[spec] clean up
[notes.git] / spec / spec-collated.tex
index e2ccd0eb7d21af85877eb08eae93d311be6d0727..4adcd53d464c205ead2d43104ef40974dc4c6f51 100644 (file)
@@ -1,39 +1,94 @@
 \documentclass[a4paper]{article}
-\usepackage[a4paper,margin=2cm]{geometry}
-\usepackage{multicol}
-\usepackage{multirow}
+\usepackage[dvipsnames, table]{xcolor}
 \usepackage{amsmath}
 \usepackage{amssymb}
-\usepackage{harpoon}
-\usepackage{tabularx}
-\usepackage[dvipsnames, table]{xcolor}
 \usepackage{blindtext}
+\usepackage{dblfloatfix}
+\usepackage{enumitem}
+\usepackage{fancyhdr}
+\usepackage[a4paper,margin=2cm]{geometry}
 \usepackage{graphicx}
-\usepackage{wrapfig}
-\usepackage{tikz}
-\usepackage{tikz-3dplot}
+\usepackage{harpoon}
+\usepackage{import}
+\usepackage{keystroke}
+\usepackage{listings}
+\usepackage{makecell}
+\usepackage{mathtools}
+\usepackage{mathtools}
+\usepackage{multicol}
+\usepackage{multirow}
 \usepackage{pgfplots}
-\usetikzlibrary{calc}
-\usetikzlibrary{angles}
-\usetikzlibrary{datavisualization.formats.functions}
-\usetikzlibrary{decorations.markings}
+\usepackage{pst-plot}
+\usepackage{subfiles}
+\usepackage{tabularx}
+\usepackage{tcolorbox}
+\usepackage{tikz-3dplot}
+\usepackage{tikz}
+\usepackage{tkz-fct}
+\usepackage[obeyspaces]{url}
+\usepackage{wrapfig}
+
+
+\usetikzlibrary{%
+  angles,
+  arrows,
+  arrows.meta,
+  calc,
+  datavisualization.formats.functions,
+  decorations,
+  decorations.markings,
+  decorations.text,
+  decorations.pathreplacing,
+  decorations.text,
+  scopes
+}
+
+\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
+
 \usepgflibrary{arrows.meta}
-\usepackage{fancyhdr}
+\pgfplotsset{compat=1.16}
+\pgfplotsset{every axis/.append style={
+  axis x line=middle,    % put the x axis in the middle
+  axis y line=middle,    % put the y axis in the middle
+  axis line style={->}, % arrows on the axis
+  xlabel={$x$},          % default put x on x-axis
+  ylabel={$y$},          % default put y on y-axis
+}}
+
+\psset{dimen=monkey,fillstyle=solid,opacity=.5}
+\def\object{%
+    \psframe[linestyle=none,fillcolor=blue](-2,-1)(2,1)
+    \psaxes[linecolor=gray,labels=none,ticks=none]{->}(0,0)(-3,-3)(3,2)[$x$,0][$y$,90]
+    \rput{*0}{%
+        \psline{->}(0,-2)%
+        \uput[-90]{*0}(0,-2){$\vec{w}$}}
+}
+
 \pagestyle{fancy}
 \fancyhead[LO,LE]{Year 12 Specialist}
 \fancyhead[CO,CE]{Andrew Lorimer}
 
-\usepackage{mathtools}
-\usepackage{xcolor} % used only to show the phantomed stuff
 \renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
-\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
 \newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}}
 \newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}}
+\linespread{1.5}
+\setlength{\parindent}{0pt}
+\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
+
 \newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X}%
 \newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}%
+
 \definecolor{cas}{HTML}{e6f0fe}
-\linespread{1.5}
-\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
+\definecolor{important}{HTML}{fc9871}
+\definecolor{dark-gray}{gray}{0.2}
+
+\newcommand{\tg}{\mathop{\mathrm{tg}}}
+\newcommand{\cotg}{\mathop{\mathrm{cotg}}}
+\newcommand{\arctg}{\mathop{\mathrm{arctg}}}
+\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
+
+\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=important, fontupper=\sffamily\bfseries}
+\newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm}
 
 \begin{document}
 
               \end{itemize}
 
               \subsubsection*{Secant}
-           \begin{center}\includegraphics[width=0.7\columnwidth]{graphics/sec.png}\end{center}
+
+\begin{tikzpicture}
+  \begin{axis}[ytick={-1,1}, yticklabels={\(-1\), \(1\)}, xmin=-7,xmax=7,ymin=-3,ymax=3,enlargelimits=true, xtick={-6.2830, -3.1415, 3.1415, 6.2830},xticklabels={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
+%    \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
+    \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
+    \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
+    \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
+    \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1});
+    \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
+    \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
+\end{axis}
+    \node [black] at (7,3.5) {\(\cos x\)};
+\end{tikzpicture}
 
                 \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
 
 
                 \subsubsection*{Cotangent}
 
-                \begin{center}\includegraphics[width=0.7\columnwidth]{graphics/cot.png}\end{center}
+\begin{tikzpicture}
+  \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
+    \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
+\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
+\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
+\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
+\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
+    \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
+    \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
+    \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
+    \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
+    \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
+\end{axis}
+\end{tikzpicture}
 
                   \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
 
 
                   \subsection*{Inverse circular functions}
 
-                  \pgfplotsset{every axis/.append style={
-                    axis x line=middle,    % put the x axis in the middle
-                    axis y line=middle,    % put the y axis in the middle
-                    axis line style={<->}, % arrows on the axis
-                    xlabel={$x$},          % default put x on x-axis
-                    ylabel={$y$},          % default put y on y-axis
-                    }}
-
-% arrows as stealth fighters
-\tikzset{>=stealth}
-
-\begin{tikzpicture}
-  \begin{axis}[domain = -1:1, samples = 500]
-    \addplot[color = red]  {rad(asin(x))} node [pos=0.25, below right] {\(\sin^{-1}x\)};
-    \addplot[color = blue] {rad(acos(x))} node [pos=0.25, below left] {\(\cos^{-1}x\)};
-  \end{axis}
-\end{tikzpicture}
+                  \begin{tikzpicture}
+                    \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
+                      \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
+                      \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
+                      \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
+                      \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
+                      \addplot[mark=*, blue] coordinates {(1,0)};
+                      \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
+                    \end{axis}
+                  \end{tikzpicture}\\
 
                   Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
 
                   \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
                   \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
 
-
+                  \begin{tikzpicture}
+                    \begin{axis}[yticklabel style={yshift=1.0pt, anchor=north east},x=0.1cm, y=1cm, ymax=2, ymin=-2, xticklabels={}, ytick={-1.5708,1.5708},yticklabels={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
+                      \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
+                      \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
+                      \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
+                    \end{axis}
+                  \end{tikzpicture}
+\columnbreak
                   \section{Differential calculus}
 
                   \subsection*{Limits}
                     \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}
                   \end{align*}
 
-                  \subsubsection*{Second derivative}
+                  \subsection*{Second derivative}
                   \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
                   \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
 
                   \(f^\prime(x)=0\)\\
                   \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
                   \(f^{\prime\prime} = 0\))
-                  %\begin{table*}[ht]
-                  %\centering
-                  %  \begin{tabularx}{\textwidth}{XXXX}
-                  %\hline
-                  %    \rowcolor{shade2}
-                  %    & \(\dfrac{d^2 y}{dx^2} > 0\)  & \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
-                  %\hline
-                  %    \(\frac{dy}{dx}>0\) & \begin{tikzpicture} \draw[domain=1:2,smooth,variable=\x,blue] plot ({\x},{(1/10)*\x*\x*\x}) plot ({\x},{0.675*\x-0.677}); \end{tikzpicture} & cell 3\\
-                  %cell 1 & cell 2 & cell 3\\
-                  %\hline
-                  %\end{tabularx}
-                  %\end{table*}
-
-
-\begin{itemize}
-  \item
+
+
+                  \begin{table*}[ht]
+                    \centering
+                    \begin{tabularx}{\textwidth}{rXXX}
+                      \hline
+                      \rowcolor{shade2}
+                      & \centering\(\dfrac{d^2 y}{dx^2} > 0\)  & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
+                      \hline
+                      \(\dfrac{dy}{dx}>0\) &
+                      \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-3,  xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x))};  \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}&
+                        \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0.1, xmax=4,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))};  \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}&
+                          \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1.5,  xmax=1.5,   scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\
+                            \hline
+                            \(\dfrac{dy}{dx}<0\) &
+                            \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {(1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}&
+                              \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0,  xmax=1.5, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(2-x*x)^(1/2)};  \addplot[red] {-x+2}; \end{axis}\end{tikzpicture} \\Falling (concave down)}&
+                                \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=1.5,  xmax=4.5,   scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {-x+3.1415}; \end{axis}\end{tikzpicture} \\Falling inflection point}\\
+                                  \hline
+                                  \(\dfrac{dy}{dx}=0\)&
+                                  \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}&                       \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x))}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}&
+                                    \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture}  \\Stationary inflection point}\\
+                                      \hline
+                    \end{tabularx}
+                  \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)
                     \item
 
                   \subsubsection*{Gradient at a point on parametric curve}
 
-                  \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0\]
+                  \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]
 
                   \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
 
 
                   \[\int^b_a f(x) \> dx = F(b) - F(a)\]
                   \hfill where \(F = \int f \> dx\)
-
+                  
                   \subsection*{Differential equations}
 
                   \noindent\textbf{Order} - highest power inside derivative\\
                   \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
                   \(e^c\) as \(A\).
 
-                  \begin{table*}[ht]
-                    \centering
-                    \includegraphics[width=0.7\textwidth]{graphics/second-derivatives.png}
-                  \end{table*}
+
 
                   \subsubsection*{Mixing problems}
 
 
                   \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
 
-                \end{multicols}
-              \end{document}
+              
+    \section{Kinematics \& Mechanics}
+
+      \subsection*{Constant acceleration}
+
+      \begin{itemize}
+        \item \textbf{Position} - relative to origin
+        \item \textbf{Displacement} - relative to starting point
+      \end{itemize}
+
+      \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}
+
+      \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
+
+        \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}
+        \end{center}
+
+        \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
+        \begin{align*}
+          \text{speed} &= |{\text{velocity}}| \\
+          &= \sqrt{v_x^2 + v_y^2 + v_z^2}
+        \end{align*}
+
+        \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
+        \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
+
+        \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
+        \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
+
+      \subsection*{Vector functions}
+
+        \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
+
+        \begin{itemize}
+          \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
+          \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
+          \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
+        \end{itemize}
+
+      \subsection*{Vector calculus}
+
+      \subsubsection*{Derivative}
+
+        Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
+        \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
+
+      \subfile{dynamics}
+      \subfile{statistics}
+  \end{multicols}
+\end{document}