From 4de6207254326a62227ef93cc0f3b6ccf30f69e2 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Tue, 5 Nov 2019 16:30:19 +1100 Subject: [PATCH] [spec] additions to complex graphs and exp identities --- spec/calculus-rules.tex | 12 +++++----- spec/spec-collated.pdf | Bin 509399 -> 509644 bytes spec/spec-collated.tex | 50 +++++++++++++++++++++------------------- 3 files changed, 32 insertions(+), 30 deletions(-) diff --git a/spec/calculus-rules.tex b/spec/calculus-rules.tex index 2b6b712..f60a82f 100644 --- a/spec/calculus-rules.tex +++ b/spec/calculus-rules.tex @@ -34,10 +34,10 @@ \flushbottom \subsubsection*{Index identities} \begin{align*} - b^{m+n} &= b^m \cdot b^n \\ - (b^m)^n &= b^{m \cdot n} \\ - (b \cdot c)^n &= b^n \cdot c^n \\ - {a^m \div a^n} &= {a^{m-n}} + a^{x+y} &= a^x \cdot a^y \\ + a^{x-y} &= a^x \div a^y \\ + (a^x)^y &= a^{x \cdot y} \\ + (a \cdot b)^x &= a^x \cdot b^x \end{align*} } @@ -82,7 +82,7 @@ Note \(\sin^{-1} \left(\dfrac{x}{a}\right) + \cos^{-1} \left(\dfrac{x}{a}\right) \begin{align*} \log_b (xy) &= \log_b x + \log_b y \\ \log_b\left(\frac{x}{y}\right) &= \log_b(x) - \log_b(y) \\ - \log_b x^n &= n \log_b x \\ - \log_b y^{x^n} &= x^n \log_b y + \log_b y^{x^n} &= x^n \log_b y \\ + \log_b x^n &= n \log_b x \end{align*} } diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 41d3d5c4543a2fba26306c6b414716be0b84465b..8ff3726d01cd238699e4b0133a35240b2881e9a4 100644 GIT binary patch delta 54595 zcmV(@K-Rz4iyzF5AF#s_1T{G{Gm~*UD1Yr;>vJSElK<|%qCR@=s6y`t7aMUf3&awz zyJoo9y8u44+dZCU+wGxk1H)f`nNsOhrK)~;V2QgAGj3O_q*N-E`OAl7(%d$r`TXLO zZ!SLk%rv&MimB$C*NvndVWKhAc3j%#o15m_%l~;{_~mM`d(m>vFDdgsU%h?NGJkWq zTCNwH^~2+fmcmnZ`SJEDRCD?Tb&px%lydLUYp4M5FAa=FRf%;@j^@a|7Rf(U6W? z)BNmO-ZjGVjxhtDH_fYys|%7(n}5rO(#}fBvWaU$I!m}G>*f93+s*11AV34`c}G8oz>j9HR*_E zjp!)t7le)Zq&j7I(`JqSy)qnr;(?iq+DJ^PJtOIkaaD};r^-l_bdtukJz`RYgp6ke zQbNWw74!OzNK&DQcSegi_kZ=c6VoYcvQFrjXzWAo;ZG}|BST>mz!bR*ENYoHQ>0PY 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z4@nrne|Cb?N7z&D<0QY6y7CxubAQm8p8>gc9k~bUo{U+XpcpInS>|nDAH~4v%jhReUZLVYrP|tU1)>VU`U_01{Y6jua&>t;t zA)nf_n~ZQy8FezrY~pT|>{GQg1fLIs*JlQuf9q(SKJLNK;&}AMbU7W375cIgT$I46 zfeu^UFL=t1haXI@FDKU;o>sui5jcb2NP4TU1$CVq~qt1byY64Ijy1k zf0E3cRqIDga2 zOO#O<2JrKo?>3sZF;nxHm&u&Q)b3enR%&S%Elcx~T`VGqKxW-83?c{$TSR|=5P~f# z3gg>In`W01LCY4k53;5CJhS=nyze*X&KP5`Zj7 zz)R$DGo6*nhRxN`vrHt4c%gW|B%5;Jy5#bP?W%&q~AaR;QFM!4C_R zF2m0ZDUHDI`>r$!f2K<53jF5}l*ZsMb|{U*&y6WfAV}w5dJ@6liqcgC*>_4)2%f%G zx`tq(Sm`=~uM0{y5Ue#T-9(sqs5FgmB>zyi5YFoV=58a*P3pM~qFYa|48mVol|ba* zQL=~{$CW%pqmPw*M9&740z~] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$}; + \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$}; + \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1); + \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)}; + \begin{scope} + \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0); + \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm); + \end{scope} + \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)}; + \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)}; + \node [brown, mydot] at (-0.25,0) {}; + \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)}; + \node [left, font=\footnotesize] at (0,-1) {\(-1\)}; + \node [below, font=\footnotesize] at (1,0) {\(1\)}; + \end{tikzpicture}\end{center} \begin{itemize} + \item \(\operatorname{Arg}(z \pm b) = \theta\) (ray) \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)} \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)} - \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)} + \item \(|z - (a+bi)|=|z - (c+di)| \\ \implies \frac{2(c-a)x + a^2 + b^2 - c^2 - d^2}{2(b-d)}\) + \item \(\operatorname{Re}(z) \pm \operatorname{Im}(z) = c\) \end{itemize} \subsubsection*{Circles} @@ -309,9 +328,10 @@ \begin{itemize} \item \(|z-z_1|^2=c^2|z_2+2|^2\) \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\) + \item \(z \overline{z} = r^2\) \end{itemize} - \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\) + \subsubsection*{Regions \qquad \(\operatorname{Arg}(z) \lessgtr \theta\)} \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}] \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$}; @@ -328,24 +348,6 @@ \node [blue, mydot] {}; \end{tikzpicture}\end{center} - \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\) - - \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}] - \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$}; - \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$}; - \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1); - \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)}; - \begin{scope} - \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0); - \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm); - \end{scope} - \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)}; - \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)}; - \node [brown, mydot] at (-0.25,0) {}; - \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)}; - \node [left, font=\footnotesize] at (0,-1) {\(-1\)}; - \node [below, font=\footnotesize] at (1,0) {\(1\)}; - \end{tikzpicture}\end{center} \section{Vectors} \begin{center}\begin{tikzpicture} @@ -1205,12 +1207,12 @@ \begin{align*} V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\ - &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy + &= 2\pi \int^{x=b}_{x=a} x|f(x)| \> dx \end{align*} - \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)} + \subsubsection*{Rotating the area between two graphs} - \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\] + \[V = \pi \int^b_a \left( f(x)^2 - g(x)^2 \right) \> dx\] \hfill where \(f(x) > g(x)\) \subsection*{Length of a curve} -- 2.43.2