-\documentclass[a4paper, twocolumn]{article}
+\documentclass[a4paper]{article}
\usepackage[dvipsnames, table]{xcolor}
\usepackage{adjustbox}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{blindtext}
+\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{longtable}
\usepackage{makecell}
\usepackage{mathtools}
+\usepackage{mathtools}
\usepackage{multicol}
\usepackage{multirow}
\usepackage{newclude}
\usepackage{pgfplots}
+\usepackage{polynom}
\usepackage{pst-plot}
\usepackage{standalone}
+\usepackage{subfiles}
\usepackage{tabularx}
\usepackage{tabu}
\usepackage{tcolorbox}
\usetikzlibrary{%
angles,
+ arrows,
+ arrows.meta,
calc,
datavisualization.formats.functions,
decorations,
decorations.markings,
+ decorations.text,
decorations.pathreplacing,
decorations.text,
+ patterns,
scopes
}
+
\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
+
\usepgflibrary{arrows.meta}
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+\pgfplotsset{compat=1.16}
+\pgfplotsset{every axis/.append style={
+ axis x line=middle, % centre axes
+ axis y line=middle,
+ axis line style={->}, % arrows on axes
+ xlabel={$x$}, % axes labels
+ ylabel={$y$}
+}}
+
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-}}
\pagestyle{fancy}
\fancyhead[LO,LE]{Year 12 Methods}
\fancyhead[CO,CE]{Andrew Lorimer}
\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
+\newcommand{\tg}{\mathop{\mathrm{tg}}}
+\newcommand{\cotg}{\mathop{\mathrm{cotg}}}
+\newcommand{\arctg}{\mathop{\mathrm{arctg}}}
+\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
+
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
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+\newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}}
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\newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}
\definecolor{cas}{HTML}{e6f0fe}
\definecolor{important}{HTML}{fc9871}
+\definecolor{highlight}{HTML}{ffb84d}
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-\definecolor{shade1}{HTML}{ffffff}
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-\definecolor{shade3}{HTML}{cce2ff}
+\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}
\begin{document}
\date{}
\maketitle
+\begin{multicols}{2}
+
\section{Functions}
\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{enumerate} \tightlist
\item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\)
- \item Find determinant of first matrix: \(\Delta = ps-qr\)
- \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
- or let \(\Delta \ne 0\) for one unique solution.
- \item Solve determinant equation to find variable \\
+ \item Find \(\det(\text{first matrix}) = ps-qr\)
+ \item Let \(\det = 0\) for \(\{0,\infty\}\) solutions
+ or \(\det \ne 0\) for 1 solution
+ \item Solve to find variable \\ \\
\textbf{For infinite/no solutions:}
\item Substitute variable into both original equations
- \item Rearrange equations so that LHS of each is the same
- \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
- \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
+ \item Rearrange so that LHS of each is the same
+ \item \(\begin{aligned}[t]
+ \infty \text{ solns: } & \text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x \\
+ 0 \text{ solns: } & \text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x
+ \end{aligned}\)
\end{enumerate}
- \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
+ \begin{cas}
+ Action \(\rightarrow\) Matrix \(\rightarrow\) Calculation \(\rightarrow\) \texttt{det}
+ \end{cas}
\subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
\textbf{Open circle:} point included\\
\textbf{Closed circle:} point not included
+\begin{cas}
+ Define piecewise functions: \\
+ \-\hspace{1em}Math3 \(\rightarrow\)
+ \begin{tikzpicture}%
+ \draw rectangle (0.5,0.5);
+ \node at (0.08,0.25) {\(\{\)};
+ \filldraw [black] (0.15, 0.4) rectangle(0.25, 0.3);
+ \draw (0.35, 0.4) rectangle(0.45, 0.3);
+ \node [font=\footnotesize] at (0.3,0.3) {\verb;,;};
+ \draw (0.15, 0.2) rectangle(0.25, 0.1);
+ \node [font=\footnotesize] at (0.3,0.1) {\verb;,;};
+ \draw (0.35, 0.2) rectangle(0.45, 0.1);
+ \end{tikzpicture}
+ % TODO: finish this section
+\end{cas}
+
\subsection*{Operations on functions}
For \(f \pm g\) and \(f \times g\):
\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 \(x^n, n \in \mathbb{Z}^+\) &
\section{Polynomials}
+ \subsection*{Factor theorem}
+
+ \begin{theorembox}{title=General form \(\beta x + \alpha\)}
+ If \(\beta x + \alpha\) is a factor of \(P(x)\), \\
+ \-\hspace{1em}then \(P(-\dfrac{\alpha}{\beta})=0\).
+ \end{theorembox}
+
+ \begin{theorembox}{title=Simple form \(x-a\)}
+ If \((x-a)\) is a factor of \(P(x)\), remainder \(R=0\). \\
+ \-\hspace{1em}\(\implies P(a)=0\)
+ \end{theorembox}
+
+ \subsection*{Remainder theorem}
+
+ \begin{theorembox}{}
+ When \(P(x)\) is divided by \(\beta x + \alpha\), the remainder is \(-\dfrac{\alpha}{\beta}\).
+ \end{theorembox}
+
+ \subsection*{Rational root theorem}
+ Let \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\) be a polynomial of degree \(n\) with \(a_i \in \mathbb{Z} \forall a\). Let \(\alpha, \beta \in \mathbb{Z}\) such that their highest common factor is 1 (i.e. relatively prime).
+
+ If \(\beta x + \alpha\) is a factor of \(P(x)\), then \(\beta\) divides \(a_n\) and \(\alpha\) divides \(a_0\) .
+
+ \subsubsection*{Discriminant}
+ \[\begin{cases}
+ b^2-4ac > 0 & \text{two solutions} \\
+ b^2-4ac = 0 & \text{one solution} \\
+ b^2-4ac < 0 & \text{no solutions}
+ \end{cases}\]
+ \begin{warning}
+ Flip inequality sign when multiplying by -1
+ \end{warning}
+
+ \subsection*{Long division}
+
+ \[ \polylongdiv{x^2+2x+4}{x-1} \]
+
+ \begin{cas}
+ Action \(\rightarrow\) Transformation \(\rightarrow\) \texttt{propFrac}
+ \end{cas}
+
\subsection*{Linear equations}
\subsubsection*{Forms}
Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
\subsection*{Quadratics}
+
\setlength{\abovedisplayskip}{1pt}
\setlength{\belowdisplayskip}{1pt}
+
+ \textbf{Linear factorisation}
\[ x^2 + bx + c = (x+m)(x+n) \]
\hfill where \(mn=c, \> m+n=b\)
\input{circ-functions}
\input{calculus}
+ \subfile{statistics-ref}
+ \end{multicols}
- \section{Statistics}
-
- \subsection*{Probability}
-
- \begin{align*}
- \Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\
- \Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\
- \Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\
- \Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime})
- \end{align*}
-
- Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\
-
- Independent events:
- \begin{flalign*}
- \quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\
- \Pr(A|B) &= \Pr(A) \\
- \Pr(B|A) &= \Pr(B)
- \end{flalign*}
-
- \subsection*{Combinatorics}
-
- \begin{itemize}
- \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\)
- \item \colorbox{important}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\)
- \item Note \({n \choose k} = {n \choose k-1}\)
- \end{itemize}
-
- \subsection*{Distributions}
-
- \subsubsection*{Mean \(\mu\)}
-
- \textbf{Mean} \(\mu\) or \textbf{expected value} \(E(X)\)
-
- \begin{align*}
- E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{\(f =\) absolute frequency} \\
- &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{discrete}\\
- &= \int_\textbf{X} (x \cdot f(x)) \> dx
- \end{align*}
-
- \subsubsection*{Mode}
-
- Most popular value (has highest probability of all \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution.
-
- \subsubsection*{Median}
-
- If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \(m = 0.5 = 0.5\), then \(m\) is halfway between this value and the next. To find \(m\), add values of \(X\) from smallest to alrgest until the sum reaches 0.5.
-
- \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
-
- \subsubsection*{Variance \(\sigma^2\)}
-
- \begin{align*}
- \operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\
- &= \sum (x-\mu)^2 \times \Pr(X=x) \\
- &= \sum x^2 \times p(x) - \mu^2 \\
- &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2
- &= E\left[(X-\mu)^2\right]
- \end{align*}
-
- \subsubsection*{Standard deviation \(\sigma\)}
-
- \begin{align*}
- \sigma &= \operatorname{sd}(X) \\
- &= \sqrt{\operatorname{Var}(X)}
- \end{align*}
-
- \subsection*{Binomial distributions}
-
- Conditions for a \textit{binomial distribution}:
- \begin{enumerate}
- \item Two possible outcomes: \textbf{success} or \textbf{failure}
- \item \(\Pr(\text{success})\) is constant across trials (also denoted \(p\))
- \item Finite number \(n\) of independent trials
- \end{enumerate}
-
-
- \subsubsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)}
-
- \begin{align*}
- \mu(X) &= np \\
- \operatorname{Var}(X) &= np(1-p) \\
- \sigma(X) &= \sqrt{np(1-p)} \\
- \Pr(X=x) &= {n \choose x} \cdot p^x \cdot (1-p)^{n-x}
- \end{align*}
-
- \begin{cas}
- Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPdf; then input
- \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=3cm, font=\normalfont]
- \item [x:] no. of successes
- \item [numtrial:] no. of trials
- \item [pos:] probability of success
- \end{description}
- \end{cas}
-
- \subsection*{Continuous random variables}
-
- A continuous random variable \(X\) has a pdf \(f\) such that:
-
- \begin{enumerate}
- \item \(f(x) \ge 0 \forall x \)
- \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
- \end{enumerate}
-
- \begin{align*}
- E(X) &= \int_\textbf{X} (x \cdot f(x)) \> dx \\
- \operatorname{Var}(X) &= E\left[(X-\mu)^2\right]
- \end{align*}
-
- \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
-
-
- \subsection*{Two random variables \(X, Y\)}
-
- If \(X\) and \(Y\) are independent:
- \begin{align*}
- \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
- \operatorname{Var}(aX \pm bY \pm c) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y)
- \end{align*}
-
- \subsection*{Linear functions \(X \rightarrow aX+b\)}
-
- \begin{align*}
- \Pr(Y \le y) &= \Pr(aX+b \le y) \\
- &= \Pr\left(X \le \dfrac{y-b}{a}\right) \\
- &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx
- \end{align*}
-
- \begin{align*}
- \textbf{Mean:} && \operatorname{E}(aX+b) & = a\operatorname{E}(X)+b \\
- \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
- \end{align*}
-
- \subsection*{Expectation theorems}
-
- For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).
-
- \begin{align*}
- E(X^2) &= \operatorname{Var}(X) - \left[E(X)\right]^2 \\
- E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear} \\
- &\ne [E(X)]^n \\
- E(aX \pm b) &= aE(X) \pm b \tag{linear} \\
- E(b) &= b \tag{\(\forall b \in \mathbb{R}\)}\\
- E(X+Y) &= E(X) + E(Y) \tag{two variables}
- \end{align*}
-
- \subsection*{Sample mean}
-
- Approximation of the \textbf{population mean} determined experimentally.
-
- \[ \overline{x} = \dfrac{\Sigma x}{n} \]
-
- where
- \begin{description}[nosep, labelindent=0.5cm]
- \item \(n\) is the size of the sample (number of sample points)
- \item \(x\) is the value of a sample point
- \end{description}
-
- \begin{cas}
- \begin{enumerate}[leftmargin=3mm]
- \item Spreadsheet
- \item In cell A1:\\ \path{mean(randNorm(sd, mean, sample size))}
- \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range
- \item Input range as A1:An where \(n\) is the number of samples
- \item Graph \(\rightarrow\) Histogram
- \end{enumerate}
- \end{cas}
-
- \subsubsection*{Sample size of \(n\)}
-
- \[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]
-
- Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).
-
- For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)
-
- \begin{cas}
-
- \begin{itemize}
- \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left
- \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
- \end{itemize}
-
- \end{cas}
-
- \subsection*{Normal distributions}
-
-
- \[ Z = \frac{X - \mu}{\sigma} \]
-
- Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\) \\
- \(\text{mean} = \text{mode} = \text{median}\)
-
- \begin{warning}
- Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
- \end{warning}
-
- \pgfmathdeclarefunction{gauss}{2}{%
- \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
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- \begin{figure*}[hb]
- \centering
- \begin{tikzpicture}
- \begin{axis}[every axis plot post/.style={
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- axis x line=bottom,
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- \addplot {gauss(0,0.75)};
- \end{axis}
- \end{tikzpicture}
- \end{figure*}
- \end{document}
+\end{document}