[methods] corrections & layout of collated notes

diff --git a/methods/calculus.tex b/methods/calculus.tex
new file mode 100644 (file)
index 0000000..4478faa
--- /dev/null
+++ b/methods/calculus.tex
@@ -0,0 +1,169 @@
+\section{Calculus}
+
+\subsection*{Average rate of change}
+
+\[m \operatorname{of} x \in [a,b] = \dfrac{f(b)-f(a)}{b - a} = \frac{dy}{dx}\]
+
+\colorbox{cas}{On CAS:} Action \(\rightarrow\) Calculation
+\(\rightarrow\) \texttt{diff}
+
+\subsection*{Average value}
+
+\[ f_{\text{avg}} = \dfrac{1}{b-a} \int^b_a f(x) \> dx \]
+
+\subsection*{Instantaneous rate of change}
+
+\textbf{Secant} - line passing through two points on a curve\\
+\textbf{Chord} - line segment joining two points on a curve
+
+\subsection*{Limit theorems}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\tightlist
+\item
+  For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
+\item
+  \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
+\item
+  \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
+\item
+  \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
+\end{enumerate}
+
+A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\).
+
+\subsection*{First principles derivative}
+
+\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\]
+
+Not differentiable at:
+\begin{itemize}
+\tightlist
+\item
+  discontinuous points
+\item
+  sharp point/cusp
+\item
+  vertical tangents (\(\infty\) gradient)
+\end{itemize}
+
+\subsection*{Tangents \& gradients}
+
+\textbf{Tangent line} - defined by \(y=mx+c\) where
+\(m={dy \over dx}\)\\
+\textbf{Normal line} - \(\perp\) tangent
+(\(m_{{tan}} \cdot m_{\operatorname{norm}} = -1\))\\
+\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\)
+
+\colorbox{cas}{On CAS:} \\ Action \(\rightarrow\) Calculation
+\(\rightarrow\) Line \(\rightarrow\) \texttt{tanLine} or \texttt{normal}
+
+\subsection*{Strictly increasing/decreasing}
+
+For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
+
+\begin{itemize}
+\tightlist
+\item
+  \textbf{strictly increasing}\\ where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
+\item
+  \textbf{strictly decreasing}\\ where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
+\item
+  Endpoints are included, even where gradient \(=0\)
+\end{itemize}
+
+\columnbreak
+
+\subsubsection*{Solving on CAS}
+
+\colorbox{cas}{\textbf{In main}}: type function. Interactive
+\(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) (Normal
+\textbar{} Tan line)\\
+\colorbox{cas}{\textbf{In graph}}: define function. Analysis
+\(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line).
+Type \(x\) value to solve for a point. Return to show equation for line.
+
+\subsection*{Stationary points}
+
+\begin{align*}
+  \textbf{Stationary point:} && f^\prime(x) &= 0 \\
+  \textbf{Point of inflection:} && f^{\prime\prime} &= 0
+\end{align*}
+
+                  \begin{tikzpicture}
+                    \begin{axis}[xmin=-21, xmax=21, ymax=1400, ymin=-1000, ticks=none, axis lines=middle]
+                      \addplot[color=red, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {x^3-3*x^2-144*x+432} node [black, pos=1, right] {\(f(x)\)};
+                      \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {3*x^2-6*x-144} node [black, pos=1, right] {\(f^\prime(x)\)};
+                      \addplot[mark=*, blue] coordinates {(1,286)} node[above right, align=left, font=\footnotesize]{inflection \\ (falling)} ;
+                      \addplot[mark=*, orange] coordinates {(-6,972)} node[above left, align=right, font=\footnotesize]{stationary \\ (local max)} ;
+                      \addplot[mark=*, orange] coordinates {(8,-400)} node[below, align=left, font=\footnotesize]{stationary \\ (local min)} ;
+                    \end{axis}
+                  \end{tikzpicture}\\
+                  \begin{tikzpicture}
+                    \begin{axis}[enlargelimits=true, xmax=3.5, ticks=none, axis lines=middle]
+                      \addplot[color=blue, smooth, thick] gnuplot [domain=0.74:3,unbounded coords=jump,samples=500] {(x-2)^3+2} node [black, pos=0.9, left] {\(f(x)\)};
+                      \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=1:3,unbounded coords=jump,samples=500] {3*(x-2)^2} node [black, pos=0.9, right] {\(f^\prime(x)\)};
+                      \addplot[mark=*, purple] coordinates {(2,2)} node[below right, align=left, font=\footnotesize]{stationary \\ inflection} ;
+                    \end{axis}
+                  \end{tikzpicture}\\
+\pagebreak
+\subsection*{Derivatives}
+
+\definecolor{shade1}{HTML}{ffffff}
+\definecolor{shade2}{HTML}{F0F9E4}
+\rowcolors{1}{shade1}{shade2}
+                  \renewcommand{\arraystretch}{1.4}
+                  \begin{tabularx}{\columnwidth}{rX}
+                    \hline
+                    \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\
+                    \hline
+                    \(\sin x\) & \(\cos x\)\\
+                    \(\sin ax\) & \(a\cos ax\)\\
+                    \(\cos x\) & \(-\sin x\)\\
+                    \(\cos ax\) & \(-a \sin ax\)\\
+                    \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
+                    \(e^x\) & \(e^x\)\\
+                    \(e^{ax}\) & \(ae^{ax}\)\\
+                    \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
+                    \(\log_e x\) & \(\dfrac{1}{x}\)\\
+                    \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
+                    \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
+                    \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
+                    \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
+                    \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\
+                    \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
+                    \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\
+                    \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\
+                    \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\
+                    \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
+                    \hline
+                  \end{tabularx}
+                  \columnbreak
+\subsection*{Antiderivatives}
+\rowcolors{1}{shade1}{cas}
+                  \renewcommand{\arraystretch}{1.4}
+                  \begin{tabularx}{\columnwidth}{rX}
+                    \hline
+                    \(f(x)\) & \(\int f(x) \cdot dx\) \\
+                    \hline
+                    \(k\) (constant) & \(kx + c\)\\
+                    \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
+                    \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
+                    \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
+                    \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
+                    \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
+                    \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
+                    \(e^k\) & \(e^kx + c\)\\
+                    \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
+                    \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
+                    \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
+                    \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
+                    \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
+                    \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
+                    \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
+                    \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\
+                    \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
+                    \hline
+                  \end{tabularx}
+

diff --git a/methods/circ-functions.tex b/methods/circ-functions.tex
index a2d85bfbf0333cb37b647aa76c2b84b468a15261..75807917c13a6af9ac95a0b4f0ff84c699451711 100644 (file)
--- a/methods/circ-functions.tex
+++ b/methods/circ-functions.tex
@@ -91,8 +91,8 @@
                   \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] {\(\tan^{-1}x\)};
                   \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}
 
                     \end{axis}
                   \end{tikzpicture}
 


@@ -122,7 +122,7 @@ where:
   \item Asymptotes at \(x=\frac{(2k+1)\pi}{2n}\)
 \end{description}
 
   \item Asymptotes at \(x=\frac{(2k+1)\pi}{2n}\)
 \end{description}
 

-\textbf{Asymptotes should always have equations and arrow pointing up}
+\textbf{Asymptotes should always have equations}

 
 \subsection*{Solving trig equations}
 
 
 \subsection*{Solving trig equations}
 

diff --git a/methods/methods-collated.pdf b/methods/methods-collated.pdf
index 808c3b5e4c720d8bce63ecbe446dba546cddd599..acdd213f7b5a629d240207140dbf1d93941fdadf 100644 (file)
Binary files a/methods/methods-collated.pdf and b/methods/methods-collated.pdf differ

diff --git a/methods/methods-collated.tex b/methods/methods-collated.tex
index b853626a19fc5b180167e9e4869a8a7e859f7a34..618f4437d4a7e90fb0de7a535fc5be954f946a90 100644 (file)
--- a/methods/methods-collated.tex
+++ b/methods/methods-collated.tex
@@ -8,6 +8,7 @@
 \usepackage{amssymb}
 \usepackage{harpoon}
 \usepackage{tabularx}
 \usepackage{amssymb}
 \usepackage{harpoon}
 \usepackage{tabularx}

+\usepackage{tabu}

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


@@ -28,6 +29,7 @@
 \pagestyle{fancy}
 \fancyhead[LO,LE]{Year 12 Methods}
 \fancyhead[CO,CE]{Andrew Lorimer}
 \pagestyle{fancy}
 \fancyhead[LO,LE]{Year 12 Methods}
 \fancyhead[CO,CE]{Andrew Lorimer}

+\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page

 \providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
 \setlength{\parindent}{0cm}
 \usepackage{mathtools}
 \providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
 \setlength{\parindent}{0cm}
 \usepackage{mathtools}


@@ -54,251 +56,241 @@
   xlabel={$x$},          % axes labels
   ylabel={$y$},
 }}
   xlabel={$x$},          % axes labels
   ylabel={$y$},
 }}

+

 \begin{document}
 
 \begin{document}
 

-\title{\vspace{-2cm}\hrule\vspace{0.4cm} Year 12 Methods}
+\title{\vspace{-20mm}Year 12 Methods}

 \author{Andrew Lorimer}
 \date{}
 \maketitle
 
 \begin{multicols}{2}
 
 \author{Andrew Lorimer}
 \date{}
 \maketitle
 
 \begin{multicols}{2}
 

-\section{Functions}
-
-\begin{itemize}
-  \tightlist
-  \item vertical line test
-  \item each \(x\) value produces only one \(y\) value
-\end{itemize}
-
-\subsection*{One to one functions}
+  \section{Functions}
+
+  \begin{itemize}
+      \tightlist
+    \item vertical line test
+    \item each \(x\) value produces only one \(y\) value
+  \end{itemize}
+
+  \subsection*{One to one functions}
+
+  \begin{itemize} \tightlist
+    \item
+      \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
+      \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
+      \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
+      \(x^3\) is)
+    \item
+      horizontal line test
+    \item
+      if not one to one, it is many to one
+  \end{itemize}
+
+      \subsection*{Odd and even functions}
+
+      \begin{align*}
+        \text{Even:}&& f(x)  &= f(-x) \\
+        \text{Odd:} && -f(x) &= f(-x)
+      \end{align*}
+
+      Even \(\implies\) symmetrical across \(y\)-axis \\
+      \(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
+      For \(x^n\), parity of \(n \equiv\) parity of function
+
+      \begin{tabularx}{\columnwidth}{XX}
+        \textbf{Even:} & \textbf{Odd:} \\
+        \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^2)};  \end{axis}\end{tikzpicture} &
+          \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^3)};  \end{axis}\end{tikzpicture}
+      \end{tabularx}
+
+  \subsection*{Inverse functions}
+
+  \begin{itemize} \tightlist
+    \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
+    \item \(f\) must be one to one
+    \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
+    \item Represents reflection across \(y=x\)
+    \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
+    \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
+      \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
+    \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
+  \end{itemize}
+
+  \subsubsection*{Finding \(f^{-1}\)}
+
+  \begin{enumerate} \tightlist
+    \item Let \(y=f(x)\)
+    \item Swap \(x\) and \(y\) (``take inverse''
+    \item Solve for \(y\) \\
+      Sqrt: state \(\pm\) solutions then restrict
+    \item State rule as \(f^{-1}(x)=\dots\)
+    \item For inverse \emph{function}, state in function notation
+  \end{enumerate}
+      
+  \subsection*{Simultaneous equations (linear)}
+
+  \begin{itemize} \tightlist
+    \item \textbf{Unique solution} - lines intersect at point
+    \item \textbf{Infinitely many solutions} - lines are equal
+    \item \textbf{No solution} - lines are parallel
+  \end{itemize}
+
+  \subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
+    where all coefficients are known except for one, and \(a, b\) are known
+
+    \begin{enumerate} \tightlist
+      \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  =  \begin{bmatrix} a \\ b \end{bmatrix}\)
+        \item Find determinant of first matrix: \(\Delta = ps-qr\)
+        \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
+          or let \(\Delta \ne 0\) for one unique solution.
+        \item Solve determinant equation to find variable \\
+          \textbf{For infinite/no solutions:}
+        \item Substitute variable into both original equations
+        \item Rearrange equations so that LHS of each is the same
+        \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
+          \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
+    \end{enumerate}
+
+    \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
+
+    \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
+
+      \begin{itemize} \tightlist
+        \item Use elimination
+        \item Generate two new equations with only two variables
+        \item Rearrange \& solve
+        \item Substitute one variable into another equation to find another variable
+      \end{itemize}
+
+\subsection*{Piecewise functions}
+
+\[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
+
+\textbf{Open circle:} point included\\
+\textbf{Closed circle:} point not included
+
+\subsection*{Operations on functions}
+
+For \(f \pm g\) and \(f \times g\):
+\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
+
+Addition of linear piecewise graphs: add \(y\)-values at key points
+
+Product functions:

 
 \begin{itemize}
 \tightlist
 \item
 
 \begin{itemize}
 \tightlist
 \item

-  \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
-  \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
-  \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
-  \(x^3\) is)
+  product will equal 0 if \(f=0\) or \(g=0\)

 \item
 \item

-  horizontal line test
-\item
-  if not one to one, it is many to one
+  \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)

 \end{itemize}
 
 \end{itemize}
 

-\subsection*{Finding inverse functions \(f^{-1}\)}
-
-\begin{itemize}
-\tightlist
-\item
-  if \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
-\item
-  reflection across \(y-x\)
-\item
-  \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
-\item
-  inverse \(\ne\) inverse \emph{function} (i.e.~inverse must pass
-  vertical line test)\\
-  \(\implies f^{-1}(x)\) exists \(\iff f(x)\) is one to one
-\item
-  \(f^{-1}(x)=f(x)\) intersections may lie on line \(y=x\)
-\end{itemize}
-
-\subsubsection*{Requirements for showing working for \(f^{-1}\)}
-
-\begin{enumerate}
-\def\labelenumi{\arabic{enumi}.}
-\tightlist
-\item
-  start with \emph{``let \(y=f(x)\)''}
-\item
-  must state \emph{``take inverse''} for line where \(y\) and \(x\) are
-  swapped
-\item
-  do all working in terms of \(y=\dots\)
-\item
-  for sqrt, state \(\pm\) solutions then show restricted
-\item
-  for inverse \emph{function}, state in function notation
-\end{enumerate}
-\subsubsection*{Solving
-\(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\)
-for \(\{0,1,\infty\}\)
-solutions}
-
-where all coefficients are known except for one, and \(a, b\) are known
-
-\begin{enumerate}
-\tightlist
-\item
-  Write as matrices:
-  \(\begin{bmatrix}p & q \\ r & s \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  =  \begin{bmatrix} a \\ b \end{bmatrix}\)
-\item
-  Find determinant of first matrix: \(\Delta = ps-qr\)
-\item
-  Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
-  or let \(\Delta \ne 0\) for one unique solution.
-\item
-  Solve determinant equation to find variable \\
-    \textbf{For infinite/no solutions:}
-\item
-  Substitute variable into both original equations
-\item
-  Rearrange equations so that LHS of each is the same
-\item
-  \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\)
-  (\(\infty\) solns)\\
-  \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0
-  solns)
-\end{enumerate}
-
-\colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
-
-\subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
-
-\begin{itemize}
-\tightlist
-\item
-  Use elimination
-\item
-  Generate two new equations with only two variables
-\item
-  Rearrange \& solve
-\item
-  Substitute one variable into another equation to find another variable
-\end{itemize}
-\subsection*{Odd and even functions}
-
-Even when \(f(x) = -f(x)\)\\
-Odd when \(-f(x) = f(-x)\)
-
-Function is even if it is symmetrical across \(y\)-axis
-\hspace{5em}\(\implies f(x)=f(-x)\)\\
-Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
-
-\begin{tabularx}{\columnwidth}{XX}
-  \textbf{Even:} & \textbf{Odd:} \\
-  \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^2)};  \end{axis}\end{tikzpicture} &
-  \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^3)};  \end{axis}\end{tikzpicture}
-\end{tabularx}
-\pagebreak
-                  \pgfplotsset{every axis/.append style={
-                    xlabel=,    % put the x axis in the middle
-                    ylabel=,    % put the y axis in the middle
-                  }}
-                  \begin{table*}[ht]
-                    \centering
-                    \begin{tabularx}{\textwidth}{r|X|X}
-                      & \(n\) is even & \(n\) is odd \\ \hline
-                      \(x^n, n \in \mathbb{Z}^+\) & 
-                      \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^2)};  \end{axis}\end{tikzpicture}} &
-                      \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^3)};  \end{axis}\end{tikzpicture}} \\
-                      \(x^n, n \in \mathbb{Z}^-\) &
-                      \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-4,  xmax=4, ymax=8, ymin=-0, scale=0.4, smooth] \addplot[orange, mark=none, samples=100] {(x^(-2))};  \end{axis}\end{tikzpicture}} &
-                        \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth] \addplot[orange, mark=none] {(x^(-1))};  \end{axis}\end{tikzpicture}} \\
-                      \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
-                      \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-1,  xmax=5, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^(1/2))};  \end{axis}\end{tikzpicture}} &
-                        \makecell{\\\begin{tikzpicture}
+\subsection*{Composite functions}
+
+\((f \circ g)(x)\) is defined iff
+\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
+
+
+      \pgfplotsset{every axis/.append style={ ticks=none, xlabel=, ylabel=, }} % remove axis labels & ticks
+      \begin{table*}[ht]
+        \centering
+        \begin{tabu} to \textwidth {@{} X[0.3,r] *2{|X[c,m]}@{}}
+          & \(n\) is even & \(n\) is odd \\ \tabucline{1pt} 
+          \(x^n, n \in \mathbb{Z}^+\) & 
+          \vspace{1em}\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^2)};  \end{axis}\end{tikzpicture} &
+            \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^3)};  \end{axis}\end{tikzpicture} \\
+              \(x^n, n \in \mathbb{Z}^-\) &
+              \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-4,  xmax=4, ymax=8, ymin=-0, scale=0.4, smooth] \addplot[orange, mark=none, samples=100] {(x^(-2))};  \end{axis}\end{tikzpicture} &
+                \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none, domain=-3:-0.1] {(x^(-1))}; \addplot[orange, mark=none, domain=0.1:3] {(x^(-1))};  \end{axis}\end{tikzpicture} \\
+                  \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
+                  \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-1,  xmax=5, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^(1/2))};  \end{axis}\end{tikzpicture} &
+                    \begin{tikzpicture}

                       \begin{axis}[enlargelimits=false, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, ymin=-3, ymax=3, smooth, scale=0.4]
                       \begin{axis}[enlargelimits=false, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, ymin=-3, ymax=3, smooth, scale=0.4]

-\addplot [orange,domain=-2:2,samples=1000,no markers] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
-\end{axis}
-                        \end{tikzpicture}}
-                    \end{tabularx}
-                  \end{table*}
-                  \pgfplotsset{every axis/.append style={
-                    xlabel=\(x\),    % put the x axis in the middle
-                    ylabel=\(y\),    % put the y axis in the middle
-                  }}
-
-\section{Polynomials}
-
-\subsection*{Quadratics}
-
-\[ x^2 + bx + c = (x+m)(x+n) \]
-\hfill where \(mn=c, \> m+n=b\)
-
-\begin{align*}
-  \hline
-  \textbf{Difference} && a^2 - b^2 &= (a-b)(a+b) \\[2ex]
-  \textbf{Perfect sq.} && a^2 \pm 2ab + b^2 &= (a \pm b^2) \\[2ex]
-  \textbf{Completing} && x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
-  && ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a} \\[2ex]
-  \textbf{Quadratic} && x &= \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \\
-  && & \text{where} \Delta=b^2-4ac \\
-  \hline
-\end{align*}
-
-\subsection*{Cubics}
-
-\textbf{Difference of cubes:} \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)\\
-\textbf{Sum of cubes:} \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)\\
-\textbf{Perfect cubes:} \(a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3\)
-
-\[ y=a(bx-h)^3 + c \]
-
-\begin{itemize}
-\tightlist
-\item
-  \(m=0\) at \emph{stationary point of inflection}
-  (i.e.~(\({h \over b}, k)\))
-\item
-  in form \(y=(x-a)^2(x-b)\), local max at \(x=a\), local min at \(x=b\)
-\item
-  in form \(y=a(x-b)(x-c)(x-d)\): \(x\)-intercepts at \(b, c, d\)
-\item
-  in form \(y=a(x-b)^2(x-c)\), touches \(x\)-axis at \(b\), intercept at
-  \(c\)
-\end{itemize}
-
-\subsection*{Linear and quadratic
-graphs}
-
-\subsubsection*{Forms of linear
-equations}
-
-\begin{itemize}
-\tightlist
-  \item \(y=mx+c\)
-  \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
-  \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
-\end{itemize}
-
-\subsection*{Line properties}
-
-Parallel lines: \(m_1 = m_2\)\\
-Perpendicular lines: \(m_1 \times m_2 = -1\)\\
-Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
-
-\subsection*{Quartic graphs}
-
-\subsubsection*{Forms of quartic
-equations}
-
-\(y=ax^4\)\\
-\(y=a(x-b)(x-c)(x-d)(x-e)\)\\
-\(y=ax^4+cd^2 (c \ge 0)\)\\
-\(y=ax^2(x-b)(x-c)\)\\
-\(y=a(x-b)^2(x-c)^2\)\\
-\(y=a(x-b)(x-c)^3\)
-
-\subsection*{Simultaneous equations
-(linear)}
-
-\begin{itemize}
-\tightlist
-\item
-  \textbf{Unique solution} - lines intersect at point
-\item
-  \textbf{Infinitely many solutions} - lines are equal
-\item
-  \textbf{No solution} - lines are parallel
-\end{itemize}
-
-
-\input{temp/transformations}
-\input{temp/stuff}
-\input{circ-functions}
-\input{temp/calculus}
-
-\end{multicols}
-\end{document}
+                        \addplot [orange,domain=-2:2,samples=1000,no markers] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
+                      \end{axis}
+                    \end{tikzpicture}
+        \end{tabu}
+        \hrule
+      \end{table*}
+      \pgfplotsset{every axis/.append style={ xlabel=\(x\), ylabel=\(y\) }} % put axis labels back
+
+      \section{Polynomials}
+
+      \subsection*{Linear equations}
+
+      \subsubsection*{Forms}
+
+      \begin{itemize}
+          \tightlist
+        \item \(y=mx+c\)
+        \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
+        \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
+      \end{itemize}
+
+      \subsubsection*{Line properties}
+
+      Parallel lines: \(m_1 = m_2\)\\
+      Perpendicular lines: \(m_1 \times m_2 = -1\)\\
+      Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
+
+      \subsection*{Quadratics}
+      \setlength{\abovedisplayskip}{1pt}
+      \setlength{\belowdisplayskip}{1pt}
+      \[ x^2 + bx + c = (x+m)(x+n) \]
+      \hfill where \(mn=c, \> m+n=b\)
+
+      \textbf{Difference of squares}
+      \[ a^2 - b^2 = (a-b)(a+b) \]
+      \textbf{Perfect squares}
+      \[ a^2 \pm 2ab + b^2 = (a \pm b^2) \]
+      \textbf{Completing the square}
+      \begin{align*}
+        x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
+        ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a}
+      \end{align*}
+      \textbf{Quadratic formula}
+      \[ x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \]
+      \hfill (Discriminant \(\Delta=b^2-4ac\))
+
+      \subsection*{Cubics}
+
+      \textbf{Difference of cubes}
+      \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]
+      \textbf{Sum of cubes}
+      \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
+      \textbf{Perfect cubes}
+      \[ a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3 \]
+
+      \[ y=a(bx-h)^3 + c \]
+
+      \begin{itemize}
+          \tightlist
+        \item
+          \(m=0\) at \emph{stationary point of inflection}
+          (i.e.~(\({h \over b}, k)\))
+        \item \(y=(x-a)^2(x-b)\) --- max at \(x=a\), min at \(x=b\)
+        \item \(y=a(x-b)(x-c)(x-d)\) --- roots at \(b, c, d\)
+        \item \(y=a(x-b)^2(x-c)\) --- roots at \(b\) (instantaneous), \(c\) (intercept)
+      \end{itemize}
+
+      \subsection*{Quartic graphs}
+
+      \subsubsection*{Forms of quartic equations}
+
+      \(y=ax^4\)\\
+      \(y=a(x-b)(x-c)(x-d)(x-e)\)\\
+      \(y=ax^4+cd^2 (c \ge 0)\)\\
+      \(y=ax^2(x-b)(x-c)\)\\
+      \(y=a(x-b)^2(x-c)^2\)\\
+      \(y=a(x-b)(x-c)^3\)
+
+      \input{transformations}
+      \input{stuff}
+      \input{circ-functions}
+      \input{calculus}
+
+    \end{multicols}
+  \end{document}

diff --git a/methods/stuff.tex b/methods/stuff.tex
new file mode 100644 (file)
index 0000000..071a828
--- /dev/null
+++ b/methods/stuff.tex
@@ -0,0 +1,118 @@
+\section{Exponentials \& Logarithms}
+
+\subsubsection*{Logarithmic identities}
+
+\begin{align*}
+  \log_b (xy) &= \log_b x + \log_b y \\
+  \log_b x^n &= n \log_b x \\
+  \log_b y^{x^n} &= x^n \log_b y \\
+  \log_a(\frac{m}{n}) &= \log_am - \log_a \\
+  \log_a(m^{-1}) & = -\log_am \\
+  \log_b c &= \frac{\log_a c}{\log_a b}
+\end{align*}
+
+\subsubsection*{Index identities}
+
+\begin{align*}
+  b^{m+n} &= b^m \cdot b^n \\
+  (b^m)^n &= b^{m \cdot n} \\
+  (b \cdot c)^n &= b^n \cdot c^n \\
+  {b^m \div a^n} &= {b^{m-n}}
+\end{align*}
+
+\subsection*{Inverse functions}
+
+For \(f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x\), inverse is:
+
+\[f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax\]
+
+\subsection*{Euler's number \(e\)}
+
+\[e= \lim_{n \rightarrow \infty} \left(1 + \dfrac{1}{n}\right)^n\]
+
+\subsection*{Modelling}
+
+\[A = A_0 e^{kt}\]
+
+\begin{itemize}
+\tightlist
+\item
+  \(A_0\) is initial value
+\item
+  \(t\) is time taken
+\item
+  \(k\) is a constant
+\item
+  For continuous growth, \(k > 0\)
+\item
+  For continuous decay, \(k < 0\)
+\end{itemize}
+
+\subsection*{Graphing exponential functions}
+
+\[f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1\]
+
+\begin{itemize}
+\tightlist
+\item
+  \textbf{\(y\)-intercept} at \((0, A \cdot a^{-kb}+c)\) as
+  \(x \rightarrow \infty\)
+\item
+  \textbf{horizontal asymptote} at \(y=c\)
+\item
+  \textbf{domain} is \(\mathbb{R}\)
+\item
+  \textbf{range} is \((c, \infty)\)
+\item
+  dilation of factor \(|A|\) from \(x\)-axis
+\item
+  dilation of factor \(1 \over k\) from \(y\)-axis
+\end{itemize}
+
+\begin{tikzpicture}
+  \begin{axis}[restrict x to domain=-0.9:0.9, axis y line = middle, yticklabels={,,}, xticklabels={,,}, enlargelimits, ticks=none]
+    \addplot[red, thick, smooth, samples=100]  plot (\x, {pow(2,x)}) node[below, pos=1] {\(2^x\)};
+    \addplot[blue, thick, smooth, samples=100] plot (\x, {pow(3,x)}) node[left, pos=1] {\(3^x\)};
+    \addplot[orange, thick, smooth, samples=100] plot (\x, {pow(e,x)}) node[below, pos=1] {\(e^x\)};
+    \addplot[mark=*] coordinates {(0,1)} node[above left]{\((0,1)\)} ;
+    \addplot[purple, ultra thick, dashed] plot (\x, 0) node[black, below, font=\footnotesize, pos=0.75] {\(y=0\)};
+  \end{axis}
+\end{tikzpicture}
+
+\subsection*{Graphing logarithmic functions}
+
+\(\log_e x\) is the inverse of \(e^x\) (reflection across \(y=x\))
+
+\[f(x)=A \log_a k(x-b) + c\]
+
+where
+
+\begin{itemize}
+\tightlist
+\item
+  \textbf{domain} is \((b, \infty)\)
+\item
+  \textbf{range} is \(\mathbb{R}\)
+\item
+  \textbf{vertical asymptote} at \(x=b\)
+\item
+  \(y\)-intercept exists if \(b<0\)
+\item
+  dilation of factor \(|A|\) from \(x\)-axis
+\item
+  dilation of factor \(1 \over k\) from \(y\)-axis
+\end{itemize}
+\begin{tikzpicture}
+  \begin{axis}[axis lines=middle, xmin=-0.5, xmax=5, ymin=-2, ymax=3, ticks=none]
+    \addplot[purple, ultra thick, dashed] coordinates {(0,-1.8) (0,2.8)} node[black, below right, pos=0.75, font=\footnotesize] {\(x=0\)};
+    \addplot[orange,thick,domain=0.01:4,smooth,samples=100] {ln(x)} node[right, pos=1] {\(\log_e x\)};
+    \addplot[red,thick,domain=0.01:4,smooth,samples=100] {log2(x)} node[right, pos=1] {\(\log_2 x\)};
+    \addplot[blue,thick,domain=0.01:4,smooth,samples=100] {ln(x)/ln(3)} node[below right, pos=1] {\(\log_3 x\)};
+    \addplot[mark=*] coordinates {(1,0)} node[above left]{\((0,1)\)} ;
+  \end{axis}
+\end{tikzpicture}
+
+\subsection*{Finding equations}
+
+\colorbox{cas}{On CAS:}
+\includegraphics[width=0.78125in]{graphics/cas-simultaneous.png}

diff --git a/methods/temp/calculus.tex b/methods/temp/calculus.tex
deleted file mode 100644 (file)
index 541a8c0..0000000
--- a/methods/temp/calculus.tex
+++ /dev/null
@@ -1,169 +0,0 @@
-\section{Calculus}
-
-\subsection*{Average rate of change}
-
-\[m \operatorname{of} x \in [a,b] = \dfrac{f(b)-f(a)}{b - a} = \frac{dy}{dx}\]
-
-\colorbox{cas}{On CAS:} Action \(\rightarrow\) Calculation
-\(\rightarrow\) \texttt{diff}
-
-\subsection*{Average value}
-
-\[ f_{\text{avg}} = \dfrac{1}{b-a} \int^b_a f(x) \> dx \]
-
-\subsection*{Instantaneous rate of change}
-
-\textbf{Secant} - line passing through two points on a curve\\
-\textbf{Chord} - line segment joining two points on a curve
-
-\subsection*{Limit theorems}
-
-\begin{enumerate}
-\def\labelenumi{\arabic{enumi}.}
-\tightlist
-\item
-  For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
-\item
-  \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
-\item
-  \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
-\item
-  \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
-\end{enumerate}
-
-A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\).
-
-\subsection*{First principles derivative}
-
-\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\]
-
-Not differentiable at:
-\begin{itemize}
-\tightlist
-\item
-  discontinuous points
-\item
-  sharp point/cusp
-\item
-  vertical tangents (\(\infty\) gradient)
-\end{itemize}
-
-\subsection*{Tangents \& gradients}
-
-\textbf{Tangent line} - defined by \(y=mx+c\) where
-\(m={dy \over dx}\)\\
-\textbf{Normal line} - \(\perp\) tangent
-(\(m_{{tan}} \cdot m_{\operatorname{norm}} = -1\))\\
-\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\)
-
-\colorbox{cas}{On CAS:} \\ Action \(\rightarrow\) Calculation
-\(\rightarrow\) Line \(\rightarrow\) \texttt{tanLine} or \texttt{normal}
-
-\subsection*{Strictly increasing/decreasing}
-
-For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
-
-\begin{itemize}
-\tightlist
-\item
-  \textbf{strictly increasing}\\ where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
-\item
-  \textbf{strictly decreasing}\\ where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
-\item
-  Endpoints are included, even where gradient \(=0\)
-\end{itemize}
-
-\columnbreak
-
-\subsubsection*{Solving on CAS}
-
-\colorbox{cas}{\textbf{In main}}: type function. Interactive
-\(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) (Normal
-\textbar{} Tan line)\\
-\colorbox{cas}{\textbf{In graph}}: define function. Analysis
-\(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line).
-Type \(x\) value to solve for a point. Return to show equation for line.
-
-\subsection*{Stationary points}
-
-\emph{Stationary point} - i.e.
-\(f^\prime(x)=0\)\\
-\emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
-\(f^{\prime\prime} = 0\))
-
-                  \begin{tikzpicture}
-                    \begin{axis}[xmin=-21, xmax=21, ymax=1400, ymin=-1000, ticks=none, axis lines=middle]
-                      \addplot[color=red, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {x^3-3*x^2-144*x+432} node [black, pos=1, right] {\(f(x)\)};
-                      \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {3*x^2-6*x-144} node [black, pos=1, right] {\(f^\prime(x)\)};
-                      \addplot[mark=*, blue] coordinates {(1,286)} node[above right, align=left, font=\footnotesize]{inflection \\ (falling)} ;
-                      \addplot[mark=*, orange] coordinates {(-6,972)} node[above left, align=right, font=\footnotesize]{stationary \\ (local max)} ;
-                      \addplot[mark=*, orange] coordinates {(8,-400)} node[below, align=left, font=\footnotesize]{stationary \\ (local min)} ;
-                    \end{axis}
-                  \end{tikzpicture}\\
-                  \begin{tikzpicture}
-                    \begin{axis}[enlargelimits=true, xmax=3.5, ticks=none, axis lines=middle]
-                      \addplot[color=blue, smooth, thick] gnuplot [domain=0.74:3,unbounded coords=jump,samples=500] {(x-2)^3+2} node [black, pos=0.9, left] {\(f(x)\)};
-                      \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=1:3,unbounded coords=jump,samples=500] {3*(x-2)^2} node [black, pos=0.9, right] {\(f^\prime(x)\)};
-                      \addplot[mark=*, purple] coordinates {(2,2)} node[below right, align=left, font=\footnotesize]{stationary \\ inflection} ;
-                    \end{axis}
-                  \end{tikzpicture}\\
-\pagebreak
-\subsection*{Derivatives}
-
-\definecolor{shade1}{HTML}{ffffff}
-\definecolor{shade2}{HTML}{F0F9E4}
-\rowcolors{1}{shade1}{shade2}
-                  \renewcommand{\arraystretch}{1.4}
-                  \begin{tabularx}{\columnwidth}{rX}
-                    \hline
-                    \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\
-                    \hline
-                    \(\sin x\) & \(\cos x\)\\
-                    \(\sin ax\) & \(a\cos ax\)\\
-                    \(\cos x\) & \(-\sin x\)\\
-                    \(\cos ax\) & \(-a \sin ax\)\\
-                    \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
-                    \(e^x\) & \(e^x\)\\
-                    \(e^{ax}\) & \(ae^{ax}\)\\
-                    \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
-                    \(\log_e x\) & \(\dfrac{1}{x}\)\\
-                    \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
-                    \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
-                    \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
-                    \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
-                    \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\
-                    \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
-                    \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\
-                    \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\
-                    \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\
-                    \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
-                    \hline
-                  \end{tabularx}
-                  \columnbreak
-\subsection*{Antiderivatives}
-\rowcolors{1}{shade1}{cas}
-                  \renewcommand{\arraystretch}{1.4}
-                  \begin{tabularx}{\columnwidth}{rX}
-                    \hline
-                    \(f(x)\) & \(\int f(x) \cdot dx\) \\
-                    \hline
-                    \(k\) (constant) & \(kx + c\)\\
-                    \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
-                    \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
-                    \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
-                    \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
-                    \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
-                    \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
-                    \(e^k\) & \(e^kx + c\)\\
-                    \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
-                    \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
-                    \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
-                    \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
-                    \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
-                    \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
-                    \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
-                    \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\
-                    \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
-                    \hline
-                  \end{tabularx}
-

diff --git a/methods/temp/inverse.tex b/methods/temp/inverse.tex
deleted file mode 100644 (file)
index f02a4d1..0000000
--- a/methods/temp/inverse.tex
+++ /dev/null
@@ -1,72 +0,0 @@
-\setstretch{1.3}
-\pagenumbering{gobble}
-
-\hypertarget{inverse-functions}{%
-\section{Inverse functions}\label{inverse-functions}}
-
-\hypertarget{functions}{%
-\subsection{Functions}\label{functions}}
-
-\begin{itemize}
-\tightlist
-\item
-  vertical line test
-\item
-  each \(x\) value produces only one \(y\) value
-\end{itemize}
-
-\hypertarget{one-to-one-functions}{%
-\subsection{One to one functions}\label{one-to-one-functions}}
-
-\begin{itemize}
-\tightlist
-\item
-  \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
-  \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
-  \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
-  \(x^3\) is)
-\item
-  horizontal line test
-\item
-  if not one to one, it is many to one
-\end{itemize}
-
-\hypertarget{deriving-f-1}{%
-\subsection{\texorpdfstring{Deriving
-\(f^{-1}\)}{Deriving f\^{}\{-1\}}}\label{deriving-f-1}}
-
-\begin{itemize}
-\tightlist
-\item
-  if \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
-\item
-  reflection across \(y-x\)
-\item
-  \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
-\item
-  inverse \(\ne\) inverse \emph{function} (i.e.~inverse must pass
-  vertical line test)\\
-  \(\implies f^{-1}(x)\) exists \(\iff f(x)\) is one to one
-\item
-  \(f^{-1}(x)=f(x)\) intersections may lie on line \(y=x\)
-\end{itemize}
-
-\hypertarget{requirements-for-showing-working-for-f-1}{%
-\subsubsection{\texorpdfstring{Requirements for showing working for
-\(f^{-1}\)}{Requirements for showing working for f\^{}\{-1\}}}\label{requirements-for-showing-working-for-f-1}}
-
-\begin{enumerate}
-\def\labelenumi{\arabic{enumi}.}
-\tightlist
-\item
-  start with \emph{``let \(y=f(x)\)''}
-\item
-  must state \emph{``take inverse''} for line where \(y\) and \(x\) are
-  swapped
-\item
-  do all working in terms of \(y=\dots\)
-\item
-  for square root, state \(\pm\) solutions then show restricted
-\item
-  for inverse \emph{function}, state in function notation
-\end{enumerate}

diff --git a/methods/temp/polynomials.tex b/methods/temp/polynomials.tex
deleted file mode 100644 (file)
index a24f1c0..0000000
--- a/methods/temp/polynomials.tex
+++ /dev/null
@@ -1,211 +0,0 @@
-% Options for packages loaded elsewhere
-\PassOptionsToPackage{unicode}{hyperref}
-\PassOptionsToPackage{hyphens}{url}
-%
-\documentclass[
-]{article}
-\usepackage{lmodern}
-\usepackage{amssymb,amsmath}
-\usepackage{ifxetex,ifluatex}
-\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
-  \usepackage[T1]{fontenc}
-  \usepackage[utf8]{inputenc}
-  \usepackage{textcomp} % provide euro and other symbols
-\else % if luatex or xetex
-  \usepackage{unicode-math}
-  \defaultfontfeatures{Scale=MatchLowercase}
-  \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
-\fi
-% Use upquote if available, for straight quotes in verbatim environments
-\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
-\IfFileExists{microtype.sty}{% use microtype if available
-  \usepackage[]{microtype}
-  \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
-}{}
-\makeatletter
-\@ifundefined{KOMAClassName}{% if non-KOMA class
-  \IfFileExists{parskip.sty}{%
-    \usepackage{parskip}
-  }{% else
-    \setlength{\parindent}{0pt}
-    \setlength{\parskip}{6pt plus 2pt minus 1pt}}
-}{% if KOMA class
-  \KOMAoptions{parskip=half}}
-\makeatother
-\usepackage{xcolor}
-\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
-\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}}
-\hypersetup{
-  pdfauthor={Andrew Lorimer},
-  hidelinks,
-  pdfcreator={LaTeX via pandoc}}
-\urlstyle{same} % disable monospaced font for URLs
-\usepackage[a4paper, margin=2cm]{geometry}
-\setlength{\emergencystretch}{3em} % prevent overfull lines
-\providecommand{\tightlist}{%
-  \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
-\setcounter{secnumdepth}{-\maxdimen} % remove section numbering
-\usepackage{setspace}
-\usepackage{fancyhdr}
-\pagestyle{fancy}
-\fancyhead[LO,LE]{Year 12 Methods}
-\fancyhead[CO,CE]{Andrew Lorimer}
-\usepackage{graphicx}
-\usepackage{tabularx}
-\usepackage[dvipsnames]{xcolor}
-
-\author{Andrew Lorimer}
-\date{}
-
-\begin{document}
-
-\hypertarget{polynomials}{%
-\section{Polynomials}\label{polynomials}}
-
-\hypertarget{quadratics}{%
-\subsection{Quadratics}\label{quadratics}}
-
-\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
-\begin{tabularx}{\columnwidth}{Rl}
-  General form& \parbox[t]{5cm}{$x^2 + bx + c = (x+m)(x+n)$\\ where $mn=c, \> m+n=b$} \\
-  \hline
-  Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\
-  \hline
-  Perfect squares & \parbox[c]{5cm}{$a^2 \pm 2ab + b^2 = (a \pm b^2)$} \\
-  \hline
-  Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\
-  \hline
-  Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\
-\end{tabularx}
-
-\hypertarget{cubics}{%
-\subsection{Cubics}\label{cubics}}
-
-\textbf{Difference of cubes:} \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)\\
-\textbf{Sum of cubes:} \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)\\
-\textbf{Perfect cubes:} \(a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3\)
-
-\[y=a(bx-h)^3 + c\]
-
-\begin{itemize}
-\tightlist
-\item
-  \(m=0\) at \emph{stationary point of inflection}
-  (i.e.~(\({h \over b}, k)\))
-\item
-  in form \(y=(x-a)^2(x-b)\), local max at \(x=a\), local min at \(x=b\)
-\item
-  in form \(y=a(x-b)(x-c)(x-d)\): \(x\)-intercepts at \(b, c, d\)
-\item
-  in form \(y=a(x-b)^2(x-c)\), touches \(x\)-axis at \(b\), intercept at
-  \(c\)
-\end{itemize}
-
-\hypertarget{linear-and-quadratic-graphs}{%
-\subsection{Linear and quadratic
-graphs}\label{linear-and-quadratic-graphs}}
-
-\hypertarget{forms-of-linear-equations}{%
-\subsubsection{Forms of linear
-equations}\label{forms-of-linear-equations}}
-
-\(y=mx+c\) where \(m\) is gradient and \(c\) is \(y\)-intercept\\
-\({x \over a} + {y \over b}=1\) where \(m\) is gradient and
-\((x_1, y_1)\) lies on the graph\\
-\(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and
-\(y\)-intercepts
-
-\hypertarget{line-properties}{%
-\subsection{Line properties}\label{line-properties}}
-
-Parallel lines: \(m_1 = m_2\)\\
-Perpendicular lines: \(m_1 \times m_2 = -1\)\\
-Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
-
-\hypertarget{quartic-graphs}{%
-\subsection{Quartic graphs}\label{quartic-graphs}}
-
-\hypertarget{forms-of-quadratic-equations}{%
-\subsubsection{Forms of quadratic
-equations}\label{forms-of-quadratic-equations}}
-
-\(y=ax^4\)\\
-\(y=a(x-b)(x-c)(x-d)(x-e)\)\\
-\(y=ax^4+cd^2 (c \ge 0)\)\\
-\(y=ax^2(x-b)(x-c)\)\\
-\(y=a(x-b)^2(x-c)^2\)\\
-\(y=a(x-b)(x-c)^3\)
-
-\hypertarget{simultaneous-equations-linear}{%
-\subsection{Simultaneous equations
-(linear)}\label{simultaneous-equations-linear}}
-
-\begin{itemize}
-\tightlist
-\item
-  \textbf{Unique solution} - lines intersect at point
-\item
-  \textbf{Infinitely many solutions} - lines are equal
-\item
-  \textbf{No solution} - lines are parallel
-\end{itemize}
-
-\hypertarget{solving-protectbegincasespx-qy-a-rx-sy-bprotectendcases-for-01infty-solutions}{%
-\subsubsection{\texorpdfstring{Solving
-\(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\)
-for \(\{0,1,\infty\}\)
-solutions}{Solving \textbackslash protect\textbackslash begin\{cases\}px + qy = a \textbackslash\textbackslash{} rx + sy = b\textbackslash protect\textbackslash end\{cases\} \textbackslash\textgreater{} for \textbackslash\{0,1,\textbackslash infty\textbackslash\} solutions}}\label{solving-protectbegincasespx-qy-a-rx-sy-bprotectendcases-for-01infty-solutions}}
-
-where all coefficients are known except for one, and \(a, b\) are known
-
-\begin{enumerate}
-\def\labelenumi{\arabic{enumi}.}
-\tightlist
-\item
-  Write as matrices:
-  \(\begin{bmatrix}p & q \\ r & s \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  =  \begin{bmatrix} a \\ b \end{bmatrix}\)
-\item
-  Find determinant of first matrix: \(\Delta = ps-qr\)
-\item
-  Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
-  or let \(\Delta \ne 0\) for one unique solution.
-\item
-  Solve determinant equation to find variable
-
-  \begin{itemize}
-  \tightlist
-  \item
-    \emph{--- for infinite/no solutions: ---}
-  \end{itemize}
-\item
-  Substitute variable into both original equations
-\item
-  Rearrange equations so that LHS of each is the same
-\item
-  \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\)
-  (\(\infty\) solns)\\
-  \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0
-  solns)
-\end{enumerate}
-
-\colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
-
-\hypertarget{solving-protectbegincasesa_1-x-b_1-y-c_1-z-d_1-a_2-x-b_2-y-c_2-z-d_2-a_3-x-b_3-y-c_3-z-d_3protectendcases}{%
-\subsubsection{\texorpdfstring{Solving
-\(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}{Solving \textbackslash protect\textbackslash begin\{cases\}a\_1 x + b\_1 y + c\_1 z = d\_1 \textbackslash\textbackslash{} a\_2 x + b\_2 y + c\_2 z = d\_2 \textbackslash\textbackslash{} a\_3 x + b\_3 y + c\_3 z = d\_3\textbackslash protect\textbackslash end\{cases\}}}\label{solving-protectbegincasesa_1-x-b_1-y-c_1-z-d_1-a_2-x-b_2-y-c_2-z-d_2-a_3-x-b_3-y-c_3-z-d_3protectendcases}}
-
-\begin{itemize}
-\tightlist
-\item
-  Use elimination
-\item
-  Generate two new equations with only two variables
-\item
-  Rearrange \& solve
-\item
-  Substitute one variable into another equation to find another variable
-\item
-  etc.
-\end{itemize}
-
-\end{document}

diff --git a/methods/temp/stuff.tex b/methods/temp/stuff.tex
deleted file mode 100644 (file)
index 2964bea..0000000
--- a/methods/temp/stuff.tex
+++ /dev/null
@@ -1,118 +0,0 @@
-\section{Exponentials \& Logarithms}
-
-\subsubsection*{Logarithmic identities}
-
-\begin{align*}
-  \log_b (xy) &= \log_b x + \log_b y \\
-  \log_b x^n &= n \log_b x \\
-  \log_b y^{x^n} &= x^n \log_b y \\
-  \log_a(\frac{m}{n}) &= \log_am - \log_a \\
-  \log_a(m^{-1}) & = -\log_am \\
-  \log_b c &= \frac{\log_a c}{\log_a b}
-\end{align*}
-
-\subsubsection*{Index identities}
-
-\begin{align*}
-  b^{m+n} &= b^m \cdot b^n \\
-  (b^m)^n &= b^{m \cdot n} \\
-  (b \cdot c)^n &= b^n \cdot c^n \\
-  {b^m \div a^n} &= {b^{m-n}}
-\end{align*}
-
-\subsection*{Inverse functions}
-
-For \(f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x\), inverse is:
-
-\[f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax\]
-
-\subsection*{Euler's number \(e\)}
-
-\[e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n\]
-
-\subsection*{Modelling}
-
-\[A = A_0 e^{kt}\]
-
-\begin{itemize}
-\tightlist
-\item
-  \(A_0\) is initial value
-\item
-  \(t\) is time taken
-\item
-  \(k\) is a constant
-\item
-  For continuous growth, \(k > 0\)
-\item
-  For continuous decay, \(k < 0\)
-\end{itemize}
-
-\subsection*{Graphing exponential functions}
-
-\[f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1\]
-
-\begin{itemize}
-\tightlist
-\item
-  \textbf{\(y\)-intercept} at \((0, A \cdot a^{-kb}+c)\) as
-  \(x \rightarrow \infty\)
-\item
-  \textbf{horizontal asymptote} at \(y=c\)
-\item
-  \textbf{domain} is \(\mathbb{R}\)
-\item
-  \textbf{range} is \((c, \infty)\)
-\item
-  dilation of factor \(|A|\) from \(x\)-axis
-\item
-  dilation of factor \(1 \over k\) from \(y\)-axis
-\end{itemize}
-
-\begin{tikzpicture}
-  \begin{axis}[restrict x to domain=-0.9:0.9, axis y line = middle, yticklabels={,,}, xticklabels={,,}, enlargelimits, ticks=none]
-    \addplot[red, thick, smooth, samples=100]  plot (\x, {pow(2,x)}) node[below, pos=1] {\(2^x\)};
-    \addplot[blue, thick, smooth, samples=100] plot (\x, {pow(3,x)}) node[left, pos=1] {\(3^x\)};
-    \addplot[orange, thick, smooth, samples=100] plot (\x, {pow(e,x)}) node[below, pos=1] {\(e^x\)};
-    \addplot[mark=*] coordinates {(0,1)} node[above left]{\((0,1)\)} ;
-    \addplot[purple, ultra thick, dashed] plot (\x, 0) node[black, below, font=\footnotesize, pos=0.75] {\(y=0\)};
-  \end{axis}
-\end{tikzpicture}
-
-\subsection*{Graphing logarithmic functions}
-
-\(\log_e x\) is the inverse of \(e^x\) (reflection across \(y=x\))
-
-\[f(x)=A \log_a k(x-b) + c\]
-
-where
-
-\begin{itemize}
-\tightlist
-\item
-  \textbf{domain} is \((b, \infty)\)
-\item
-  \textbf{range} is \(\mathbb{R}\)
-\item
-  \textbf{vertical asymptote} at \(x=b\)
-\item
-  \(y\)-intercept exists if \(b<0\)
-\item
-  dilation of factor \(|A|\) from \(x\)-axis
-\item
-  dilation of factor \(1 \over k\) from \(y\)-axis
-\end{itemize}
-\begin{tikzpicture}
-  \begin{axis}[axis lines=middle, xmin=-0.5, xmax=5, ymin=-2, ymax=3, ticks=none]
-    \addplot[purple, ultra thick, dashed] coordinates {(0,-1.8) (0,2.8)} node[black, below right, pos=0.75, font=\footnotesize] {\(x=0\)};
-    \addplot[orange,thick,domain=0.01:4,smooth,samples=100] {ln(x)} node[right, pos=1] {\(\log_e x\)};
-    \addplot[red,thick,domain=0.01:4,smooth,samples=100] {log2(x)} node[right, pos=1] {\(\log_2 x\)};
-    \addplot[blue,thick,domain=0.01:4,smooth,samples=100] {ln(x)/ln(3)} node[below right, pos=1] {\(\log_3 x\)};
-    \addplot[mark=*] coordinates {(1,0)} node[above left]{\((0,1)\)} ;
-  \end{axis}
-\end{tikzpicture}
-
-\subsection*{Finding equations}
-
-\colorbox{cas}{On CAS:}
-\includegraphics[width=0.78125in]{graphics/cas-simultaneous.png}

diff --git a/methods/temp/transformations.tex b/methods/temp/transformations.tex
deleted file mode 100644 (file)
index d2b253c..0000000
--- a/methods/temp/transformations.tex
+++ /dev/null
@@ -1,168 +0,0 @@
-\definecolor{shade1}{HTML}{ffffff}
-\definecolor{shade2}{HTML}{e6f2ff}
-\definecolor{shade3}{HTML}{cce2ff}
-\section{Transformations}
-
-\textbf{Order of operations:} DRT
-
-\begin{center}dilations --- reflections --- translations\end{center}
-
-\subsection*{Transforming \(x^n\) to \(a(x-h)^n+K\)}
-
-\begin{itemize}
-\tightlist
-\item
-  dilation factor of \(|a|\) units parallel to \(y\)-axis or from
-  \(x\)-axis
-\item
-  if \(a<0\), graph is reflected over \(x\)-axis
-\item
-  translation of \(k\) units parallel to \(y\)-axis or from \(x\)-axis
-\item
-  translation of \(h\) units parallel to \(x\)-axis or from \(y\)-axis
-\item
-  for \((ax)^n\), dilation factor is \(1 \over a\) parallel to
-  \(x\)-axis or from \(y\)-axis
-\item
-  when \(0 < |a| < 1\), graph becomes closer to axis
-\end{itemize}
-
-\subsection*{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)}
-
-Applies to exponential, log, trig, \(e^x\), polynomials.\\
-Functions must be written in form \(y=Af[n(x+c)]+b\)
-
-\begin{itemize}
-\tightlist
-\item
-  dilation by factor \(|A|\) from \(x\)-axis (if \(A<0\), reflection
-  across \(y\)-axis)
-\item
-  dilation by factor \(1 \over n\) from \(y\)-axis (if \(n<0\),
-  reflection across \(x\)-axis)
-\item
-  translation of \(c\) units from \(y\)-axis (\(x\)-shift)
-\item
-  translation of \(b\) units from \(x\)-axis (\(y\)-shift)
-\end{itemize}
-
-\subsection*{Dilations}
-
-Two pairs of equivalent processes for \(y=f(x)\):
-
-\begin{enumerate}
-\def\labelenumi{\arabic{enumi}.}
-\item
-  \begin{itemize}
-  \tightlist
-  \item
-    Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\)
-  \item
-    Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\)
-  \end{itemize}
-\item
-  \begin{itemize}
-  \tightlist
-  \item
-    Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\)
-  \item
-    Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\)
-  \end{itemize}
-\end{enumerate}
-
-For graph of \(y={1 \over x}\), horizontal \& vertical dilations are
-equivalent (symmetrical). If \(y={a \over x}\), graph is contracted
-rather than dilated.
-
-\subsection*{Matrix transformations}
-
-Find new point \((x^\prime, y^\prime)\). Substitute these into original
-equation to find image with original variables \((x, y)\).
-
-\subsection*{Reflections}
-
-\begin{itemize}
-\tightlist
-\item
-  Reflection \textbf{in} axis = reflection \textbf{over} axis =
-  reflection \textbf{across} axis
-\item
-  Translations do not change
-\end{itemize}
-
-\subsection*{Translations}
-
-For \(y = f(x)\), these processes are equivalent:
-
-\begin{itemize}
-\tightlist
-\item
-  applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the
-  graph of \(y = f(x)\)
-\item
-  replacing \(x\) with \(x-h\) and \(y\) with \(y-k\) to obtain
-  \(y-k = f(x-h)\)
-\end{itemize}
-
-\subsection*{Power functions}
-
-\textbf{Strictly increasing:} \(f(x_2) > f(x_1)\) where \(x_2 > x_1\)
-(including \(x=0\))
-
-
-\subsubsection*{\(x^{-1 \over n}\) where \(n \in \mathbb{Z}^+\)}
-
-Mostly only on CAS.
-
-We can write
-\(x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}\)n.\\
-Domain is:
-\(\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\)
-
-If \(n\) is odd, it is an odd function.
-
-\subsubsection*{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)}
-
-\[x^{p \over q} = \sqrt[q]{x^p}\]
-
-\begin{itemize}
-\tightlist
-\item
-  if \(p > q\), the shape of \(x^p\) is dominant
-\item
-  if \(p < q\), the shape of \(x^{1 \over q}\) is dominant
-\item
-  points \((0, 0)\) and \((1, 1)\) will always lie on graph
-\item
-  Domain is:
-  \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
-\end{itemize}
-
-\subsection*{Piecewise functions}
-
-\[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
-
-\textbf{Open circle:} point included\\
-\textbf{Closed circle:} point not included
-
-\subsection*{Operations on functions}
-
-For \(f \pm g\) and \(f \times g\):
-\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
-
-Addition of linear piecewise graphs: add \(y\)-values at key points
-
-Product functions:
-
-\begin{itemize}
-\tightlist
-\item
-  product will equal 0 if \(f=0\) or \(g=0\)
-\item
-  \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
-\end{itemize}
-
-\subsection*{Composite functions}
-
-\((f \circ g)(x)\) is defined iff
-\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)

diff --git a/methods/transformations.tex b/methods/transformations.tex
new file mode 100644 (file)
index 0000000..00428e3
--- /dev/null
+++ b/methods/transformations.tex
@@ -0,0 +1,134 @@
+\definecolor{shade1}{HTML}{ffffff}
+\definecolor{shade2}{HTML}{e6f2ff}
+\definecolor{shade3}{HTML}{cce2ff}
+\section{Transformations}
+
+\textbf{Order of operations:} DRT
+
+\begin{center}dilations --- reflections --- translations\end{center}
+
+\subsection*{Transforming \(x^n\) to \(a(x-h)^n+K\)}
+
+\begin{itemize}
+\tightlist
+\item
+  dilation factor of \(|a|\) units parallel to \(y\)-axis or from
+  \(x\)-axis
+\item
+  if \(a<0\), graph is reflected over \(x\)-axis
+\item
+  translation of \(k\) units parallel to \(y\)-axis or from \(x\)-axis
+\item
+  translation of \(h\) units parallel to \(x\)-axis or from \(y\)-axis
+\item
+  for \((ax)^n\), dilation factor is \(1 \over a\) parallel to
+  \(x\)-axis or from \(y\)-axis
+\item
+  when \(0 < |a| < 1\), graph becomes closer to axis
+\end{itemize}
+
+\subsection*{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)}
+
+Applies to exponential, log, trig, \(e^x\), polynomials.\\
+Functions must be written in form \(y=Af[n(x+c)]+b\)
+
+\begin{itemize}
+\tightlist
+\item
+  dilation by factor \(|A|\) from \(x\)-axis (if \(A<0\), reflection
+  across \(y\)-axis)
+\item
+  dilation by factor \(1 \over n\) from \(y\)-axis (if \(n<0\),
+  reflection across \(x\)-axis)
+\item
+  translation of \(c\) units from \(y\)-axis (\(x\)-shift)
+\item
+  translation of \(b\) units from \(x\)-axis (\(y\)-shift)
+\end{itemize}
+
+\subsection*{Dilations}
+
+Two pairs of equivalent processes for \(y=f(x)\):
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  \begin{itemize}
+  \tightlist
+  \item
+    Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\)
+  \item
+    Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\)
+  \end{itemize}
+\item
+  \begin{itemize}
+  \tightlist
+  \item
+    Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\)
+  \item
+    Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\)
+  \end{itemize}
+\end{enumerate}
+
+For graph of \(y={1 \over x}\), horizontal \& vertical dilations are
+equivalent (symmetrical). If \(y={a \over x}\), graph is contracted
+rather than dilated.
+
+\subsection*{Matrix transformations}
+
+Find new point \((x^\prime, y^\prime)\). Substitute these into original
+equation to find image with original variables \((x, y)\).
+
+\subsection*{Reflections}
+
+\begin{itemize}
+\tightlist
+\item
+  Reflection \textbf{in} axis = reflection \textbf{over} axis =
+  reflection \textbf{across} axis
+\item
+  Translations do not change
+\end{itemize}
+
+\subsection*{Translations}
+
+For \(y = f(x)\), these processes are equivalent:
+
+\begin{itemize}
+\tightlist
+\item
+  applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the
+  graph of \(y = f(x)\)
+\item
+  replacing \(x\) with \(x-h\) and \(y\) with \(y-k\) to obtain
+  \(y-k = f(x-h)\)
+\end{itemize}
+
+\subsection*{Power functions}
+
+Mostly only on CAS.
+
+We can write
+\(x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}\)n.\\
+Domain is:
+\(\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\)
+
+If \(n\) is odd, it is an odd function.
+
+\subsubsection*{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)}
+
+\[x^{p \over q} = \sqrt[q]{x^p}\]
+
+\begin{itemize}
+\tightlist
+\item
+  if \(p > q\), the shape of \(x^p\) is dominant
+\item
+  if \(p < q\), the shape of \(x^{1 \over q}\) is dominant
+\item
+  points \((0, 0)\) and \((1, 1)\) will always lie on graph
+\item
+  Domain is:
+  \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
+\end{itemize}
+