[spec] clean up
[notes.git] / spec / spec-collated.tex
index 0acbd2480314ce95ed711dab066e8768e3844e10..e747466de93d642db8a9f7785a72ac874570398d 100644 (file)
@@ -1,19 +1,45 @@
 \documentclass[a4paper]{article}
 \usepackage[a4paper,margin=2cm]{geometry}
 \usepackage{multicol}
+\usepackage{dblfloatfix}
 \usepackage{multirow}
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{harpoon}
 \usepackage{tabularx}
 \usepackage{makecell}
+\usepackage{enumitem}
+\usepackage[obeyspaces]{url}
 \usepackage[dvipsnames, table]{xcolor}
 \usepackage{blindtext}
 \usepackage{graphicx}
 \usepackage{wrapfig}
 \usepackage{tikz}
+\usepackage{tkz-fct}
 \usepackage{tikz-3dplot}
 \usepackage{pgfplots}
+\usetikzlibrary{arrows,
+    decorations,
+    decorations.markings,
+    decorations.text,
+    decorations.pathreplacing,
+    scopes
+}
+\usetikzlibrary{datavisualization.formats.functions}
+\usetikzlibrary{decorations.markings}
+\usepgflibrary{arrows.meta}
+\usetikzlibrary{decorations.markings}
+\usepgflibrary{arrows.meta}
+\usepackage{pst-plot}
+\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){$\vec{w}$}}
+}
+
 \usetikzlibrary{calc}
 \usetikzlibrary{angles}
 \usetikzlibrary{datavisualization.formats.functions}
@@ -23,7 +49,6 @@
 \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!
 \newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X}%
 \newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}%
 \definecolor{cas}{HTML}{e6f0fe}
+\definecolor{important}{HTML}{fc9871}
+\definecolor{dark-gray}{gray}{0.2}
 \linespread{1.5}
 \newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
 \newcommand{\tg}{\mathop{\mathrm{tg}}}
 \newcommand{\cotg}{\mathop{\mathrm{cotg}}}
 \newcommand{\arctg}{\mathop{\mathrm{arctg}}}
 \newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
-
-
-                  \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
-                  }}
+\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
+}}
+\usepackage{tcolorbox}
+\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}
+\usepackage{keystroke}
+\usepackage{listings}
+\usepackage{mathtools}
+\pgfplotsset{compat=1.16}
+\usepackage{subfiles}
+\usepackage{import}
+\setlength{\parindent}{0pt}
 \begin{document}
 
 \begin{multicols}{2}
                   \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};
-                      \addplot[->, gray, dotted, thick, domain=-35:35] {-1.5708};
+                      \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
                   \(f^{\prime\prime} = 0\))
 
 
-                  \pgfplotsset{every axis/.append style={
-                    axis x line=none,    % put the x axis in the middle
-                    axis y line=none,    % put the y axis in the middle
-                  }}
                   \begin{table*}[ht]
                     \centering
                     \begin{tabularx}{\textwidth}{rXXX}
                       & \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}[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}[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}[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}\\
+                      \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}[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}[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}[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}\\
+                            \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}[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}[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}[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}[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}\\
+                                  \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*}
 
       \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 }  % TODO need to fix centering here
+          \begin{tabular}{ l r }
               \hline & no \\ \hline
               \(v=u+at\) & \(x\) \\
               \(v^2 = u^2+2as\) & \(t\) \\
           &= \sqrt{v_x^2 + v_y^2 + v_z^2}
         \end{align*}
 
-        \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
+        \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}
 
         \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
         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}