#pjOhRw!Cq+?rP4_ay43sJZmUI~=6}JKlAfK~1F9>u_t$Jvh$J zI*U@z#4lT%js6{X+FRa><1!A-~Rjv`w z-^IGNOiRi=EZe(lgzsJ7Ezgu6N7OF+th{(m-gvb&0h(#2L_9V@!{E+lka+jC3-xRE zDn2rM&SZPVODji~M0z=5@>yzTz#X8WMx>nQITX*&IUZYZVZxV&CDC^y*sgwsH^mEK zjW#5~809Q|E|gW27ch?WP??1HI+!XintqsiLVJxbcf+O_;xM&>xX*JCQp;it$7{

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+\newcommand{\tikzcircle}[2][red,fill=red]{\tikz[baseline=-0.5ex]\draw[#1,radius=#2] (0,0) circle ;}% + + + + \setlength\parindent{0pt} \everymath{\displaystyle}