methods / midyear-lecture.texon commit [englsih] typos in script (efaaf47)
   1\documentclass[a4paper]{article}
   2\usepackage[a4paper,margin=2cm]{geometry}
   3\usepackage{multicol}
   4\usepackage{amsmath}
   5\usepackage{amssymb}
   6\usepackage{enumitem}
   7\usepackage{tcolorbox}
   8\usepackage{fancyhdr}
   9\usepackage{pgfplots}
  10\usepackage{tabularx}
  11
  12\pagestyle{fancy}
  13\fancyhead[LO,LE]{Unit 3 Methods Revision Lecture}
  14\fancyhead[CO,CE]{Andrew Lorimer}
  15
  16\setlength\parindent{0pt}
  17
  18\begin{document}
  19
  20  \title{\large Year 12 Methods \\ \huge Unit 3 Revision Lecture \\ \large Monash University \\ presented by Kevin McMenamin}
  21  \author{Andrew Lorimer}
  22  \date{5 July 2019}
  23  \renewcommand{\abstractname}{}
  24  \maketitle
  25
  26  \section{Graphs}
  27
  28  \textbf{16 types of graph}---put in reference book:
  29  \begin{multicols}{2}
  30  \begin{enumerate}
  31    \item truncus
  32    \item hyperbola
  33    \item sqrt
  34    \item parabola
  35    \item cubic
  36    \item quartic
  37    \item linear
  38    \item circle
  39    \item semicircle
  40    \item tan
  41    \item sin
  42    \item cos
  43    \item log
  44    \item exp
  45    \item $x^{a \over b}$
  46    \item $x^{-a \over b}$
  47  \end{enumerate}
  48  \end{multicols}
  49
  50  \subsection{Power functions}
  51  
  52  \begin{itemize}
  53    \item In first quadrant, shape of graph for $x>0 \cap y>0$ is either $\sqrt{x}$ or $x^2$
  54  \end{itemize}
  55
  56  \subsection{Features of graphs}
  57  
  58  \begin{itemize}
  59    \item Asymptotes
  60    \item Intercepts
  61    \item Stationary points
  62    \item Endpoints
  63    \item Other critical points
  64    \item Continuous or discontinuous
  65  \end{itemize}
  66
  67  \begin{tcolorbox}[title=Key points]
  68  \begin{itemize}
  69    \item All transformations can be described by matrices
  70    \item Inverse is a transformation
  71    \item Memorise approximate values of $e,\>\pi,\>\sqrt{2},\>\sqrt{3}$
  72    \item Put 16 base graphs in reference book
  73  \end{itemize}
  74  \end{tcolorbox}
  75
  76  \section{Transformations}
  77
  78  Order: \qquad \textbf{Reflect $\longrightarrow$ Dilate $\longrightarrow$ Translate}
  79
  80  \subsection{Two forms}
  81  
  82  \begin{itemize}
  83    \item note $a$ and $b$ can be positive or negative
  84    \item check validity of solutions for logarithms
  85    \item results in transformed equation $y^\prime = f^\prime(x)$
  86  \end{itemize}
  87
  88  \[ y^\prime = a \cdot f(\dfrac{1}{b} (x^\prime - c)) + d \]
  89  \[
  90    \begin{bmatrix}
  91      x^\prime \\ y^\prime
  92    \end{bmatrix}
  93    =
  94    \begin{bmatrix}
  95      b & 0 \\
  96      0 & a
  97    \end{bmatrix}
  98    \begin{bmatrix}
  99      x \\ y
 100    \end{bmatrix}
 101    +
 102    \begin{bmatrix}
 103      c \\d
 104    \end{bmatrix}
 105  \]
 106
 107  \begin{tcolorbox}[title=Key points]
 108  \begin{itemize}
 109    \item All transformations can be described by matrices
 110    \item Inverse is a transformation
 111    \item Check validity of $\log_a x$ solutions/transformations
 112  \end{itemize}
 113  \end{tcolorbox}
 114
 115  \section{Calculus}
 116
 117  Possible questions:
 118  \begin{itemize}
 119    \item Average rate of change
 120    \item Instantaneous rate of change
 121    \item Tangent line
 122    \item Normal line
 123    \item Features of gradient function
 124      \begin{itemize}
 125        \item Degree
 126        \item Orientation
 127        \item Format
 128        \item Turning points
 129        \item Inflection points
 130        \item Asymptotes
 131      \end{itemize}
 132    \item Find original function from derivative\\
 133      $\longrightarrow$ \textit{Use information to find unknowns}
 134    \item Application questions - e.g. Pythagoras, trig. functions, measurement, given eqn
 135  \end{itemize}
 136
 137  \subsection{Integration}
 138
 139  \subsubsection{Polynomials}
 140
 141  \[ f(x) = \int ax^n \> dx = \dfrac{ax^{n+1}}{n+1}+c \>, \quad n \ne -1 \]
 142  \[f(x) = \int (ax+b)^n \> dx = \dfrac{(ax+b)^{n+1}}{a(n+1)}+c \>, \quad n \ne -1 \]
 143
 144  \subsubsection{Exponentials}
 145
 146  \[ f(x) = \int e^{ax+b} \> dx = \dfrac{e^{ax+b}}{a}+c \]
 147
 148  \subsubsection{Logarithms}
 149
 150  \textit{ignore modulus for methods}
 151  \[ f(x) = \int \dfrac{1}{x} \> dx = \ln|x| + c \]
 152  \[ f(x) = \int \dfrac{1}{ax+b} \> dx = \dfrac{1}{a} \ln|ax+b| + c \]
 153  \[ f(x) - \int \dfrac{h^\prime (x)}{h(x)} \> dx = \ln|h(x)|+c \tag{general form}\]
 154
 155  \subsubsection{Trigonometric functions}
 156
 157  \[ f(x) = \int \cos(ax+b) \> dx = \dfrac{1}{a} \sin (ax+b) + c \]
 158  \[ f(x) = \int \sin(ax+b) \> dx = -\dfrac{1}{a}\sin(ax+b) + c \]
 159  \[ f(x) = \int \sec^2(ax+b) \> dx = \dfrac{1}{a}\tan(ax+b) + c \]
 160
 161  \subsection{Area under curves}
 162
 163  \begin{itemize}
 164    \item \textbf{Upper rectangles} (overestimate) vs. \textbf{lower rectangles} (underestimate)
 165    \item Rotate (invert) graph to make it easier, e.g. $y=\sqrt{x} \longrightarrow x=y^2$
 166  \end{itemize}
 167
 168  \begin{tcolorbox}[title=Key points]
 169  \begin{itemize}
 170    \item For \textit{an} antiderivative, \qquad $+c \quad \forall \> c \in \mathbb{R}$ \qquad is also acceptable
 171    \item Practice multi-part problems e.g:
 172      \begin{enumerate}[label={\alph*)}]
 173        \item Let $f:\mathbb{R}\rightarrow\mathbb{R},\quad f(x)=x\sin x$. Find $f^\prime(x)$.
 174        \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 $a\pi + b$.
 175      \end{enumerate}
 176  \end{itemize}
 177  \end{tcolorbox}
 178
 179  \section{Probability}
 180
 181  \[ \Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B) \]
 182  \[ \Pr(A \cup B) = 0 \tag{mutually exclusive} \]
 183
 184  \subsection{Conditional probability}
 185
 186  \[ \Pr(A|B)=\dfrac{\Pr(A \cap B)}{\Pr(B)} \]
 187  \[ \Pr(A \cap B) = \Pr(A|B) \times \Pr(B) \tag{multiplication theorem} \]
 188  \[ \Pr(A \cap B) = \Pr(A) \times \Pr(B) \tag{independent events} \]
 189
 190  \subsection{Discrete random distributions}
 191
 192  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}.
 193
 194  \subsubsection{Discrete probability distributions}
 195  
 196  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.
 197
 198  \begin{itemize}
 199    \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.
 200    \item \textbf{Mean $\mu$} - measure of central tendency. \textit{Balance point} or \textit{expected value} of a distribution. Centre of a symmetrical distribution.
 201    \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: $\sigma^2 = \operatorname{Var}(X) = \sum x^2 \times p(x) - \mu^2$
 202    \item \textbf{Standard deviation $\sigma$} - measure of spread in the original magnitude of the data. Found by taking square root of the variance: $\sigma =\operatorname{sd}(X)=\sqrt{\operatorname{Var}(X)}$
 203  \end{itemize}
 204
 205  \subsection{Binomial distribution (Bernoulli trials)}
 206  
 207  A type of discrete probability distribution. This distribution has the following characteristics:
 208  
 209  \begin{enumerate}
 210    \item Samples are taken from a population size that remains constant (\textit{sampling with replacement})
 211    \item Every result or trial can be classed as either a \textit{success} or \textit{failure}
 212    \item The probability of a succcess is the same from one trial to the next, notated by $p$
 213    \item The probability of a failure is the complement of the probability of a success, notated by $1-p$
 214    \item There are a finite number of trials that define the sample size, notated by $n$
 215  \end{enumerate}
 216  
 217  \subsubsection{Bernoulli trials}
 218  
 219  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:
 220  \[X \sim \operatorname{Bi}(n,p) \]
 221  
 222  Then, the probability values for each value of $X$ follow the rule:
 223  \[ p(x) = \begin{bmatrix}n\\x\end{bmatrix}(p)^x(1-p)^{n-x} \]
 224
 225  \subsection{Continuous random distributions}
 226
 227  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).
 228
 229  The probability of a single \textit{outcome value} does not exist for continuous probability distributions.
 230
 231  \subsection{Continuous probability distributions}
 232
 233  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}.
 234
 235  Example probability density function: $f(x)=\begin{cases}k(9-x^2), & 0\le x \le 3\\0, &\text{elsewhere}\end{cases}$
 236
 237  \subsection{Normal distributions}
 238  
 239  A very specific and special continuous probability distribution. Characteristics:
 240  \begin{itemize}
 241    \item Many sets of data occurring naturally and taken randomly will have a normal distribution
 242    \item No single outcome value can be calculated
 243    \item Probabilities are found between certain outcome values of the distribution
 244    \item The values of the distribution are symmetrical around the mean ($\mu$) and form a bell-shaped curve
 245    \item The distribution is best described using its central or mean value, $\mu$, and its measure of spread, $\sigma$
 246    \item The distribution can take the form $X\sim N(\mu, \sigma^2)$
 247  \end{itemize}
 248
 249  \pgfmathdeclarefunction{gauss}{2}{%
 250    \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
 251  }
 252  \pgfmathdeclarefunction{sndist}{0}{%
 253    \pgfmathparse{(1/sqrt(2*pi))*exp((-x^2)/2)}%
 254  }
 255
 256  \begin{figure}
 257    \begin{center}
 258      \begin{tikzpicture}
 259        \begin{axis}[every axis plot post/.append style={
 260          mark=none,domain=-2:3,samples=50,smooth}, % All plots: from -2:2, 50 samples, smooth, no marks
 261          axis x line=bottom, % no box around the plot, only x and y axis
 262          axis y line=left, % the * suppresses the arrow tips
 263          enlargelimits=upper] % extend the axes a bit to the right and top
 264          \addplot {gauss(0,0.5)};
 265          \addplot [mark=*, blue] coordinates {(-1,0.4)} node[text width=1cm, font=\footnotesize]{$\mu=0$ \\ $\sigma=0.5$};
 266          \addplot [mark=*, red] coordinates {(2.5,0.4)} node[text width=2cm, font=\footnotesize]{$\mu=1$ \\ $\sigma=0.75$};
 267          \addplot {gauss(1,0.75)};
 268        \end{axis}
 269      \end{tikzpicture}
 270    \end{center}
 271    \caption{Two \textit{general} normal distributions}
 272  \end{figure}
 273
 274  \begin{tabularx}{\textwidth}{X|X}
 275    \hline
 276    \begin{center}General normal distribution\end{center} & \begin{center}Standard normal distribution\end{center} \\ \hline
 277    \[ f(x) = \dfrac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left(\dfrac{x-\mu}{\sigma}\right)^2} \] & \[ f(x) = \dfrac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} \] \\ \hline
 278    \begin{tikzpicture}
 279      \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
 280        axis x line=bottom, % no box around the plot, only x and y axis
 281        axis y line=none, % the * suppresses the arrow tips
 282        enlargelimits=upper] % extend the axes a bit to the right and top
 283        \addplot [orange] {gauss(0,0.75)};
 284      \end{axis}
 285    \end{tikzpicture}
 286    &
 287    \begin{tikzpicture}
 288      \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
 289        axis x line=bottom, % no box around the plot, only x and y axis
 290        axis y line=none, % the * suppresses the arrow tips
 291        enlargelimits=upper] % extend the axes a bit to the right and top
 292        \addplot [purple] {sndist};
 293      \end{axis}
 294    \end{tikzpicture}
 295    \\ \hline
 296  \end{tabularx}
 297
 298\end{document}