\usepackage{amssymb}
\usepackage{harpoon}
\usepackage{tabularx}
+\usepackage{tabu}
\usepackage{makecell}
\usepackage[dvipsnames, table]{xcolor}
\usepackage{blindtext}
\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}
xlabel={$x$}, % axes labels
ylabel={$y$},
}}
+
\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}
-\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
- \(f(x)\) 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
- 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}
-\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]
-\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}