(\(m_{{tan}} \cdot m_{\operatorname{norm}} = -1\))\\
\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}
\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*}
\usepackage{dblfloatfix}
\usepackage{enumitem}
\usepackage{fancyhdr}
-\usepackage[a4paper,margin=2cm]{geometry}
+\usepackage[a4paper,margin=1.8cm]{geometry}
\usepackage{graphicx}
\usepackage{harpoon}
+\usepackage{keystroke}
\usepackage{listings}
\usepackage{makecell}
\usepackage{mathtools}
\definecolor{peach}{HTML}{e6beb2}
\definecolor{lblue}{HTML}{e5e9f0}
-\newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm}
-\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=important, fontupper=\sffamily\bfseries}
+\newtcolorbox{cas}{colframe=cas!75!black, fonttitle=\sffamily\bfseries, title=On CAS, left*=3mm}
+\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=darkgray, fontupper=\sffamily\bfseries}
\newtcolorbox{theorembox}[1]{colback=green!10!white, colframe=blue!20!white, coltitle=black, fontupper=\sffamily, fonttitle=\sffamily, #1}
\subsection*{One to one functions}
\begin{itemize} \tightlist
- \item
- \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
- \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
- \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
- \(x^3\) is)
- \item
- horizontal line test
- \item
- if not one to one, it is many to one
+ \item \(f(x)\) is 1:1 if \(f(a) \ne f(b) \> \forall \>\{a,b\} \in \operatorname{dom}(f)\) \\
+ \(\implies\) 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}
\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 \(x^n, n \in \mathbb{Z}^+\) &
\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}
\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}
\documentclass[a4paper]{article}
\usepackage[dvipsnames, table]{xcolor}
+\usepackage{adjustbox}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{array}
\usepackage{dblfloatfix}
\usepackage{enumitem}
\usepackage{fancyhdr}
-\usepackage[a4paper,margin=2cm]{geometry}
+\usepackage[a4paper,margin=1.8cm]{geometry}
\usepackage{graphicx}
\usepackage{harpoon}
\usepackage{hhline}
\usepgflibrary{arrows.meta}
\pgfplotsset{compat=1.16}
\pgfplotsset{every axis/.append style={
- axis x line=middle, % put the x axis in the middle
- axis y line=middle, % put the y axis in the middle
- axis line style={->}, % arrows on the axis
- xlabel={$x$}, % default put x on x-axis
- ylabel={$y$}, % default put y on y-axis
+ axis x line=middle,
+ axis y line=middle,
+ axis line style={->},
+ xlabel={$x$},
+ ylabel={$y$},
}}
\psset{dimen=monkey,fillstyle=solid,opacity=.5}
\newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X}%
\newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}%
+\newcolumntype{Y}{>{\centering\arraybackslash}X}
\definecolor{cas}{HTML}{e6f0fe}
\definecolor{important}{HTML}{fc9871}
\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)}&