From 41eacf2b65512b5d0bfdedd85b5f2231b77e6c3e Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 29 Sep 2019 21:08:38 +1000 Subject: [PATCH 1/1] [spec/methods] formatting fixes --- methods/calculus.tex | 19 ++++++++----------- methods/methods-collated.pdf | Bin 482358 -> 494956 bytes methods/methods-collated.tex | 25 ++++++++++++------------- methods/statistics-ref.tex | 16 +++++++++++----- methods/transformations.tex | 7 +++---- spec/spec-collated.pdf | Bin 462756 -> 462687 bytes spec/spec-collated.tex | 30 ++++++++++++++++++++++-------- 7 files changed, 56 insertions(+), 41 deletions(-) diff --git a/methods/calculus.tex b/methods/calculus.tex index 72b8bdb..05bab2b 100644 --- a/methods/calculus.tex +++ b/methods/calculus.tex @@ -53,8 +53,14 @@ Not differentiable at: (\(m_{{tan}} \cdot m_{\operatorname{norm}} = -1\))\\ \textbf{Secant} \(={{f(x+h)-f(x)} \over h}\) -\colorbox{cas}{On CAS:} \\ Action \(\rightarrow\) Calculation -\(\rightarrow\) Line \(\rightarrow\) \texttt{tanLine} or \texttt{normal} +\begin{cas} + \textbf{In main}: Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \\ + \-\hspace{1em} \texttt{tanLine(f(x), x, p)} \\ + \-\hspace{1em} \texttt{normal(f(x), x, p)} \\ + where \texttt{p} is the \(x\)-value of the coordinate + + \textbf{In graph}: define function, then Analysis \(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line). Type \(x\) value to solve for a point. Return to show equation for line. +\end{cas} \subsection*{Strictly increasing/decreasing} @@ -67,15 +73,6 @@ For \(x_2\) and \(x_1\) where \(x_2 > x_1\): \item Endpoints are included, even where gradient \(=0\) \end{itemize} -\begin{cas} - \colorbox{cas}{\textbf{In main}}: type function. Interactive - \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) (Normal - \textbar{} Tan line)\\ - \colorbox{cas}{\textbf{In graph}}: define function. Analysis - \(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line). - Type \(x\) value to solve for a point. Return to show equation for line. -\end{cas} - \subsection*{Stationary points} \begin{align*} diff --git a/methods/methods-collated.pdf b/methods/methods-collated.pdf index 35e900eddb3c27d47ecc19c8f006267cf35a6eb8..6737a92da15778afc00115feb3b6ba902af4bdca 100644 GIT binary patch delta 257755 zcmZsCV~j2g59Qdl%{#Vj&mG&gJ@brh+q`4jwr$(?zF+p|?vExXP1-a~ik!49ZTMH2 z1W4qHqT+N+^lUKXb4$Z(FkA^pXdp~1X~gm%RDf$;YiF`n*Y6qq`TMfm9%TU#^f#YC`8YcW zR&Q5F0=>NN=Vf~Eo=HK_gK725><8dVNLzN16x7oWuz4hx3fv`ruN=_$d+X3N>69A`W-|!lWe_02eUsb(ZyeHG!+s3_l+aqF6 zQRU~|EzrAlEu#!4*csz0lvxwwglm3K=Vq3qW1h|K_k138rRJ_))_S_C{&2v~6Wu4b z7KhO2d`4^ARg>dW?ZRu3L?L&CdiHRouaCuO2=Mmn=w0Mg&97&k_IkJS9dGWcj$Dj8kJ#Y7X%T+)@G{*>8b)lA`DoZO+_;%S+`hVw-^jaj zgt+*+-WSMT;52sr@>d^FY^>~ueKLRng1UbPTIzHu`VbqT^Q&!l(#vI%1fbN5^tbiW z@>WUrryEz+?Nu5OAH3Krup4)`0C|a{@uRi9YI<~c!*)tfakN3ZVCBlUBA66;>MfUP z|5*s1#(QTC#&PytD`8RuuLH>tD(?t~Mbh=KQV~-?r8P&&Qxi!O~$i zapuXgC=<$S3bl9CW7V*Hsu~8ttma7+`lh&<5(JFrm~|OfgG!!B!@;b2ZdaX 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zx=2Q*9147Nm(q{1lFpB_#;sQI8RTAIojJ&DwArHai;@g8fC?(eL33y)3^H1$i~1N(!o*R-qF?mpD`p8+s_Tj07* \forall \>\{a,b\} \in \operatorname{dom}(f)\) \\ + \(\implies\) unique \(y\) for each \(x\) + \item e.g. \(\sin x\) is not 1:1, \(x^3\) is + \item horizontal line test + \item if not one to one, it is many to one \end{itemize} \subsection*{Odd and even functions} @@ -258,8 +255,10 @@ For \(x^n\), parity of \(n \equiv\) parity of function \begin{figure*}[ht] \centering - \begin{tabularx}{\textwidth}{r|Y|Y} + \begin{tabularx}{\textwidth}{|r|Y|Y|} + \hline + \rowcolor{lblue} & \(n\) is even & \(n\) is odd \\ \hline \centering \(x^n, n \in \mathbb{Z}^+\) & diff --git a/methods/statistics-ref.tex b/methods/statistics-ref.tex index 8731999..ec728d8 100644 --- a/methods/statistics-ref.tex +++ b/methods/statistics-ref.tex @@ -24,11 +24,17 @@ Independent events: \subsection*{Combinatorics} -\begin{itemize} \tightlist - \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\) - \item \colorbox{highlight}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\) - \item Note \({n \choose k} = {n \choose k-1}\) -\end{itemize} +\begin{align*} + \text{Arrangements} && {n \choose k} & = \frac{n!}{(n-k)} \\ + \text{Combinations} && {n \choose k} & = \frac{n!}{k!(n-k)!} +\end{align*} + +Note \({n \choose k} = {n \choose k-1}\) + +\begin{cas} + Keyboard \(\rightarrow\) Advance \(\rightarrow\) \keystroke{nCr}/\keystroke{nPr} \\ + \-\hspace{1em} \texttt{nCr(n, r)} or \texttt{nPr(n, r)} +\end{cas} \subsection*{Distributions} diff --git a/methods/transformations.tex b/methods/transformations.tex index 00428e3..96b520a 100644 --- a/methods/transformations.tex +++ b/methods/transformations.tex @@ -17,12 +17,11 @@ \item if \(a<0\), graph is reflected over \(x\)-axis \item - translation of \(k\) units parallel to \(y\)-axis or from \(x\)-axis + translation of \(k\) units \(\parallel y\)-axis/from \(x\)-axis \item - translation of \(h\) units parallel to \(x\)-axis or from \(y\)-axis + translation of \(h\) units \(\parallel x\)-axis/from \(y\)-axis \item - for \((ax)^n\), dilation factor is \(1 \over a\) parallel to - \(x\)-axis or from \(y\)-axis + for \((ax)^n\), dilation factor is \(1 \over a \> \parallel x\)-axis/from \(y\)-axis \item when \(0 < |a| < 1\), graph becomes closer to axis \end{itemize} diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 7b998a92be779cb72a149b61be13c27b80ef4a20..1a674192724f96f6140f8f7b6ef89768db68d21e 100644 GIT binary patch delta 147948 zcmV)YK&-!{pB&$w9FQXeI50Su0X7CGf6W}*a@)rBU0;E>Qscp9@0TX&)N!3mGjZ+O z>P*@=4>UzPbR<%xBq!re`|Ukvaa#b31wc}9UMw11?w&pOD`@sEquDoSUtOGi_O;C% zahmJw;%%lFp<< z%l;9DNLzjVDDwLj2yZt14FCjbRJG!dku88AqF%849MGl>U?*}LO^h(re@Y&~nMv&8^>N)X%7df(q)6DjKNSDV|&9*h%XYXrWgyrcq(10zl; zFAhLKK{SX^E+jEZ7OS7%f0P`*yxZ)yw|x->xzoTZnpxt6ONl6~Hn2vN66z(An-G_d zSb#-ykWuEkWbB>HP~`P#jt0RcQg+UMaSmF}Q)x&s;n7B>PUeyk4F}Wk=r)Ccy_v~0 zRUp-++W1%y2%ZvWge_zqx5*iiAPc3g>Whjd+KMb#^qNvVz_y~|e~3w^@nOPUVft3I^=(`-;ifWu 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\newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}% +\newcolumntype{Y}{>{\centering\arraybackslash}X} \definecolor{cas}{HTML}{e6f0fe} \definecolor{important}{HTML}{fc9871} @@ -876,13 +878,25 @@ \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e. \(f^{\prime\prime} = 0\)) + \subsubsection*{Strictly increasing/decreasing} + + For \(x_2\) and \(x_1\) where \(x_2 > x_1\): + + \textbf{Strictly increasing}\\ + \hspace{1em}where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\) + \textbf{strictly decreasing}\\ + \hspace{1em}where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\) + \begin{warning} + Endpoints are included, even where gradient \(=0\) + \end{warning} + \begin{table*}[ht] \centering - \begin{tabularx}{\textwidth}{rXXX} + \begin{tabularx}{\textwidth}{rYYY} \hline \rowcolor{shade2} - & \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\[1.5em] + & \adjustbox{margin=0 1ex, valign=m}{\centering\(\dfrac{d^2 y}{dx^2} > 0\)} & \adjustbox{margin=0 1ex, valign=m}{\centering \(\dfrac{d^2y}{dx^2}<0\)} & \adjustbox{margin=0 1ex, valign=m}{\(\dfrac{d^2y}{dx^2}=0\) (inflection)} \\ \hline \(\dfrac{dy}{dx}>0\) & \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-3, xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x)}; \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}& -- 2.47.1

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