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+\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} -- 2.47.1