[spec/methods] tidy up

diff --git a/methods/calculus.tex b/methods/calculus.tex
index 05bab2be0fadd4352cd7d2062f86a55a192ce9bb..b33091ed440fa81a8ce94664e3f623089a3c27b6 100644 (file)
--- 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}
 
   \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}
 
 \end{document}

diff --git a/methods/circ-functions.tex b/methods/circ-functions.tex
index 2b615cd7f8f3011cd58e04d22b8317841b4d93e9..c7c48526a75014a6aceef78daa4c79e488f3b07a 100644 (file)
--- a/methods/circ-functions.tex
+++ b/methods/circ-functions.tex
@@ -70,38 +70,6 @@
 
 \[\cos^2\theta+\sin^2\theta=1\]
 
 
 \[\cos^2\theta+\sin^2\theta=1\]
 

-                  \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}
 
 \[ f(x)=a\sin(bx-c)+d \]
 \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 (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 b7accd12f1a571031926ac971b9abe565108ed05..c4d21f10e356277e211572b0c50915794d520009 100644 (file)
--- a/methods/methods-collated.tex
+++ b/methods/methods-collated.tex
@@ -216,6 +216,22 @@ For \(x^n\), parity of \(n \equiv\) parity of function
       \textbf{Open circle:} point included\\
       \textbf{Closed circle:} point not included
 
       \textbf{Open circle:} point included\\
       \textbf{Closed circle:} point not included
 

+\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*{Operations on functions}
 
       For \(f \pm g\) and \(f \times g\):
       \subsection*{Operations on functions}
 
       For \(f \pm g\) and \(f \times g\):

diff --git a/methods/statistics-ref.tex b/methods/statistics-ref.tex
index ec728d8e52d0c226f98e1b1a3ae8e4459e6c6a38..ae4153ec75e8a4c1656ac484a741d36371e9b45f 100644 (file)
--- a/methods/statistics-ref.tex
+++ b/methods/statistics-ref.tex
@@ -157,22 +157,6 @@ A continuous random variable \(X\) has a pdf \(f\) such that:
 
 \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
 
 
 \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
 

-\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:
 \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 9ea94430f41d761bd5ac0b146dd163173c6a9728..ec3372640c264410ddebc10a532a561902233b36 100644 (file)
--- a/spec/calculus-rules.tex
+++ b/spec/calculus-rules.tex
@@ -31,6 +31,14 @@
 
 \vfill
 
 
 \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}
 \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 \(\forall \> x \in (-a, a)\)
 
 \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 (file)
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 785a7fdcb6da91ca2a2b82ef125294daf4e1b9a4..06cb97454a4eb7d37b40de70704bc1a950c0cbaa 100644 (file)
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -236,6 +236,11 @@
       \hline
     \end{tabularx}
 
       \hline
     \end{tabularx}
 

+    \begin{theorembox}{title=Factor theorem}
+      If \(\beta z + \alpha\) is a factor of \(P(z)\), \\
+      \-\hspace{1em}then \(P(-\dfrac{\alpha}{\beta})=0\).
+    \end{theorembox}
+

     \subsection*{\(n\)th roots}
 
     \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
     \subsection*{\(n\)th roots}
 
     \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:


@@ -816,6 +821,8 @@
 \columnbreak
                   \section{Differential calculus}
 
 \columnbreak
                   \section{Differential calculus}
 

+                  \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
+

                   \subsection*{Limits}
 
                   \[\lim_{x \rightarrow a}f(x)\]
                   \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}
 
                       \(f(x)\) 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
 
 
                   \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.
                   \subsubsection*{Points of Inflection}
 
                   \emph{Stationary point} - i.e.


@@ -898,6 +875,32 @@
                   \end{warning}
 
 
                   \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|}
                   \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}
 
                       \-\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}
 
 
                   \subsection*{Parametric equations}
 


@@ -982,19 +980,6 @@
 
                 \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
 
 
                 \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
 

-                  \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*}
                   \subsubsection*{Properties}
 
                   \begin{align*}


@@ -1052,6 +1037,12 @@
                       \(\sin 2x = 2 \sin x \cos x\)
                   \end{itemize}
 
                       \(\sin 2x = 2 \sin x \cos x\)
                   \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 \(f(x) = \frac{\delta}{\alpha \cdot \beta}\):
                   \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}
 
                     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
                   \subsection*{Solids of revolution}
 
                   Approximate as sum of infinitesimally-thick cylinders


@@ -1138,6 +1110,25 @@
                     \end{enumerate}
                   \end{cas}
 
                     \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}
                   \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}
 
                     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*{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}

 
                   \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
 
 
                   \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
 

diff --git a/spec/statistics.tex b/spec/statistics.tex
index fc4165c7a8631ca23a7d4db8382643e3fde52b0b..dadaff7a8849fd2a6550aea4bc5c624306fe3b5f 100644 (file)
--- a/spec/statistics.tex
+++ b/spec/statistics.tex
@@ -86,10 +86,11 @@
 
   \begin{cas}
   
 
   \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}
   
 
   \end{cas}
   


@@ -112,7 +113,9 @@
 
   \subsection*{Central limit theorem}
 
 
   \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}
 
 
   \subsection*{Confidence intervals}