methods / methods-collated.texon commit [methods] re-render methods notes (1dadb9e)
   1\documentclass[a4paper]{article}
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  22\usepackage{polynom}
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  34
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  71
  72\pagestyle{fancy}
  73\fancyhead[LO,LE]{Year 12 Methods}
  74\fancyhead[CO,CE]{Andrew Lorimer}
  75\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
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 100\newtcolorbox{cas}{colframe=cas!75!black, fonttitle=\sffamily\bfseries, title=On CAS, left*=3mm}
 101\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=darkgray, fontupper=\sffamily\bfseries}
 102\newtcolorbox{theorembox}[1]{colback=green!10!white, colframe=blue!20!white, coltitle=black, fontupper=\sffamily, fonttitle=\sffamily, #1}
 103
 104
 105\begin{document}
 106
 107\title{\vspace{-20mm}Year 12 Methods}
 108\author{Andrew Lorimer}
 109\date{}
 110\maketitle
 111
 112\begin{multicols}{2}
 113
 114
 115\section{Functions}
 116
 117\begin{itemize} \tightlist
 118  \item vertical line test
 119  \item each \(x\) value produces only one \(y\) value
 120\end{itemize}
 121
 122\subsection*{One to one functions}
 123
 124\begin{itemize} \tightlist
 125  \item \(f(x)\) is 1:1 if \(f(a) \ne f(b) \> \forall \>\{a,b\} \in \operatorname{dom}(f)\) \\
 126        \(\implies\) unique \(y\) for each \(x\)
 127  \item e.g. \(\sin x\) is not 1:1, \(x^3\) is
 128  \item horizontal line test
 129  \item if not one to one, it is many to one
 130\end{itemize}
 131
 132\subsection*{Odd and even functions}
 133
 134\begin{align*}
 135  \text{Even:}&& f(x)  &= f(-x) \\
 136  \text{Odd:} && -f(x) &= f(-x)
 137\end{align*}
 138
 139Even \(\implies\) symmetrical across \(y\)-axis \\
 140\(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
 141For \(x^n\), parity of \(n \equiv\) parity of function
 142
 143\begin{tabularx}{\columnwidth}{XX}
 144  \textbf{Even:} & \textbf{Odd:} \\
 145  \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} &
 146    \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}
 147\end{tabularx}
 148
 149\subsection*{Inverse functions}
 150
 151\begin{itemize} \tightlist
 152  \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
 153  \item \(f\) must be one to one
 154  \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
 155  \item Represents reflection across \(y=x\)
 156  \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
 157  \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
 158    \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
 159  \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
 160\end{itemize}
 161
 162\subsubsection*{Finding \(f^{-1}\)}
 163
 164\begin{enumerate} \tightlist
 165  \item Let \(y=f(x)\)
 166  \item Swap \(x\) and \(y\) (``take inverse''
 167  \item Solve for \(y\) \\
 168    Sqrt: state \(\pm\) solutions then restrict
 169  \item State rule as \(f^{-1}(x)=\dots\)
 170  \item For inverse \emph{function}, state in function notation
 171\end{enumerate}
 172
 173\subsection*{Simultaneous equations (linear)}
 174
 175\begin{itemize} \tightlist
 176  \item \textbf{Unique solution} - lines intersect at point
 177  \item \textbf{Infinitely many solutions} - lines are equal
 178  \item \textbf{No solution} - lines are parallel
 179\end{itemize}
 180
 181\subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
 182  where all coefficients are known except for one, and \(a, b\) are known
 183
 184  \begin{enumerate} \tightlist
 185    \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  =  \begin{bmatrix} a \\ b \end{bmatrix}\)
 186      \item Find \(\det(\text{first matrix}) = ps-qr\)
 187      \item Let \(\det = 0\) for \(\{0,\infty\}\) solutions
 188        or \(\det \ne 0\) for 1 solution
 189      \item Solve to find variable \\ \\
 190        \textbf{For infinite/no solutions:}
 191      \item Substitute variable into both original equations
 192      \item Rearrange so that LHS of each is the same
 193      \item \(\begin{aligned}[t]
 194          \infty \text{ solns: } & \text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x \\
 195          0 \text{ solns: } & \text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x
 196      \end{aligned}\)
 197  \end{enumerate}
 198
 199  \begin{cas}
 200    Action \(\rightarrow\) Matrix \(\rightarrow\) Calculation \(\rightarrow\) \texttt{det}
 201  \end{cas}
 202
 203  \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
 204
 205    \begin{itemize} \tightlist
 206      \item Use elimination
 207      \item Generate two new equations with only two variables
 208      \item Rearrange \& solve
 209      \item Substitute one variable into another equation to find another variable
 210    \end{itemize}
 211
 212    \subsection*{Piecewise functions}
 213
 214    \[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
 215
 216      \textbf{Open circle:} point included\\
 217      \textbf{Closed circle:} point not included
 218
 219\begin{cas}
 220  Define piecewise functions: \\
 221  \-\hspace{1em}Math3 \(\rightarrow\)
 222  \begin{tikzpicture}%
 223    \draw rectangle (0.5,0.5); 
 224    \node at (0.08,0.25) {\(\{\)};
 225    \filldraw [black] (0.15, 0.4) rectangle(0.25, 0.3);
 226    \draw (0.35, 0.4) rectangle(0.45, 0.3);
 227    \node [font=\footnotesize] at (0.3,0.3) {\verb;,;};
 228    \draw (0.15, 0.2) rectangle(0.25, 0.1);
 229    \node [font=\footnotesize] at (0.3,0.1) {\verb;,;};
 230    \draw (0.35, 0.2) rectangle(0.45, 0.1);
 231  \end{tikzpicture}
 232  % TODO: finish this section
 233\end{cas}
 234
 235      \subsection*{Operations on functions}
 236
 237      For \(f \pm g\) and \(f \times g\):
 238      \quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
 239
 240      Addition of linear piecewise graphs: add \(y\)-values at key points
 241
 242      Product functions:
 243
 244      \begin{itemize}
 245          \tightlist
 246        \item
 247          product will equal 0 if \(f=0\) or \(g=0\)
 248        \item
 249          \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
 250      \end{itemize}
 251
 252      \subsection*{Composite functions}
 253
 254      \((f \circ g)(x)\) is defined iff
 255      \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
 256
 257      \pgfplotsset{
 258        blank/.append style={%
 259          enlargelimits=true,
 260          ticks=none,
 261          yticklabels={,,}, xticklabels={,,},
 262          xlabel=, ylabel=,
 263          scale=0.4,
 264          samples=100, smooth, unbounded coords=jump
 265        }
 266      }
 267      \tikzset{
 268        blankplot/.append style={orange, mark=none}
 269      }
 270
 271      \begin{figure*}[ht]
 272        \centering
 273
 274        \begin{tabularx}{\textwidth}{|r|Y|Y|}
 275
 276          \hline
 277          \rowcolor{lblue}
 278          & \(n\) is even & \(n\) is odd \\ \hline
 279
 280          \centering \(x^n, n \in \mathbb{Z}^+\) & 
 281
 282          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 283            \begin{axis}[blank, xmin=-3,  xmax=3]
 284              \addplot[blankplot] {(x^2)};
 285            \end{axis}
 286          \end{tikzpicture}} &
 287
 288          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 289            \begin{axis}[blank, xmin=-3,  xmax=3]
 290              \addplot[blankplot, domain=-3:3] {(x^3)};
 291            \end{axis}
 292          \end{tikzpicture}} \\ \hline
 293
 294          \centering \(x^n, n \in \mathbb{Z}^-\) &
 295
 296          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 297            \begin{axis}[blank, xmin=-4, xmax=4, ymax=8, ymin=-0]
 298              \addplot[blankplot, samples=100] {(x^(-2))};
 299            \end{axis}
 300          \end{tikzpicture}} &
 301
 302          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 303            \begin{axis}[blank, xmin=-3, xmax=3]
 304              \addplot[blankplot, domain=-3:-0.1] {(x^(-1))};
 305              \addplot[blankplot, domain=0.1:3] {(x^(-1))};
 306            \end{axis}
 307          \end{tikzpicture}} \\ \hline
 308
 309          \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
 310
 311          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 312            \begin{axis}[blank, xmin=-1,  xmax=5]
 313              \addplot[blankplot] {(x^(1/2))};
 314            \end{axis}
 315          \end{tikzpicture}} &
 316
 317          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 318            \begin{axis}[blank, xmin=-3, xmax=3, ymin=-3, ymax=3]
 319              \addplot [blankplot, domain=-2:2] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
 320            \end{axis}
 321          \end{tikzpicture}} \\ \hline
 322
 323        \end{tabularx}
 324      \end{figure*}
 325
 326      \section{Polynomials}
 327
 328      \subsection*{Factor theorem}
 329
 330      \begin{theorembox}{title=General form \(\beta x + \alpha\)}
 331        If \(\beta x + \alpha\) is a factor of \(P(x)\), \\
 332        \-\hspace{1em}then \(P(-\dfrac{\alpha}{\beta})=0\).
 333      \end{theorembox}
 334
 335      \begin{theorembox}{title=Simple form \(x-a\)}
 336        If \((x-a)\) is a factor of \(P(x)\), remainder \(R=0\). \\
 337        \-\hspace{1em}\(\implies P(a)=0\)
 338      \end{theorembox}
 339
 340      \subsection*{Remainder theorem}
 341
 342      \begin{theorembox}{}
 343        When \(P(x)\) is divided by \(\beta x + \alpha\), the remainder is \(-\dfrac{\alpha}{\beta}\).
 344      \end{theorembox}
 345
 346      \subsection*{Rational root theorem}
 347      Let \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\) be a polynomial of degree \(n\) with \(a_i \in \mathbb{Z} \forall a\). Let \(\alpha, \beta \in \mathbb{Z}\) such that their highest common factor is 1 (i.e. relatively prime).
 348
 349      If \(\beta x + \alpha\) is a factor of \(P(x)\), then \(\beta\) divides \(a_n\) and \(\alpha\) divides \(a_0\) .
 350
 351      \subsubsection*{Discriminant}
 352      \[\begin{cases}
 353        b^2-4ac > 0 & \text{two solutions} \\
 354        b^2-4ac = 0 & \text{one solution} \\
 355        b^2-4ac < 0 & \text{no solutions}
 356      \end{cases}\]
 357      \begin{warning}
 358        Flip inequality sign when multiplying by -1
 359      \end{warning}
 360
 361      \subsection*{Long division}
 362
 363      \[ \polylongdiv{x^2+2x+4}{x-1} \]
 364
 365      \begin{cas}
 366        Action \(\rightarrow\) Transformation \(\rightarrow\) \texttt{propFrac}
 367      \end{cas}
 368
 369      \subsection*{Linear equations}
 370
 371      \subsubsection*{Forms}
 372
 373      \begin{itemize}
 374          \tightlist
 375        \item \(y=mx+c\)
 376        \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
 377        \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
 378      \end{itemize}
 379
 380      \subsubsection*{Line properties}
 381
 382      Parallel lines: \(m_1 = m_2\)\\
 383      Perpendicular lines: \(m_1 \times m_2 = -1\)\\
 384      Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
 385
 386      \subsection*{Quadratics}
 387
 388      \setlength{\abovedisplayskip}{1pt}
 389      \setlength{\belowdisplayskip}{1pt}
 390
 391      \textbf{Linear factorisation}
 392      \[ x^2 + bx + c = (x+m)(x+n) \]
 393      \hfill where \(mn=c, \> m+n=b\)
 394
 395      \textbf{Difference of squares}
 396      \[ a^2 - b^2 = (a-b)(a+b) \]
 397      \textbf{Perfect squares}
 398      \[ a^2 \pm 2ab + b^2 = (a \pm b^2) \]
 399      \textbf{Completing the square}
 400      \begin{align*}
 401        x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
 402        ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a}
 403      \end{align*}
 404      \textbf{Quadratic formula}
 405      \[ x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \]
 406      \hfill (Discriminant \(\Delta=b^2-4ac\))
 407
 408      \subsection*{Cubics}
 409
 410      \textbf{Difference of cubes}
 411      \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]
 412      \textbf{Sum of cubes}
 413      \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
 414      \textbf{Perfect cubes}
 415      \[ a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3 \]
 416
 417      \[ y=a(bx-h)^3 + c \]
 418
 419      \begin{itemize}
 420          \tightlist
 421        \item
 422          \(m=0\) at \emph{stationary point of inflection}
 423          (i.e.~(\({h \over b}, k)\))
 424        \item \(y=(x-a)^2(x-b)\) --- max at \(x=a\), min at \(x=b\)
 425        \item \(y=a(x-b)(x-c)(x-d)\) --- roots at \(b, c, d\)
 426        \item \(y=a(x-b)^2(x-c)\) --- roots at \(b\) (instantaneous), \(c\) (intercept)
 427      \end{itemize}
 428
 429      \subsection*{Quartic graphs}
 430
 431      \subsubsection*{Forms of quartic equations}
 432
 433      \(y=ax^4\)\\
 434      \(y=a(x-b)(x-c)(x-d)(x-e)\)\\
 435      \(y=ax^4+cd^2 (c \ge 0)\)\\
 436      \(y=ax^2(x-b)(x-c)\)\\
 437      \(y=a(x-b)^2(x-c)^2\)\\
 438      \(y=a(x-b)(x-c)^3\)
 439
 440      \input{transformations}
 441      \input{stuff}
 442      \input{circ-functions}
 443      \input{calculus}
 444
 445      \subfile{statistics-ref}
 446
 447    \end{multicols}
 448
 449\end{document}