From: Andrew Lorimer Date: Sun, 29 Sep 2019 12:28:54 +0000 (+1000) Subject: [spec] condense X-Git-Tag: yr12~22 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/8d92e2ded2bb847951c3e478ec0629e07fc2c09d?ds=sidebyside [spec] condense --- diff --git a/spec/calculus-rules.tex b/spec/calculus-rules.tex index 5371aaf..9ea9443 100644 --- a/spec/calculus-rules.tex +++ b/spec/calculus-rules.tex @@ -59,6 +59,7 @@ \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\ \hline \end{tabularx} +\rowcolors{2}{white}{white} \vspace{1em} Note \(\sin^{-1} \left(\dfrac{x}{a}\right) + \cos^{-1} \left(\dfrac{x}{a}\right)\) is constant \(\forall \> x \in (-a, a)\) diff --git a/spec/dynamics.tex b/spec/dynamics.tex index ac373c3..f8deab1 100644 --- a/spec/dynamics.tex +++ b/spec/dynamics.tex @@ -30,13 +30,13 @@ To convert force \(||\vec{OA}\) to angle-magnitude form, find component \(\perp\ \subsection*{Newton's laws} -\begin{tcolorbox} +\begin{theorembox}{} \begin{enumerate}[leftmargin=1mm] \item Velocity is constant without \(\Sigma F\) \item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\) \item Equal and opposite forces \end{enumerate} -\end{tcolorbox} +\end{theorembox} \subsubsection*{Weight} A mass of \(m\) kg has force of \(mg\) acting on it @@ -206,9 +206,14 @@ A mass of \(m\) kg has force of \(mg\) acting on it \begin{itemize} \item \textbf{Suspended pulley:} tension in both sections of rope are equal \\ - \(|a| = g \frac{m_1 - m_2}{m_1 + m_2}\) where \(m_1\) accelerates down \\ - With tension: - \[ \begin{cases}m_1 g - T = m_1 a\\ T - m_2 g = m_2 a\end{cases} \\ \implies m_1 g - m_2 g = m_1 a + m_2 a \] + \(|a| = g \dfrac{m_1 - m_2}{m_1 + m_2}\) where \(m_1\) accelerates down \\ + \[ + \left\{\begin{array}{lr} + m_1g-T = m_1a\\ + T-m_2g = m_2a + \end{array}\right\} + \implies m_1 g - m_2 g = m_1 a + m_2 a + \] \item \textbf{String pulling mass on inclined pane:} Resolve parallel to plane \[ T-mg \sin \theta = ma \] \item \textbf{Linear connection:} find acceleration of system first diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 1a67419..6f36688 100644 Binary files a/spec/spec-collated.pdf and b/spec/spec-collated.pdf differ diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex index e084adb..0b1d56e 100644 --- a/spec/spec-collated.tex +++ b/spec/spec-collated.tex @@ -70,6 +70,7 @@ } \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} @@ -96,11 +97,17 @@ \newcommand{\arctg}{\mathop{\mathrm{arctg}}} \newcommand{\arccotg}{\mathop{\mathrm{arccotg}}} -\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{-23mm}Year 12 Specialist\vspace{-5mm}} +\author{Andrew Lorimer} +\date{} +\maketitle +\vspace{-10mm} \begin{multicols}{2} \section{Complex numbers} @@ -114,10 +121,9 @@ \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 \(a+bi\)\\ @@ -128,7 +134,8 @@ \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} @@ -145,7 +152,6 @@ \overline{z} &= a \mp bi\\ &= r \operatorname{cis}(-\theta) \end{align*} - \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}} \subsubsection*{Properties} @@ -153,7 +159,7 @@ \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 @@ -185,7 +191,7 @@ \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} @@ -199,11 +205,14 @@ \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{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \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} \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\] @@ -882,20 +891,21 @@ 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 \(\dfrac{dy}{dx}>0\) & @@ -916,16 +926,17 @@ \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} @@ -939,7 +950,7 @@ \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} @@ -963,12 +974,10 @@ \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*} @@ -1066,7 +1075,7 @@ \begin{cas} Action \(\rightarrow\) Transformation:\\ - \hspace{1em} \texttt{expand(..., x)} + \-\hspace{1em} \texttt{expand(..., x)} To reverse, use \texttt{combine(...)} \end{cas} @@ -1075,7 +1084,7 @@ \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} @@ -1142,21 +1151,6 @@ \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\] - \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 @@ -1170,10 +1164,9 @@ \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} @@ -1213,23 +1206,23 @@ \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} \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]