}
\pagestyle{fancy}
+\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
\fancyhead[LO,LE]{Year 12 Specialist}
\fancyhead[CO,CE]{Andrew Lorimer}
\newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}%
\newcolumntype{Y}{>{\centering\arraybackslash}X}
-\definecolor{cas}{HTML}{e6f0fe}
+\definecolor{cas}{HTML}{cde1fd}
\definecolor{important}{HTML}{fc9871}
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\definecolor{light-gray}{HTML}{cccccc}
\newcommand{\arctg}{\mathop{\mathrm{arctg}}}
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-\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=darkgray, fontupper=\sffamily\bfseries}
\newtcolorbox{cas}{colframe=cas!75!black, fonttitle=\sffamily\bfseries, title=On CAS, left*=3mm}
+\newtcolorbox{theorembox}[1]{colback=green!10!white, colframe=blue!20!white, coltitle=black, fontupper=\sffamily, fonttitle=\sffamily, #1}
+\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=darkgray, fontupper=\sffamily\bfseries}
\begin{document}
+\title{\vspace{-22mm}Year 12 Specialist\vspace{-4mm}}
+\author{Andrew Lorimer}
+\date{}
+\maketitle
+\vspace{-9mm}
\begin{multicols}{2}
\section{Complex numbers}
\[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
-
\begin{align*}
\text{Cartesian form: } & a+bi\\
\text{Polar form: } & r\operatorname{cis}\theta
\subsection*{Operations}
- \definecolor{shade1}{HTML}{ffffff}
- \definecolor{shade2}{HTML}{e6f2ff}
- \definecolor{shade3}{HTML}{cce2ff}
- \begin{tabularx}{\columnwidth}{r|X|X}
+ \begin{tabularx}{\columnwidth}{|r|X|X|}
+ \hline
+ \rowcolor{cas}
& \textbf{Cartesian} & \textbf{Polar} \\
\hline
\(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
\hline
\(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
\hline
- \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
+ \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\) \\
+ \hline
\end{tabularx}
\subsubsection*{Scalar multiplication in polar form}
\[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
\subsection*{Conjugate}
-
+ \vspace{-7mm} \hfill \colorbox{cas}{\texttt{conjg(a+bi)}}
\begin{align*}
\overline{z} &= a \mp bi\\
&= r \operatorname{cis}(-\theta)
\end{align*}
- \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
-
\subsubsection*{Properties}
\begin{align*}
\overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
\overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
- \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
+ \overline{kz} &= k\overline{z} \> \forall \> k \in \mathbb{R}\\
z\overline{z} &= (a+bi)(a-bi)\\
&= a^2 + b^2\\
&= |z|^2
\frac{z_1}{z_2}&=z_1z_2^{-1}\\
&=\frac{z_1\overline{z_2}}{|z_2|^2}\\
&=\frac{(a+bi)(c-di)}{c^2+d^2}\\
- & \qquad \text{(rationalise denominator)}
+ & \text{then rationalise denominator}
\end{align*}
\subsection*{Polar form}
- \begin{align*}
- z&=r\operatorname{cis}\theta\\
- &=r(\cos \theta + i \sin \theta)
- \end{align*}
+ \[ r \operatorname{cis} \theta = r\left( \cos \theta + i \sin \theta \right) \]
\begin{itemize}
\item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
- \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
+ \item{\(\theta = \operatorname{arg}(z)\) \hfill \colorbox{cas}{\texttt{arg(a+bi)}}}
\item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
- \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
\item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
\item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
\end{itemize}
+ \begin{cas}
+ \-\hspace{1em}\verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|
+ \end{cas}
+
\subsection*{de Moivres' theorem}
- \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
+ \begin{theorembox}{}
+ \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
+ \end{theorembox}
\subsection*{Complex polynomials}
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 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 \(=0\)
+ Endpoints are included, even where \(\boldsymbol{\frac{dy}{dx}=0}\)
\end{warning}
\begin{table*}[ht]
\centering
- \begin{tabularx}{\textwidth}{rYYY}
+ \begin{tabularx}{\textwidth}{|r|Y|Y|Y|}
\hline
- \rowcolor{shade2}
+ \rowcolor{lblue}
& \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
\(\dfrac{dy}{dx}>0\) &
\end{table*}
\begin{itemize}
\item
- if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
- \((a, f(a))\) is a local min (curve is concave up)
+ \(f^\prime (a) = 0, \> f^{\prime\prime}(a) > 0\) \\
+ \textbf{local min} at \((a, f(a))\) (concave up)
\item
- if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
- \((a, f(a))\) is local max (curve is concave down)
+ \(f^\prime (a) = 0, \> f^{\prime\prime} (a) < 0\) \\
+ \textbf{local max} at \((a, f(a))\) (concave down)
\item
- if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
- inflection
+ \(f^{\prime\prime}(a) = 0\) \\
+ \textbf{point of inflection} at \((a, f(a))\)
\item
- if also \(f^\prime(a)=0\), then it is a stationary point of inflection
+ \(f^{\prime\prime}(a) = 0, \> f^\prime(a)=0\) \\
+ stationary point of inflection at \((a, f(a)\)
\end{itemize}
\subsection*{Implicit Differentiation}
\begin{cas}
Action \(\rightarrow\) Calculation \\
- \hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)} \hfill(returns \(y^\prime= \dots\))
+ \-\hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}
\end{cas}
\subsection*{Slope fields}
\subsection*{Parametric equations}
- For each point on \(\left( f(t), g(t) \right)\):
\begin{align*}
\dfrac{dy}{dt} &= \dfrac{dy}{dx} \cdot \dfrac{dx}{dt} \\
\therefore \dfrac{dy}{dx} &= \dfrac{\left(\dfrac{dy}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ provided } \dfrac{dx}{dt} \ne 0 \\
- \text{Also...} \\
\dfrac{d^2y}{dx^2} &= \dfrac{\left(\dfrac{dy^\prime}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ where } y^\prime = \dfrac{dy}{dx}
\end{align*}
\begin{cas}
Action \(\rightarrow\) Transformation:\\
- \hspace{1em} \texttt{expand(..., x)}
+ \-\hspace{1em} \texttt{expand(..., x)}
To reverse, use \texttt{combine(...)}
\end{cas}
\begin{cas}
\textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\
- Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
+ For restrictions, \texttt{Define\ f(x)=...} then \texttt{f(x)\textbar{}x\textgreater{}...}
\end{cas}
\subsection*{Applications of antidifferentiation}
\subsection*{Length of a curve}
- \[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
-
- \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
+ For length of \(f(x)\) from \(x=a \rightarrow x=b\):
+ \begin{align*}
+ &\text{Cartesian} \> & L &= \int^b_a \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} \> dx \\
+ &\text{Parametric} \> & L & = \int^b_a \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} \> dt
+ \end{align*}
\begin{cas}
\begin{enumerate}[label=\alph*), leftmargin=5mm]
\[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
- \subsubsection*{Addition of ordinates}
-
- \begin{itemize}
-
- \item
- when two graphs have the same ordinate, \(y\)-coordinate is double the
- ordinate
- \item
- when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
- (\(x\)-intercept)
- \item
- when one of the ordinates is 0, the resulting ordinate is equal to the
- other ordinate
- \end{itemize}
-
\subsection*{Fundamental theorem of calculus}
If \(f\) is continuous on \([a, b]\), then
\textbf{Degree} - highest power of highest derivative\\
e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
- \subsubsection*{Verifying solutions}
-
- Start with \(y=\dots\), and differentiate. Substitute into original
- equation.
+ \begin{warning}
+ To verify solutions, find \(\frac{dy}{dx}\) from \(y\) and substitute into original
+ \end{warning}
\subsubsection*{Function of the dependent
variable}
\subsubsection*{Velocity-time graphs}
- \begin{itemize}
- \item Displacement: \textit{signed} area between graph and \(t\) axis
- \item Distance travelled: \textit{total} area between graph and \(t\) axis
- \end{itemize}
+ \begin{description}[nosep, labelindent=0.5cm, leftmargin=0.5\columnwidth]
+ \item[Displacement:] \textit{signed} area
+ \item[Distance travelled:] \textit{total} area
+ \end{description}
\[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
\begin{center}
\renewcommand{\arraystretch}{1}
\begin{tabular}{ l r }
- \hline & no \\ \hline
- \(v=u+at\) & \(x\) \\
- \(v^2 = u^2+2as\) & \(t\) \\
- \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
- \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
- \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
- \end{tabular}
+ \hline & no \\ \hline
+ \(v=u+at\) & \(x\) \\
+ \(v^2 = u^2+2as\) & \(t\) \\
+ \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
+ \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
+ \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
+ \end{tabular}
\end{center}
\[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]