[spec] finalise reference notes before SAC

diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf
index 6e6dc208768da9be5b522be916a5644b69d5ccf9..9714796099c07445488f0159bfd3c9f571620a91 100644 (file)
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 e2ccd0eb7d21af85877eb08eae93d311be6d0727..39ab74ecb586908cb18c2851ac08382cc756a136 100644 (file)
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -6,6 +6,7 @@
 \usepackage{amssymb}
 \usepackage{harpoon}
 \usepackage{tabularx}
 \usepackage{amssymb}
 \usepackage{harpoon}
 \usepackage{tabularx}

+\usepackage{makecell}

 \usepackage[dvipsnames, table]{xcolor}
 \usepackage{blindtext}
 \usepackage{graphicx}
 \usepackage[dvipsnames, table]{xcolor}
 \usepackage{blindtext}
 \usepackage{graphicx}


@@ -34,7 +35,19 @@
 \definecolor{cas}{HTML}{e6f0fe}
 \linespread{1.5}
 \newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
 \definecolor{cas}{HTML}{e6f0fe}
 \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
+                  }}

 \begin{document}
 
 \begin{multicols}{2}
 \begin{document}
 
 \begin{multicols}{2}


@@ -607,7 +620,19 @@
               \end{itemize}
 
               \subsubsection*{Secant}
               \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}

 
                 \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
 
 
                 \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
 


@@ -628,7 +653,20 @@
 
                 \subsubsection*{Cotangent}
 
 
                 \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}

 
                   \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
 
 
                   \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
 


@@ -689,23 +727,16 @@
 
                   \subsection*{Inverse circular functions}
 
 
                   \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)
 
 
                   Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
 


@@ -718,7 +749,14 @@
                   \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
                   \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
 
                   \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
                   \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};
+                      \addplot[->, gray, dotted, thick, domain=-35:35] {-1.5708};
+                    \end{axis}
+                  \end{tikzpicture}
+\columnbreak

                   \section{Differential calculus}
 
                   \subsection*{Limits}
                   \section{Differential calculus}
 
                   \subsection*{Limits}


@@ -810,7 +848,7 @@
                     \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}
                   \end{align*}
 
                     \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*}
 
                   \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*}
 


@@ -822,22 +860,37 @@
                   \(f^\prime(x)=0\)\\
                   \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
                   \(f^{\prime\prime} = 0\))
                   \(f^\prime(x)=0\)\\
                   \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
+
+
+                  \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}
+                      \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
+                      \(\frac{dy}{dx}>0\) &
+                      \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}\\
+                            \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}\\
+                                  \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}\\
+                                      \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
                       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


@@ -1066,7 +1119,7 @@
 
                   \[\int^b_a f(x) \> dx = F(b) - F(a)\]
                   \hfill where \(F = \int f \> dx\)
 
                   \[\int^b_a f(x) \> dx = F(b) - F(a)\]
                   \hfill where \(F = \int f \> dx\)

-
+                  

                   \subsection*{Differential equations}
 
                   \noindent\textbf{Order} - highest power inside derivative\\
                   \subsection*{Differential equations}
 
                   \noindent\textbf{Order} - highest power inside derivative\\


@@ -1085,10 +1138,7 @@
                   \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
                   \(e^c\) as \(A\).
 
                   \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). 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}
 
 
                   \subsubsection*{Mixing problems}