From 2fcea49ca330ca7577a91f29d3732a704a8e6e2c Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Wed, 2 Oct 2019 11:29:32 +1000 Subject: [PATCH] [spec/methods] tidy up --- methods/calculus.tex | 60 +------------ methods/circ-functions.tex | 32 ------- methods/methods-collated.pdf | Bin 494956 -> 455528 bytes methods/methods-collated.tex | 16 ++++ methods/statistics-ref.tex | 16 ---- spec/calculus-rules.tex | 16 ++++ spec/spec-collated.pdf | Bin 495113 -> 496019 bytes spec/spec-collated.tex | 166 +++++++++++++++-------------------- spec/statistics.tex | 13 +-- 9 files changed, 113 insertions(+), 206 deletions(-) diff --git a/methods/calculus.tex b/methods/calculus.tex index 05bab2b..b33091e 100644 --- a/methods/calculus.tex +++ b/methods/calculus.tex @@ -97,64 +97,6 @@ For \(x_2\) and \(x_1\) where \(x_2 > x_1\): \end{axis} \end{tikzpicture} -\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} - -\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} +\include{../spec/calculus-rules} \end{document} diff --git a/methods/circ-functions.tex b/methods/circ-functions.tex index 2b615cd..c7c4852 100644 --- a/methods/circ-functions.tex +++ b/methods/circ-functions.tex @@ -70,38 +70,6 @@ \[\cos^2\theta+\sin^2\theta=1\] - \subsection*{Inverse circular functions} - - \begin{tikzpicture} - \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}] - \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)}; - \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)}; - \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ; - \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ; - \addplot[mark=*, blue] coordinates {(1,0)}; - \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ; - \end{axis} - \end{tikzpicture}\\ - - Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain) - - \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\] - \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\) - - \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\] - \hfill where \(\cos y = x, \> y \in [0, \pi]\) - - \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\] - \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\) - - \begin{tikzpicture} - \begin{axis}[yticklabel style={yshift=1.0pt, anchor=north east},x=0.1cm, y=1cm, ymax=2, ymin=-2, xticklabels={}, ytick={-1.5708,1.5708},yticklabels={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}] - \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)}; - \addplot[gray, dotted, thick, domain=-35:35] {1.5708} 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} - \subsection*{\(\sin\) and \(\cos\) graphs} \[ f(x)=a\sin(bx-c)+d \] diff --git a/methods/methods-collated.pdf b/methods/methods-collated.pdf index 6737a92da15778afc00115feb3b6ba902af4bdca..e8c38a3e33e4b6b2e6a5ea9a93c40ae592b765a4 100644 GIT binary patch delta 162229 zcmV(=K-s_SnjYxp8?X!{12r)-m(hU&D1YT#Taz2Nv3}oQ!Mvon8s14Yj6PKB2>$+e{X`qK>zFu)KqBv+Rsy(;%}dSE32dpQE{Rx1`4?9p*~tw)%58P zPq3k%{_BVD|N8Zl=lI~&UtV1Oaz&wJHKCnul1R-bi}lq%{#i|y@VCEBD#ERqymBSi z6Ja@F%)s~6iUSZG2gSrJ(Ma 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zL&#rk9xnmiq$@>*83*4&u-xfUHXQynf0drxwn@qIhv>mkY4V!(nASm~bT(S;O&`0# zi{ND7gCh`h_U>d_GC%BAtd|d7X=VHxlcMFj=cfL}UMwU2m4$xKLx55-p%i0Fz(RAm zMo0lej2m7lXUg_8ug;9&r_)4J!F@*&_L`qSnu{*t3R~CIqgEZppp`PGNeRqQSnqIu zE9SNZhdLBhf1*@->&6FMrlrsAxcLaU*|3kB`m}aXhId0E>9(>Ppi(^WZJ|<{{x~de z$Z1^XIGw(70dmi7NW?$4DLrUhTg{N0)Mg~G#Y~!?sw!h{&3TC7T+ys<<6DSyh^n&LGJB dx \] -\begin{cas} - Define piecewise functions: \\ - \-\hspace{1em}Math3 \(\rightarrow\) - \begin{tikzpicture}% - \draw rectangle (0.5,0.5); - \node at (0.08,0.25) {\(\{\)}; - \filldraw [black] (0.15, 0.4) rectangle(0.25, 0.3); - \draw (0.35, 0.4) rectangle(0.45, 0.3); - \node [font=\footnotesize] at (0.3,0.3) {\verb;,;}; - \draw (0.15, 0.2) rectangle(0.25, 0.1); - \node [font=\footnotesize] at (0.3,0.1) {\verb;,;}; - \draw (0.35, 0.2) rectangle(0.45, 0.1); - \end{tikzpicture} - % TODO: finish this section -\end{cas} - \subsection*{Two random variables \(X, Y\)} If \(X\) and \(Y\) are independent: diff --git a/spec/calculus-rules.tex b/spec/calculus-rules.tex index 9ea9443..ec33726 100644 --- a/spec/calculus-rules.tex +++ b/spec/calculus-rules.tex @@ -31,6 +31,14 @@ \vfill +\subsubsection*{Index identities} + +\(b^{m+n}=b^m \cdot b^n\)\\ +\((b^m)^n=b^{m \cdot n}\)\\ +\((b \cdot c)^n = b^n \cdot c^n\)\\ +\({a^m \div a^n} = {a^{m-n}}\) + + \subsection*{Antiderivatives} \rowcolors{1}{white}{lblue} @@ -63,3 +71,11 @@ \vspace{1em} Note \(\sin^{-1} \left(\dfrac{x}{a}\right) + \cos^{-1} \left(\dfrac{x}{a}\right)\) is constant \(\forall \> x \in (-a, a)\) + +\vfill + +\subsubsection*{Logarithmic identities} + +\(\log_b (xy)=\log_b x + \log_b y\)\\ +\(\log_b x^n = n \log_b x\)\\ +\(\log_b y^{x^n} = x^n \log_b y\) diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 21130bc279b924c900d00c087b43513d957c294c..174123300d682eed2ce36cbdcf45de357ef04085 100644 GIT binary patch delta 90650 zcmV)0K+eC3oF0>-9Q>!{yc@9;EitT+ zY(vzBy&>sgFc^X(0|RORNTR-eGpo9;?&_Wkl*L#-kihh0S7l}8ommCaTsNe7dG_fy 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0}{\delta y \over \delta x}={\frac{dy}{dx}}\] + \subsection*{Limits} \[\lim_{x \rightarrow a}f(x)\] @@ -842,41 +849,11 @@ \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\) \end{enumerate} - \subsection*{Gradients of secants and tangents} + \subsection*{Gradients} \textbf{Secant (chord)} - line joining two points on curve\\ \textbf{Tangent} - line that intersects curve at one point - \subsection*{First principles derivative} - - \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\] - - \subsubsection*{Logarithmic identities} - - \(\log_b (xy)=\log_b x + \log_b y\)\\ - \(\log_b x^n = n \log_b x\)\\ - \(\log_b y^{x^n} = x^n \log_b y\) - - \subsubsection*{Index identities} - - \(b^{m+n}=b^m \cdot b^n\)\\ - \((b^m)^n=b^{m \cdot n}\)\\ - \((b \cdot c)^n = b^n \cdot c^n\)\\ - \({a^m \div a^n} = {a^{m-n}}\) - - \subsection*{Reciprocal derivatives} - - \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\] - - \subsection*{Differentiating \(x=f(y)\)} - 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)\\ - \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*} - - \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken - \subsubsection*{Points of Inflection} \emph{Stationary point} - i.e. @@ -898,6 +875,32 @@ \end{warning} + \subsection*{Second derivative} + \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\ + \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*} + + \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken + + + \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} + \begin{table*}[ht] \centering \begin{tabularx}{\textwidth}{|r|Y|Y|Y|} @@ -950,24 +953,19 @@ \-\hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)} \end{cas} - \subsection*{Slope fields} + \subsection*{Function of the dependent + variable} - \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} + If \({\frac{dy}{dx}}=g(y)\), then + \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express + \(e^c\) as \(A\). + + \subsection*{Reciprocal derivatives} + + \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\] + + \subsection*{Differentiating \(x=f(y)\)} + Find \(\dfrac{dx}{dy}\), then \(\dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)}\) \subsection*{Parametric equations} @@ -982,19 +980,6 @@ \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\] - \subsubsection*{Definite integrals} - - \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\] - - \begin{itemize} - - \item - Signed area enclosed by\\ - \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\). - \item - \emph{Integrand} is \(f\). - \end{itemize} - \subsubsection*{Properties} \begin{align*} @@ -1052,6 +1037,12 @@ \(\sin 2x = 2 \sin x \cos x\) \end{itemize} + \subsection*{Separation of variables} + + If \({\frac{dy}{dx}}=f(x)g(y)\), then: + + \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\] + \subsection*{Partial fractions} To factorise \(f(x) = \frac{\delta}{\alpha \cdot \beta}\): @@ -1084,25 +1075,6 @@ For restrictions, \texttt{Define\ f(x)=...} then \texttt{f(x)\textbar{}x\textgreater{}...} \end{cas} - \subsection*{Applications of antidifferentiation} - - \begin{itemize} - - \item - \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of - stationary points on \(y=F(x)\) - \item - nature of stationary points is determined by sign of \(y=f(x)\) on - either side of its \(x\)-intercepts - \item - if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree - \(n+1\) - \end{itemize} - - To find stationary points of a function, substitute \(x\) value of given - point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find - original function. - \subsection*{Solids of revolution} Approximate as sum of infinitesimally-thick cylinders @@ -1138,6 +1110,25 @@ \end{enumerate} \end{cas} + \subsection*{Applications of antidifferentiation} + + \begin{itemize} + + \item + \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of + stationary points on \(y=F(x)\) + \item + nature of stationary points is determined by sign of \(y=f(x)\) on + either side of its \(x\)-intercepts + \item + if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree + \(n+1\) + \end{itemize} + + To find stationary points of a function, substitute \(x\) value of given + point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find + original function. + \subsection*{Rates} \subsubsection*{Gradient at a point on parametric curve} @@ -1167,26 +1158,13 @@ To verify solutions, find \(\frac{dy}{dx}\) from \(y\) and substitute into original \end{warning} - \subsubsection*{Function of the dependent - variable} - - If \({\frac{dy}{dx}}=g(y)\), then - \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express - \(e^c\) as \(A\). - \subsubsection*{Mixing problems} \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\] - \subsubsection*{Separation of variables} - - If \({\frac{dy}{dx}}=f(x)g(y)\), then: - - \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\] - - \subsubsection*{Euler's method for solving DEs} + \subsection*{Euler's method} \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\] diff --git a/spec/statistics.tex b/spec/statistics.tex index fc4165c..dadaff7 100644 --- a/spec/statistics.tex +++ b/spec/statistics.tex @@ -86,10 +86,11 @@ \begin{cas} - \begin{itemize} - \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left - \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable - \end{itemize} + \hspace{1em} Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); \\ + where \verb;n; is the number of samples. Show histogram with Histogram key in top left. + + To calculate parameters of a dataset: \\ + \-\hspace{1em}Calc \(\rightarrow\) One-variable \end{cas} @@ -112,7 +113,9 @@ \subsection*{Central limit theorem} - If \(X\) is randomly distributed with mean \(\mu\) and sd \(\sigma\), then with an adequate sample size \(n\) the distribution of the sample mean \(\overline{X}\) is approximately normal with mean \(E(\overline{X})\) and \(\operatorname{sd}(\overline{X}) = \frac{\sigma}{\sqrt{n}}\). + \begin{theorembox}{} + If \(X\) is randomly distributed with mean \(\mu\) and sd \(\sigma\), then with an adequate sample size \(n\) the distribution of the sample mean \(\overline{X}\) is approximately normal with mean \(E(\overline{X})\) and \(\operatorname{sd}(\overline{X}) = \frac{\sigma}{\sqrt{n}}\). + \end{theorembox} \subsection*{Confidence intervals} -- 2.47.1