From: Andrew Lorimer Date: Sat, 28 Sep 2019 12:22:52 +0000 (+1000) Subject: [methods] general notes additions X-Git-Tag: yr12~24 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/3a71a25c6117f902910486df7e54207be01f0ef2 [methods] general notes additions --- diff --git a/methods/longdiv.tex b/methods/longdiv.tex new file mode 100644 index 0000000..4c9a1d4 --- /dev/null +++ b/methods/longdiv.tex @@ -0,0 +1,53 @@ +% longdiv.tex v.1 (1994) Donald Arseneau +% +% Work out and print integer long division problems. Use: +% \longdiv{numerator}{denominator} +% The numerator and denominator (divisor and dividend) must be integers, and +% the quotient is an integer too. \longdiv leaves a remainder. +% Use this in any type of TeX. + +\newcount\gpten % (global) power-of-ten -- tells which digit we are doing +\countdef\rtot2 % running total -- remainder so far +\countdef\LDscratch4 % scratch + +\def\longdiv#1#2{% + \vtop{\normalbaselines \offinterlineskip + \setbox\strutbox\hbox{\vrule height 2.1ex depth .5ex width0ex}% + \def\showdig{$\underline{\the\LDscratch\strut}$\cr\the\rtot\strut\cr + \noalign{\kern-.2ex}}% + \global\rtot=#1\relax + \count0=\rtot\divide\count0by#2\edef\quotient{\the\count0}%\show\quotient + % make list macro out of digits in quotient: + \def\temp##1{\ifx##1\temp\else \noexpand\dodig ##1\expandafter\temp\fi}% + \edef\routine{\expandafter\temp\quotient\temp}% + % process list to give power-of-ten: + \def\dodig##1{\global\multiply\gpten by10 }\global\gpten=1 \routine + % to display effect of one digit in quotient (zero ignored): + \def\dodig##1{\global\divide\gpten by10 + \LDscratch =\gpten + \multiply\LDscratch by##1% + \multiply\LDscratch by#2% + \global\advance\rtot-\LDscratch \relax + \ifnum\LDscratch>0 \showdig \fi % must hide \cr in a macro to skip it + }% + \tabskip=0pt + \halign{\hfil##\cr % \halign for entire division problem + $\quotient$\strut\cr + #2$\,\overline{\vphantom{\big)}% + \hbox{\smash{\raise3.5\fontdimen8\textfont3\hbox{$\big)$}}}% + \mkern2mu \the\rtot}$\cr\noalign{\kern-.2ex} + \routine \cr % do each digit in quotient +}}} + +\endinput % Demonstration below: + +\noindent Here are some long division problems + +\indent +\longdiv{12345}{13} \quad +\longdiv{123}{1234} \quad +\longdiv{31415926}{2} \quad +\longdiv{81}{3} \quad +\longdiv{1132}{99} \quad +\longdiv{86491}{94} +\bye diff --git a/methods/methods-collated.pdf b/methods/methods-collated.pdf index 776aa11..35e900e 100644 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 a3a1e79..4a0403a 100644 --- a/methods/methods-collated.tex +++ b/methods/methods-collated.tex @@ -18,6 +18,7 @@ \usepackage{multirow} \usepackage{newclude} \usepackage{pgfplots} +\usepackage{polynom} \usepackage{pst-plot} \usepackage{standalone} \usepackage{subfiles} @@ -97,6 +98,7 @@ \newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm} \newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=important, fontupper=\sffamily\bfseries} +\newtcolorbox{theorembox}[1]{colback=green!10!white, colframe=blue!20!white, coltitle=black, fontupper=\sffamily, fonttitle=\sffamily, #1} \begin{document} @@ -184,18 +186,22 @@ For \(x^n\), parity of \(n \equiv\) parity of function \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 \\ + \item Find \(\det(\text{first matrix}) = ps-qr\) + \item Let \(\det = 0\) for \(\{0,\infty\}\) solutions + or \(\det \ne 0\) for 1 solution + \item Solve 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) + \item Rearrange so that LHS of each is the same + \item \(\begin{aligned}[t] + \infty \text{ solns: } & \text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x \\ + 0 \text{ solns: } & \text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x + \end{aligned}\) \end{enumerate} - \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det} + \begin{cas} + Action \(\rightarrow\) Matrix \(\rightarrow\) Calculation \(\rightarrow\) \texttt{det} + \end{cas} \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)} @@ -304,6 +310,47 @@ For \(x^n\), parity of \(n \equiv\) parity of function \section{Polynomials} + \subsection*{Factor theorem} + + \begin{theorembox}{title=General form \(\beta x + \alpha\)} + If \(\beta x + \alpha\) is a factor of \(P(x)\), \\ + \-\hspace{1em}then \(P(-\dfrac{\alpha}{\beta})=0\). + \end{theorembox} + + \begin{theorembox}{title=Simple form \(x-a\)} + If \((x-a)\) is a factor of \(P(x)\), remainder \(R=0\). \\ + \-\hspace{1em}\(\implies P(a)=0\) + \end{theorembox} + + \subsection*{Remainder theorem} + + \begin{theorembox}{} + When \(P(x)\) is divided by \(\beta x + \alpha\), the remainder is \(-\dfrac{\alpha}{\beta}\). + \end{theorembox} + + \subsection*{Rational root theorem} + 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). + + If \(\beta x + \alpha\) is a factor of \(P(x)\), then \(\beta\) divides \(a_n\) and \(\alpha\) divides \(a_0\) . + + \subsubsection*{Discriminant} + \[\begin{cases} + b^2-4ac > 0 & \text{two solutions} \\ + b^2-4ac = 0 & \text{one solution} \\ + b^2-4ac < 0 & \text{no solutions} + \end{cases}\] + \begin{warning} + Flip inequality sign when multiplying by -1 + \end{warning} + + \subsection*{Long division} + + \[ \polylongdiv{x^2+2x+4}{x-1} \] + + \begin{cas} + Action \(\rightarrow\) Transformation \(\rightarrow\) \texttt{propFrac} + \end{cas} + \subsection*{Linear equations} \subsubsection*{Forms} @@ -322,8 +369,11 @@ For \(x^n\), parity of \(n \equiv\) parity of function Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) \subsection*{Quadratics} + \setlength{\abovedisplayskip}{1pt} \setlength{\belowdisplayskip}{1pt} + + \textbf{Linear factorisation} \[ x^2 + bx + c = (x+m)(x+n) \] \hfill where \(mn=c, \> m+n=b\) diff --git a/spec/calculus-rules.tex b/spec/calculus-rules.tex new file mode 100644 index 0000000..5371aaf --- /dev/null +++ b/spec/calculus-rules.tex @@ -0,0 +1,64 @@ +\subsection*{Derivatives} + +\rowcolors{1}{white}{peach} +\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} + +\vfill + +\subsection*{Antiderivatives} + +\rowcolors{1}{white}{lblue} +\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} + +\vspace{1em} +Note \(\sin^{-1} \left(\dfrac{x}{a}\right) + \cos^{-1} \left(\dfrac{x}{a}\right)\) is constant \(\forall \> x \in (-a, a)\) diff --git a/spec/normal-dist-graph.tex b/spec/normal-dist-graph.tex index b0b558a..073bee8 100644 --- a/spec/normal-dist-graph.tex +++ b/spec/normal-dist-graph.tex @@ -52,9 +52,9 @@ \fill[lightgray!30] (-2,0) -- plot[id=f3,domain=-2:2,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (2,0) -- cycle; \fill[white!30] (-1,0) -- plot[id=f3,domain=-1:1,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (1,0) -- cycle; \begin{scope}[<->] - \draw (-1,0.35) -- (1,0.35) node [midway, fill=white] {68.3\%}; - \draw (-2,0.25) -- (2,0.25) node [midway, fill=white] {95.5\%}; - \draw (-3,0.15) -- (3,0.15) node [midway, fill=white] {99.7\%}; + \draw (-1,0.35) -- (1,0.35) node [midway, fill=white] {68.27\%}; + \draw (-2,0.25) -- (2,0.25) node [midway, fill=white] {95.35\%}; + \draw (-3,0.15) -- (3,0.15) node [midway, fill=white] {99.73\%}; \end{scope} \begin{scope}[-, dashed, gray] \draw (-1,0) -- (-1, 0.35); diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 705bf5d..7b998a9 100644 Binary files a/spec/spec-collated.pdf and b/spec/spec-collated.pdf differ diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex index 4adcd53..94d6187 100644 --- a/spec/spec-collated.tex +++ b/spec/spec-collated.tex @@ -2,6 +2,7 @@ \usepackage[dvipsnames, table]{xcolor} \usepackage{amsmath} \usepackage{amssymb} +\usepackage{array} \usepackage{blindtext} \usepackage{dblfloatfix} \usepackage{enumitem} @@ -9,6 +10,7 @@ \usepackage[a4paper,margin=2cm]{geometry} \usepackage{graphicx} \usepackage{harpoon} +\usepackage{hhline} \usepackage{import} \usepackage{keystroke} \usepackage{listings} @@ -19,6 +21,7 @@ \usepackage{multirow} \usepackage{pgfplots} \usepackage{pst-plot} +\usepackage{rotating} \usepackage{subfiles} \usepackage{tabularx} \usepackage{tcolorbox} @@ -43,6 +46,7 @@ scopes } +\newcommand\given[1][]{\:#1\vert\:} \newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} \usepgflibrary{arrows.meta} @@ -81,14 +85,17 @@ \definecolor{cas}{HTML}{e6f0fe} \definecolor{important}{HTML}{fc9871} \definecolor{dark-gray}{gray}{0.2} +\definecolor{light-gray}{HTML}{cccccc} +\definecolor{peach}{HTML}{e6beb2} +\definecolor{lblue}{HTML}{e5e9f0} \newcommand{\tg}{\mathop{\mathrm{tg}}} \newcommand{\cotg}{\mathop{\mathrm{cotg}}} \newcommand{\arctg}{\mathop{\mathrm{arctg}}} \newcommand{\arccotg}{\mathop{\mathrm{arccotg}}} -\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=important, fontupper=\sffamily\bfseries} -\newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm} +\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=darkgray, fontupper=\sffamily\bfseries} +\newtcolorbox{cas}{colframe=cas!75!black, fonttitle=\sffamily\bfseries, title=On CAS, left*=3mm} \begin{document} @@ -849,46 +856,12 @@ \((b \cdot c)^n = b^n \cdot c^n\)\\ \({a^m \div a^n} = {a^{m-n}}\) - \subsection*{Derivative rules} - - \renewcommand{\arraystretch}{1.4} - \begin{tabularx}{\columnwidth}{rX} - \hline - \(f(x)\) & \(f^\prime(x)\)\\ - \hline - \(\sin x\) & \(\cos x\)\\ - \(\sin ax\) & \(a\cos ax\)\\ - \(\cos x\) & \(-\sin x\)\\ - \(\cos ax\) & \(-a \sin ax\)\\ - \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\ - \(e^x\) & \(e^x\)\\ - \(e^{ax}\) & \(ae^{ax}\)\\ - \(ax^{nx}\) & \(an \cdot e^{nx}\)\\ - \(\log_e x\) & \(\dfrac{1}{x}\)\\ - \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\ - \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\ - \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\ - \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\ - \(\cos^{-1} x\) & \(\dfrac{-1}{sqrt{1-x^2}}\)\\ - \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\ - \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) (reciprocal)\\ - \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx} (product rule)\)\\ - \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) (quotient rule)\\ - \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\ - \hline - \end{tabularx} - \subsection*{Reciprocal derivatives} \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\] \subsection*{Differentiating \(x=f(y)\)} - \begin{align*} - \text{Find }& \frac{dx}{dy}\\ - \text{Then, }\frac{dx}{dy} &= \frac{1}{\frac{dy}{dx}} \\ - \implies {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}\\ - \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}} - \end{align*} + Find \(\dfrac{dx}{dy}\), then \(\dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)}\) \subsection*{Second derivative} \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\ @@ -909,21 +882,21 @@ \begin{tabularx}{\textwidth}{rXXX} \hline \rowcolor{shade2} - & \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\ + & \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\[1.5em] \hline \(\dfrac{dy}{dx}>0\) & - \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-3, xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x))}; \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}& + \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-3, xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x)}; \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}& \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0.1, xmax=4, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))}; \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}& \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1.5, xmax=1.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\ \hline \(\dfrac{dy}{dx}<0\) & - \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {(1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}& + \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}& \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0, xmax=1.5, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(2-x*x)^(1/2)}; \addplot[red] {-x+2}; \end{axis}\end{tikzpicture} \\Falling (concave down)}& \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=1.5, xmax=4.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {-x+3.1415}; \end{axis}\end{tikzpicture} \\Falling inflection point}\\ \hline \(\dfrac{dy}{dx}=0\)& - \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}& \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x))}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}& - \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Stationary inflection point}\\ + \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}& \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x)}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}& + \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Stationary inflection point}\\ \hline \end{tabularx} \end{table*} @@ -950,44 +923,46 @@ \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\] - \noindent \colorbox{cas}{\textbf{On CAS:}}\\ - Action \(\rightarrow\) Calculation \(\rightarrow\) \texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}\\ - Returns \(y^\prime= \dots\). - - \subsection*{Integration} - - \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\] - - \subsection*{Integral laws} - - \renewcommand{\arraystretch}{1.4} - \begin{tabularx}{\columnwidth}{rX} - \hline - \(f(x)\) & \(\int f(x) \cdot dx\) \\ - \hline - \(k\) (constant) & \(kx + c\)\\ - \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\ - \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\ - \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\ - \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\ - \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\ - \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\ - \(e^k\) & \(e^kx + c\)\\ - \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\ - \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\ - \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\ - \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ - \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ - \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\ - \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\ - \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) (substitution)\\ - \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\ - \hline - \end{tabularx} - - Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant \(\forall x \in (-a, a)\) - - \subsection*{Definite integrals} + \begin{cas} + Action \(\rightarrow\) Calculation \\ + \hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)} \hfill(returns \(y^\prime= \dots\)) + \end{cas} + + \subsection*{Slope fields} + + \begin{tikzpicture}[declare function={diff(\x,\y) = \x+\y;}] + \begin{axis}[axis equal, ymin=-4, ymax=4, xmin=-4, xmax=4, ticks=none, enlargelimits=true, ] + \addplot[thick, orange, domain=-4:2] {e^(x)-x-1}; + \pgfplotsinvokeforeach{-4,...,4}{% + \draw[gray] ( {#1 -0.1}, {4 - diff(#1, 4) *0.1}) -- ( {#1 +0.1}, {4 + diff(#1, 4) *0.1}); + \draw[gray] ( {#1 -0.1}, {3 - diff(#1, 3) *0.1}) -- ( {#1 +0.1}, {3 + diff(#1, 3) *0.1}); + \draw[gray] ( {#1 -0.1}, {2 - diff(#1, 2) *0.1}) -- ( {#1 +0.1}, {2 + diff(#1, 2) *0.1}); + \draw[gray] ( {#1 -0.1}, {1 - diff(#1, 1) *0.1}) -- ( {#1 +0.1}, {1 + diff(#1, 1) *0.1}); + \draw[gray] ( {#1 -0.1}, {0 - diff(#1, 0) *0.1}) -- ( {#1 +0.1}, {0 + diff(#1, 0) *0.1}); + \draw[gray] ( {#1 -0.1}, {-1 - diff(#1, -1) *0.1}) -- ( {#1 +0.1}, {-1 + diff(#1, -1) *0.1}); + \draw[gray] ( {#1 -0.1}, {-2 - diff(#1, -2) *0.1}) -- ( {#1 +0.1}, {-2 + diff(#1, -2) *0.1}); + \draw[gray] ( {#1 -0.1}, {-3 - diff(#1, -3) *0.1}) -- ( {#1 +0.1}, {-3 + diff(#1, -3) *0.1}); + \draw[gray] ( {#1 -0.1}, {-4 - diff(#1, -4) *0.1}) -- ( {#1 +0.1}, {-4 + diff(#1, -4) *0.1}); + } + \end{axis} + \end{tikzpicture} + + \subsection*{Parametric equations} + + For each point on \(\left( f(t), g(t) \right)\): + + \begin{align*} + \dfrac{dy}{dt} &= \dfrac{dy}{dx} \cdot \dfrac{dx}{dt} \\ + \therefore \dfrac{dy}{dx} &= \dfrac{\left(\dfrac{dy}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ provided } \dfrac{dx}{dt} \ne 0 \\ + \text{Also...} \\ + \dfrac{d^2y}{dx^2} &= \dfrac{\left(\dfrac{dy^\prime}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ where } y^\prime = \dfrac{dy}{dx} + \end{align*} + + \subsection*{Integration} + + \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\] + + \subsubsection*{Definite integrals} \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\] @@ -1002,21 +977,21 @@ \subsubsection*{Properties} - \[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\] - - \[\int^a_a f(x) \> dx = 0\] - - \[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\] - - \[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\] - - \[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\] + \begin{align*} + \int^b_a f(x) \> dx &= \int^c_a f(x) \> dx + \int^b_c f(x) \> dx \\ + \int^a_a f(x) \> dx &= 0 \\ + \int^b_a k \cdot f(x) \> dx &= k \int^b_a f(x) \> dx \\ + \int^b_a f(x) \pm g(x) \> dx &= \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx \\ + \int^b_a f(x) \> dx &= - \int^a_b f(x) \> dx \\ + \end{align*} \subsection*{Integration by substitution} \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\] - \noindent Note \(f(u)\) must be 1:1 \(\implies\) one \(x\) for each \(y\) + \begin{warning} + \(\boldsymbol{f(u)}\) must be 1:1 \(\boldsymbol{\implies}\) one \(\boldsymbol{x}\) for each \(\boldsymbol{y}\) + \end{warning} \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\ \text{let } u&=x+4\\ \implies& {\frac{du}{dx}} = 1\\ @@ -1059,17 +1034,35 @@ \subsection*{Partial fractions} - \colorbox{cas}{On CAS:}\\ - \indent Action \(\rightarrow\) Transformation \(\rightarrow\) - \texttt{expand/combine}\\ - \indent Interactive \(\rightarrow\) Transformation \(\rightarrow\) - Expand \(\rightarrow\) Partial + To factorise \(f(x) = \frac{\delta}{\alpha \cdot \beta}\): + \begin{align*} + \dfrac{\delta}{\alpha \cdot \beta \cdot \gamma} &= \dfrac{A}{\alpha} + \dfrac{B}{\beta} + \dfrac{C}{\gamma} \tag{1} \\ + \text{Multiply by } & (\alpha \cdot \beta \cdot \gamma) \text{:} \\ + \delta &= \beta\gamma A + \alpha\gamma B +\alpha\beta C \tag{2} \\ + \text{Substitute } x &= \{\alpha, \beta, \gamma\} \text{ into (2) to find denominators} + \end{align*} + + \subsubsection*{Repeated linear factors} + + \[ \dfrac{p(x)}{(x-a)^n} = \dfrac{A_1}{(x-a)} + \dfrac{A_2}{(x-a)^2} + \dots + \dfrac{A_n}{(x-a)^n} \] + + \subsubsection*{Irreducible quadratic factors} + + \[ \text{e.g. } \dfrac{3x-4}{(2x-3)(x^2+5)} = \dfrac{A}{2x-3} + \dfrac{Bx+C}{x^2+5} \] + + \begin{cas} + Action \(\rightarrow\) Transformation:\\ + \hspace{1em} \texttt{expand(..., x)} + + To reverse, use \texttt{combine(...)} + \end{cas} \subsection*{Graphing integrals on CAS} - \colorbox{cas}{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) - \(\int\) (\(\rightarrow\) Definite)\\ - Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..} + \begin{cas} + \textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\ + Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..} + \end{cas} \subsection*{Applications of antidifferentiation} @@ -1094,21 +1087,18 @@ Approximate as sum of infinitesimally-thick cylinders - \subsubsection*{Rotation about \(x\)-axis} + \subsubsection*{Rotation about \(\boldsymbol{x}\)-axis} - \begin{align*} - V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\ - &= \pi \int^b_a (f(x))^2 \> dx - \end{align*} + \[ V = \pi\int^{x=b}_{x=a} f(x)^2 \> dx \] - \subsubsection*{Rotation about \(y\)-axis} + \subsubsection*{Rotation about \(\boldsymbol{y}\)-axis} \begin{align*} - V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\ - &= \pi \int^b_a (f(y))^2 \> dy + V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\ + &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy \end{align*} - \subsubsection*{Regions not bound by \(y=0\)} + \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)} \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\] \hfill where \(f(x) > g(x)\) @@ -1119,10 +1109,12 @@ \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\] - \noindent \colorbox{cas}{On CAS:}\\ - \indent Evaluate formula,\\ - \indent or Interactive \(\rightarrow\) Calculation - \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen} + \begin{cas} + \begin{enumerate}[label=\alph*), leftmargin=5mm] + \item Evaluate formula + \item Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen} + \end{enumerate} + \end{cas} \subsection*{Rates} @@ -1194,7 +1186,8 @@ \[\implies f(x+h) \approx f(x) + hf^\prime(x)\] - + \include{calculus-rules} + \section{Kinematics \& Mechanics} \subsection*{Constant acceleration} diff --git a/spec/statistics.pdf b/spec/statistics.pdf index 5408f9e..d756a7f 100644 Binary files a/spec/statistics.pdf and b/spec/statistics.pdf differ diff --git a/spec/statistics.tex b/spec/statistics.tex index 323ec2a..fc4165c 100644 --- a/spec/statistics.tex +++ b/spec/statistics.tex @@ -167,40 +167,41 @@ Note hypotheses are always expressed in terms of population parameters \end{warning} - \subsection*{Null hypothesis \(H_0\)} + \subsection*{Null hypothesis \(\textbf{H}_0\)} Sample drawn from population has same mean as control population, and any difference can be explained by sample variations. - \subsection*{Alternative hypothesis \(H_1\)} + \subsection*{Alternative hypothesis \(\textbf{H}_1\)} Amount of variation from control is significant, despite standard sample variations. \subsection*{\(p\)-value} + Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true. + For one-tail tests: \begin{align*} - p &= \Pr(\overline{X} \lessgtr \mu(H_1)) \\ - &= 2 \cdot \Pr(\overline{X} <> \mu(H_1) | \mu = 8) + p\text{-value} &= \Pr\left( \> \overline{X} \lessgtr \mu(\textbf{H}_1) \> \given \> \mu = \mu(\textbf{H}_0)\> \right) \\ + &= \Pr\left( Z \lessgtr \dfrac{\left( \mu(\textbf{H}_1) - \mu(\textbf{H}_0) \right) \cdot \sqrt{n} }{\operatorname{sd}(X)} \right) \\ + &\text{then use \texttt{normCdf} with std. norm.} \end{align*} - Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true. - \vspace{0.5em} \begin{tabularx}{23em}{|l|X|} \hline \rowcolor{cas} \(\boldsymbol{p}\) & \textbf{Conclusion} \\ \hline - \(> 0.05\) & insufficient evidence against \(H_0\) \\ - \(< 0.05\) (5\%) & good evidence against \(H_0\) \\ - \(< 0.01\) (1\%) & strong evidence against \(H_0\) \\ - \(< 0.001\) (0.1\%) & very strong evidence against \(H_0\) \\ + \(> 0.05\) & insufficient evidence against \(\textbf{H}_0\) \\ + \(< 0.05\) (5\%) & good evidence against \(\textbf{H}_0\) \\ + \(< 0.01\) (1\%) & strong evidence against \(\textbf{H}_0\) \\ + \(< 0.001\) (0.1\%) & very strong evidence against \(\textbf{H}_0\) \\ \hline \end{tabularx} - \subsection*{Statistical significance} + \subsection*{Significance level \(\alpha\)} - Significance level is denoted by \(\alpha\). + The condition for rejecting the null hypothesis. \-\hspace{1em} If \(p<\alpha\), null hypothesis is \textbf{rejected} \\ \-\hspace{1em} If \(p>\alpha\), null hypothesis is \textbf{accepted} @@ -213,7 +214,7 @@ Menu \(\rightarrow\) Statistics \(\rightarrow\) Calc \(\rightarrow\) Test. \\ Select \textit{One-Sample Z-Test} and \textit{Variable}, then input: \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=2cm, font=\normalfont] - \item[\(\mu\) cond:] same operator as \(H_1\) + \item[\(\mu\) cond:] same operator as \(\textbf{H}_1\) \item[\(\mu_0\):] expected sample mean (null hypothesis) \item[\(\sigma\):] standard deviation (null hypothesis) \item[\(\overline{x}\):] sample mean @@ -222,27 +223,36 @@ \end{cas} \subsection*{One-tail and two-tail tests} + + \[ p\text{-value (two-tail)} = 2 \times p\text{-value (one-tail)} \] \subsubsection*{One tail} \begin{itemize} \item \(\mu\) has changed in one direction - \item State ``\(H_1: \mu \lessgtr \) known population mean'' + \item State ``\(\textbf{H}_1: \mu \lessgtr \) known population mean'' \end{itemize} \subsubsection*{Two tail} \begin{itemize} \item Direction of \(\Delta \mu\) is ambiguous - \item State ``\(H_1: \mu \ne\) known population mean'' + \item State ``\(\textbf{H}_1: \mu \ne\) known population mean'' \end{itemize} - For two tail tests: \begin{align*} p\text{-value} &= \Pr(|\overline{X} - \mu| \ge |\overline{x}_0 - \mu|) \\ - &= \left( |Z| \ge \left|\dfrac{\overline{x}_0 - \mu}{\sigma \div \sqrt{n}} \right| \right) + &= \left( |Z| \ge \left|\dfrac{\overline{x}_0 - \mu}{\sigma \div \sqrt{n}} \right| \right) \\ \end{align*} + where + \begin{description}[nosep, labelindent=0.5cm] + \item [\(\mu\)] is the population mean under \(\textbf{H}_0\) + \item [\(\overline{x}_0\)] is the observed sample mean + \item [\(\sigma\)] is the population s.d. + \item [\(n\)] is the sample size + \end{description} + \subsection*{Modulus notation for two tail} \(\Pr(|\overline{X} - \mu| \ge a) \implies\) ``the probability that the distance between \(\overline{\mu}\) and \(\mu\) is \(\ge a\)'' @@ -256,10 +266,22 @@ \subsection*{Errors} \begin{description}[labelwidth=2.5cm, labelindent=0.5cm] - \item [Type I error] \(H_0\) is rejected when it is \textbf{true} - \item [Type II error] \(H_0\) is \textbf{not} rejected when it is \textbf{false} + \item [Type I error] \(\textbf{H}_0\) is rejected when it is \textbf{true} + \item [Type II error] \(\textbf{H}_0\) is \textbf{not} rejected when it is \textbf{false} \end{description} + \begin{tabularx}{\columnwidth}{|X|l|l|} + \rowcolor{cas}\hline + \cellcolor{white}&\multicolumn{2}{c|}{\textbf{Actual result}} \\ + \hline + \cellcolor{cas}\(\boldsymbol{z}\)\textbf{-test} & \cellcolor{light-gray}\(\textbf{H}_0\) true & \cellcolor{light-gray}\(\textbf{H}_0\) false \\ + \hline + \cellcolor{light-gray}Reject \(\textbf{H}_0\) & Type I error & Correct \\ + \hline + \cellcolor{light-gray}Do not reject \(\textbf{H}_0\) & Correct& Type II error \\ + \hline + \end{tabularx} + % \subsection*{Using c.i. to find \(p\)} % need more here