\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{makecell}
-\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}
- \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
- }}
\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*}
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}