+\documentclass[a4paper]{article}
+\usepackage[a4paper,margin=2cm]{geometry}
+\usepackage{multicol}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{enumitem}
+\usepackage{tcolorbox}
+\usepackage{fancyhdr}
+\usepackage{pgfplots}
+\usepackage{tabularx}
+
+\pagestyle{fancy}
+\fancyhead[LO,LE]{Unit 3 Methods Revision Lecture}
+\fancyhead[CO,CE]{Andrew Lorimer}
+
+\setlength\parindent{0pt}
+
+\begin{document}
+
+ \title{\large Year 12 Methods \\ \huge Unit 3 Revision Lecture \\ \large Monash University \\ presented by Kevin McMenamin}
+ \author{Andrew Lorimer}
+ \date{5 July 2019}
+ \renewcommand{\abstractname}{}
+ \maketitle
+
+ \section{Graphs}
+
+ \textbf{16 types of graph}---put in reference book:
+ \begin{multicols}{2}
+ \begin{enumerate}
+ \item truncus
+ \item hyperbola
+ \item sqrt
+ \item parabola
+ \item cubic
+ \item quartic
+ \item linear
+ \item circle
+ \item semicircle
+ \item tan
+ \item sin
+ \item cos
+ \item log
+ \item exp
+ \item $x^{a \over b}$
+ \item $x^{-a \over b}$
+ \end{enumerate}
+ \end{multicols}
+
+ \subsection{Power functions}
+
+ \begin{itemize}
+ \item In first quadrant, shape of graph for $x>0 \cap y>0$ is either $\sqrt{x}$ or $x^2$
+ \end{itemize}
+
+ \subsection{Features of graphs}
+
+ \begin{itemize}
+ \item Asymptotes
+ \item Intercepts
+ \item Stationary points
+ \item Endpoints
+ \item Other critical points
+ \item Continuous or discontinuous
+ \end{itemize}
+
+ \begin{tcolorbox}[title=Key points]
+ \begin{itemize}
+ \item All transformations can be described by matrices
+ \item Inverse is a transformation
+ \item Memorise approximate values of $e,\>\pi,\>\sqrt{2},\>\sqrt{3}$
+ \item Put 16 base graphs in reference book
+ \end{itemize}
+ \end{tcolorbox}
+
+ \section{Transformations}
+
+ Order: \qquad \textbf{Reflect $\longrightarrow$ Dilate $\longrightarrow$ Translate}
+
+ \subsection{Two forms}
+
+ \begin{itemize}
+ \item note $a$ and $b$ can be positive or negative
+ \item check validity of solutions for logarithms
+ \item results in transformed equation $y^\prime = f^\prime(x)$
+ \end{itemize}
+
+ \[ y^\prime = a \cdot f(\dfrac{1}{b} (x^\prime - c)) + d \]
+ \[
+ \begin{bmatrix}
+ x^\prime \\ y^\prime
+ \end{bmatrix}
+ =
+ \begin{bmatrix}
+ b & 0 \\
+ 0 & a
+ \end{bmatrix}
+ \begin{bmatrix}
+ x \\ y
+ \end{bmatrix}
+ +
+ \begin{bmatrix}
+ c \\d
+ \end{bmatrix}
+ \]
+
+ \begin{tcolorbox}[title=Key points]
+ \begin{itemize}
+ \item All transformations can be described by matrices
+ \item Inverse is a transformation
+ \item Check validity of $\log_a x$ solutions/transformations
+ \end{itemize}
+ \end{tcolorbox}
+
+ \section{Calculus}
+
+ Possible questions:
+ \begin{itemize}
+ \item Average rate of change
+ \item Instantaneous rate of change
+ \item Tangent line
+ \item Normal line
+ \item Features of gradient function
+ \begin{itemize}
+ \item Degree
+ \item Orientation
+ \item Format
+ \item Turning points
+ \item Inflection points
+ \item Asymptotes
+ \end{itemize}
+ \item Find original function from derivative\\
+ $\longrightarrow$ \textit{Use information to find unknowns}
+ \item Application questions - e.g. Pythagoras, trig. functions, measurement, given eqn
+ \end{itemize}
+
+ \subsection{Integration}
+
+ \subsubsection{Polynomials}
+
+ \[ f(x) = \int ax^n \> dx = \dfrac{ax^{n+1}}{n+1}+c \>, \quad n \ne -1 \]
+ \[f(x) = \int (ax+b)^n \> dx = \dfrac{(ax+b)^{n+1}}{a(n+1)}+c \>, \quad n \ne -1 \]
+
+ \subsubsection{Exponentials}
+
+ \[ f(x) = \int e^{ax+b} \> dx = \dfrac{e^{ax+b}}{a}+c \]
+
+ \subsubsection{Logarithms}
+
+ \textit{ignore modulus for methods}
+ \[ f(x) = \int \dfrac{1}{x} \> dx = \ln|x| + c \]
+ \[ f(x) = \int \dfrac{1}{ax+b} \> dx = \dfrac{1}{a} \ln|ax+b| + c \]
+ \[ f(x) - \int \dfrac{h^\prime (x)}{h(x)} \> dx = \ln|h(x)|+c \tag{general form}\]
+
+ \subsubsection{Trigonometric functions}
+
+ \[ f(x) = \int \cos(ax+b) \> dx = \dfrac{1}{a} \sin (ax+b) + c \]
+ \[ f(x) = \int \sin(ax+b) \> dx = -\dfrac{1}{a}\sin(ax+b) + c \]
+ \[ f(x) = \int \sec^2(ax+b) \> dx = \dfrac{1}{a}\tan(ax+b) + c \]
+
+ \subsection{Area under curves}
+
+ \begin{itemize}
+ \item \textbf{Upper rectangles} (overestimate) vs. \textbf{lower rectangles} (underestimate)
+ \item Rotate (invert) graph to make it easier, e.g. $y=\sqrt{x} \longrightarrow x=y^2$
+ \end{itemize}
+
+ \begin{tcolorbox}[title=Key points]
+ \begin{itemize}
+ \item For \textit{an} antiderivative, \qquad $+c \quad \forall \> c \in \mathbb{R}$ \qquad is also acceptable
+ \item Practice multi-part problems e.g:
+ \begin{enumerate}[label={\alph*)}]
+ \item Let $f:\mathbb{R}\rightarrow\mathbb{R},\quad f(x)=x\sin x$. Find $f^\prime(x)$.
+ \item Use the result of (a) to find the value of $\int^{\frac{\pi}{2}}_{\frac{\pi}{6}} x \cos x \> dx$ in the form $a\pi + b$.
+ \end{enumerate}
+ \end{itemize}
+ \end{tcolorbox}
+
+ \section{Probability}
+
+ \[ \Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B) \]
+ \[ \Pr(A \cup B) = 0 \tag{mutually exclusive} \]
+
+ \subsection{Conditional probability}
+
+ \[ \Pr(A|B)=\dfrac{\Pr(A \cap B)}{\Pr(B)} \]
+ \[ \Pr(A \cap B) = \Pr(A|B) \times \Pr(B) \tag{multiplication theorem} \]
+ \[ \Pr(A \cap B) = \Pr(A) \times \Pr(B) \tag{independent events} \]
+
+ \subsection{Discrete random distributions}
+
+ Any experiment or activity involving chance will have a probability associated with each result or \textit{outcome}. If the outcomes have a reference to \textbf{discrete numeric values} (outcomes that can be counted), and the result is unknown, then the activity is a \textit{discrete random probability distribution}.
+
+ \subsubsection{Discrete probability distributions}
+
+ If an activity has outcomes whose probability values are all positive and less than one ($\implies 0 \le p(x) \le 1$), and for which the sum of all outcome probabilities is unity ($\implies \sum p(x) = 1$), then it is called a \textit{probability distribution} or \textit{probability mass} function.
+
+ \begin{itemize}
+ \item \textbf{Probability distribution graph} - a series of points on a cartesian axis representing results of outcomes. $\Pr(X=x)$ is on $y$-axis, $x$ is on $x$ axis.
+ \item \textbf{Mean $\mu$} - measure of central tendency. \textit{Balance point} or \textit{expected value} of a distribution. Centre of a symmetrical distribution.
+ \item \textbf{Variance $\sigma^2$} - measure of spread of data around the mean. Not the same magnitude as the original data. Represented by $\sigma^2=\operatorname{Var}(x) = \sum (x=\mu)^2 \times p(x) = \sum (x-\mu)^2 \times \Pr(X=x)$. Alternatively: $\sigma^2 = \operatorname{Var}(X) = \sum x^2 \times p(x) - \mu^2$
+ \item \textbf{Standard deviation $\sigma$} - measure of spread in the original magnitude of the data. Found by taking square root of the variance: $\sigma =\operatorname{sd}(X)=\sqrt{\operatorname{Var}(X)}$
+ \end{itemize}
+
+ \subsection{Binomial distribution (Bernoulli trials)}
+
+ A type of discrete probability distribution. This distribution has the following characteristics:
+
+ \begin{enumerate}
+ \item Samples are taken from a population size that remains constant (\textit{sampling with replacement})
+ \item Every result or trial can be classed as either a \textit{success} or \textit{failure}
+ \item The probability of a succcess is the same from one trial to the next, notated by $p$
+ \item The probability of a failure is the complement of the probability of a success, notated by $1-p$
+ \item There are a finite number of trials that define the sample size, notated by $n$
+ \end{enumerate}
+
+ \subsubsection{Bernoulli trials}
+
+ Same properties as above. Number of successes in a finite number of Bernoulli trials is defined as the \textbf{binomial distribution}. The distribution can take the form:
+ \[X \sim \operatorname{Bi}(n,p) \]
+
+ Then, the probability values for each value of $X$ follow the rule:
+ \[ p(x) = \begin{bmatrix}n\\x\end{bmatrix}(p)^x(1-p)^{n-x} \]
+
+ \subsection{Continuous random distributions}
+
+ If the outcomes of an activity have a reference to \textit{continuous numeric} values (outcomes that can be measured), then the activity is associated with a \textbf{continuous probability distribution}. The probabilities are calculuated by finding the area under the graph between the required $x$ values (integrate).
+
+ The probability of a single \textit{outcome value} does not exist for continuous probability distributions.
+
+ \subsection{Continuous probability distributions}
+
+ If an experiment or activity has a \textbf{function} whose values are all positive ($\implies f(x) \ge 0 \forall x$), and for which the area under the graph between the lowest outcome value and the greatest outcome value is unity ($\implies \int^{\text{upper}}_{\text{lower}} f(x) \ dx = 1$), then it is called a \textbf{probability density function}.
+
+ Example probability density function: $f(x)=\begin{cases}k(9-x^2), & 0\le x \le 3\\0, &\text{elsewhere}\end{cases}$
+
+ \subsection{Normal distributions}
+
+ A very specific and special continuous probability distribution. Characteristics:
+ \begin{itemize}
+ \item Many sets of data occurring naturally and taken randomly will have a normal distribution
+ \item No single outcome value can be calculated
+ \item Probabilities are found between certain outcome values of the distribution
+ \item The values of the distribution are symmetrical around the mean ($\mu$) and form a bell-shaped curve
+ \item The distribution is best described using its central or mean value, $\mu$, and its measure of spread, $\sigma$
+ \item The distribution can take the form $X\sim N(\mu, \sigma^2)$
+ \end{itemize}
+
+ \pgfmathdeclarefunction{gauss}{2}{%
+ \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
+ }
+ \pgfmathdeclarefunction{sndist}{0}{%
+ \pgfmathparse{(1/sqrt(2*pi))*exp((-x^2)/2)}%
+ }
+
+ \begin{figure}
+ \begin{center}
+ \begin{tikzpicture}
+ \begin{axis}[every axis plot post/.append style={
+ mark=none,domain=-2:3,samples=50,smooth}, % All plots: from -2:2, 50 samples, smooth, no marks
+ axis x line=bottom, % no box around the plot, only x and y axis
+ axis y line=left, % the * suppresses the arrow tips
+ enlargelimits=upper] % extend the axes a bit to the right and top
+ \addplot {gauss(0,0.5)};
+ \addplot [mark=*, blue] coordinates {(-1,0.4)} node[text width=1cm, font=\footnotesize]{$\mu=0$ \\ $\sigma=0.5$};
+ \addplot [mark=*, red] coordinates {(2.5,0.4)} node[text width=2cm, font=\footnotesize]{$\mu=1$ \\ $\sigma=0.75$};
+ \addplot {gauss(1,0.75)};
+ \end{axis}
+ \end{tikzpicture}
+ \end{center}
+ \caption{Two \textit{general} normal distributions}
+ \end{figure}
+
+ \begin{tabularx}{\textwidth}{X|X}
+ \hline
+ \begin{center}General normal distribution\end{center} & \begin{center}Standard normal distribution\end{center} \\ \hline
+ \[ f(x) = \dfrac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left(\dfrac{x-\mu}{\sigma}\right)^2} \] & \[ f(x) = \dfrac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} \] \\ \hline
+ \begin{tikzpicture}
+ \begin{axis}[xtick={-2,0,2}, xticklabels={$\mu-3\sigma$,$\mu$,$\mu+3\sigma$}, every axis plot post/.append style={mark=none,domain=-3:3,samples=50,smooth}, % All plots: from -2:2, 50 samples, smooth, no marks
+ axis x line=bottom, % no box around the plot, only x and y axis
+ axis y line=none, % the * suppresses the arrow tips
+ enlargelimits=upper] % extend the axes a bit to the right and top
+ \addplot [orange] {gauss(0,0.75)};
+ \end{axis}
+ \end{tikzpicture}
+ &
+ \begin{tikzpicture}
+ \begin{axis}[every axis plot post/.append style={mark=none,domain=-4:4,samples=50,smooth}, % All plots: from -2:2, 50 samples, smooth, no marks
+ axis x line=bottom, % no box around the plot, only x and y axis
+ axis y line=none, % the * suppresses the arrow tips
+ enlargelimits=upper] % extend the axes a bit to the right and top
+ \addplot [purple] {sndist};
+ \end{axis}
+ \end{tikzpicture}
+ \\ \hline
+ \end{tabularx}
+
+\end{document}