[spec] condense

diff --git a/spec/calculus-rules.tex b/spec/calculus-rules.tex
index 5371aaf6a76d64692bf9f276f87be5762514e2b5..9ea94430f41d761bd5ac0b146dd163173c6a9728 100644 (file)
--- a/spec/calculus-rules.tex
+++ b/spec/calculus-rules.tex
@@ -59,6 +59,7 @@
   \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
   \hline
 \end{tabularx}
   \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
   \hline
 \end{tabularx}

+\rowcolors{2}{white}{white}

 
 \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 \(\forall \> x \in (-a, a)\)

diff --git a/spec/dynamics.tex b/spec/dynamics.tex
index ac373c314afc99003ad741aa060975bb61a8c4bb..f8deab1434ff16adc8d8e7cd6a52e1cd57f2dc51 100644 (file)
--- a/spec/dynamics.tex
+++ b/spec/dynamics.tex
@@ -30,13 +30,13 @@ To convert force \(||\vec{OA}\) to angle-magnitude form, find component \(\perp\
 
 \subsection*{Newton's laws}
 
 
 \subsection*{Newton's laws}
 

-\begin{tcolorbox}
+\begin{theorembox}{}

   \begin{enumerate}[leftmargin=1mm]
     \item Velocity is constant without \(\Sigma F\)
     \item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\)
     \item Equal and opposite forces
   \end{enumerate}
   \begin{enumerate}[leftmargin=1mm]
     \item Velocity is constant without \(\Sigma F\)
     \item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\)
     \item Equal and opposite forces
   \end{enumerate}

-\end{tcolorbox}
+\end{theorembox}

 
 \subsubsection*{Weight}
 A mass of \(m\) kg has force of \(mg\) acting on it
 
 \subsubsection*{Weight}
 A mass of \(m\) kg has force of \(mg\) acting on it


@@ -206,9 +206,14 @@ A mass of \(m\) kg has force of \(mg\) acting on it
 
 \begin{itemize}
   \item \textbf{Suspended pulley:} tension in both sections of rope are equal \\
 
 \begin{itemize}
   \item \textbf{Suspended pulley:} tension in both sections of rope are equal \\

-    \(|a| = g \frac{m_1 - m_2}{m_1 + m_2}\) where \(m_1\) accelerates down \\
-    With tension:
-    \[ \begin{cases}m_1 g - T = m_1 a\\ T - m_2 g = m_2 a\end{cases} \\ \implies m_1 g - m_2 g = m_1 a + m_2 a \]
+    \(|a| = g \dfrac{m_1 - m_2}{m_1 + m_2}\) where \(m_1\) accelerates down \\
+    \[
+    \left\{\begin{array}{lr}
+      m_1g-T = m_1a\\
+      T-m_2g = m_2a
+    \end{array}\right\}
+    \implies m_1 g - m_2 g = m_1 a + m_2 a
+    \]

   \item \textbf{String pulling mass on inclined pane:} Resolve parallel to plane
     \[ T-mg \sin \theta = ma \]
   \item \textbf{Linear connection:} find acceleration of system first
   \item \textbf{String pulling mass on inclined pane:} Resolve parallel to plane
     \[ T-mg \sin \theta = ma \]
   \item \textbf{Linear connection:} find acceleration of system first

diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf
index 1a674192724f96f6140f8f7b6ef89768db68d21e..6f36688793e0fcd53aff9fa4782f029198c39614 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 e084adbe3b00bfbf96abe5203c14a0d184bcbb29..0b1d56eda2635b43dbecf8de3cf3f5d4a228f667 100644 (file)
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -70,6 +70,7 @@
 }
 
 \pagestyle{fancy}
 }
 
 \pagestyle{fancy}

+\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page

 \fancyhead[LO,LE]{Year 12 Specialist}
 \fancyhead[CO,CE]{Andrew Lorimer}
 
 \fancyhead[LO,LE]{Year 12 Specialist}
 \fancyhead[CO,CE]{Andrew Lorimer}
 


@@ -96,11 +97,17 @@
 \newcommand{\arctg}{\mathop{\mathrm{arctg}}}
 \newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
 
 \newcommand{\arctg}{\mathop{\mathrm{arctg}}}
 \newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
 

-\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}
 \newtcolorbox{cas}{colframe=cas!75!black, fonttitle=\sffamily\bfseries, title=On CAS, left*=3mm}

+\newtcolorbox{theorembox}[1]{colback=green!10!white, colframe=blue!20!white, coltitle=black, fontupper=\sffamily, fonttitle=\sffamily, #1}
+\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=darkgray, fontupper=\sffamily\bfseries}

 
 \begin{document}
 
 
 \begin{document}
 

+\title{\vspace{-23mm}Year 12 Specialist\vspace{-5mm}}
+\author{Andrew Lorimer}
+\date{}
+\maketitle
+\vspace{-10mm}

 \begin{multicols}{2}
 
   \section{Complex numbers}
 \begin{multicols}{2}
 
   \section{Complex numbers}


@@ -114,10 +121,9 @@
 
   \subsection*{Operations}
 
 
   \subsection*{Operations}
 

-  \definecolor{shade1}{HTML}{ffffff}
-  \definecolor{shade2}{HTML}{e6f2ff}
-  \definecolor{shade3}{HTML}{cce2ff}
-  \begin{tabularx}{\columnwidth}{r|X|X}
+  \begin{tabularx}{\columnwidth}{|r|X|X|}
+    \hline
+    \rowcolor{cas}

     & \textbf{Cartesian} & \textbf{Polar} \\
     \hline
     \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
     & \textbf{Cartesian} & \textbf{Polar} \\
     \hline
     \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\


@@ -128,7 +134,8 @@
     \hline
     \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
     \hline
     \hline
     \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
     \hline

-    \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
+    \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\) \\
+    \hline

   \end{tabularx}
 
   \subsubsection*{Scalar multiplication in polar form}
   \end{tabularx}
 
   \subsubsection*{Scalar multiplication in polar form}


@@ -145,7 +152,6 @@
       \overline{z} &= a \mp bi\\
       &= r \operatorname{cis}(-\theta)
     \end{align*}
       \overline{z} &= a \mp bi\\
       &= r \operatorname{cis}(-\theta)
     \end{align*}

-

     \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
 
     \subsubsection*{Properties}
     \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
 
     \subsubsection*{Properties}


@@ -153,7 +159,7 @@
     \begin{align*}
       \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
       \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
     \begin{align*}
       \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
       \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\

-      \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
+      \overline{kz} &= k\overline{z} \> \forall \>  k \in \mathbb{R}\\

       z\overline{z} &= (a+bi)(a-bi)\\
       &= a^2 + b^2\\
       &= |z|^2
       z\overline{z} &= (a+bi)(a-bi)\\
       &= a^2 + b^2\\
       &= |z|^2


@@ -185,7 +191,7 @@
       \frac{z_1}{z_2}&=z_1z_2^{-1}\\
       &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
       &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
       \frac{z_1}{z_2}&=z_1z_2^{-1}\\
       &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
       &=\frac{(a+bi)(c-di)}{c^2+d^2}\\

-      & \qquad \text{(rationalise denominator)}
+      & \text{then rationalise denominator}

     \end{align*}
 
     \subsection*{Polar form}
     \end{align*}
 
     \subsection*{Polar form}


@@ -199,11 +205,14 @@
       \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
       \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
       \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
       \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
       \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
       \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}

-      \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}

       \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
       \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
     \end{itemize}
 
       \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
       \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
     \end{itemize}
 

+    \begin{cas}
+      \-\hspace{1em}\verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|
+    \end{cas}
+

     \subsection*{de Moivres' theorem}
 
     \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
     \subsection*{de Moivres' theorem}
 
     \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]


@@ -882,20 +891,21 @@
 
                   For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
 
 
                   For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
 

-                  \textbf{Strictly increasing}\\
-                  \hspace{1em}where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
+                  \textbf{strictly increasing}\\
+                  \-\hspace{1em}where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
+

                   \textbf{strictly decreasing}\\
                   \hspace{1em}where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
                   \begin{warning}
                   \textbf{strictly decreasing}\\
                   \hspace{1em}where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
                   \begin{warning}

-                    Endpoints are included, even where gradient \(=0\)
+                    Endpoints are included, even where \(\boldsymbol{\frac{dy}{dx}=0}\)

                   \end{warning}
 
 
                   \begin{table*}[ht]
                     \centering
                   \end{warning}
 
 
                   \begin{table*}[ht]
                     \centering

-                    \begin{tabularx}{\textwidth}{rYYY}
+                    \begin{tabularx}{\textwidth}{|r|Y|Y|Y|}

                       \hline
                       \hline

-                      \rowcolor{shade2}
+                      \rowcolor{lblue}

                       & \adjustbox{margin=0 1ex, valign=m}{\centering\(\dfrac{d^2 y}{dx^2} > 0\)}  & \adjustbox{margin=0 1ex, valign=m}{\centering \(\dfrac{d^2y}{dx^2}<0\)} & \adjustbox{margin=0 1ex, valign=m}{\(\dfrac{d^2y}{dx^2}=0\) (inflection)} \\
                       \hline
                       \(\dfrac{dy}{dx}>0\) &
                       & \adjustbox{margin=0 1ex, valign=m}{\centering\(\dfrac{d^2 y}{dx^2} > 0\)}  & \adjustbox{margin=0 1ex, valign=m}{\centering \(\dfrac{d^2y}{dx^2}<0\)} & \adjustbox{margin=0 1ex, valign=m}{\(\dfrac{d^2y}{dx^2}=0\) (inflection)} \\
                       \hline
                       \(\dfrac{dy}{dx}>0\) &


@@ -916,16 +926,17 @@
                   \end{table*}
                   \begin{itemize}
                     \item
                   \end{table*}
                   \begin{itemize}
                     \item

-                      if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
-                      \((a, f(a))\) is a local min (curve is concave up)
+                      \(f^\prime (a) = 0, \> f^{\prime\prime}(a) > 0\) \\
+                      \textbf{local min} at \((a, f(a))\) (concave up)

                     \item
                     \item

-                      if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
-                      \((a, f(a))\) is local max (curve is concave down)
+                      \(f^\prime (a) = 0, \>  f^{\prime\prime} (a) < 0\) \\
+                      \textbf{local max} at \((a, f(a))\) (concave down)

                     \item
                     \item

-                      if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
-                      inflection
+                      \(f^{\prime\prime}(a) = 0\) \\
+                      \textbf{point of inflection} at \((a, f(a))\)

                     \item
                     \item

-                      if also \(f^\prime(a)=0\), then it is a stationary point of inflection
+                      \(f^{\prime\prime}(a) = 0, \> f^\prime(a)=0\) \\
+                      stationary point of inflection at \((a, f(a)\)

                   \end{itemize}
 
                   \subsection*{Implicit Differentiation}
                   \end{itemize}
 
                   \subsection*{Implicit Differentiation}


@@ -939,7 +950,7 @@
 
                   \begin{cas}
                     Action \(\rightarrow\) Calculation \\
 
                   \begin{cas}
                     Action \(\rightarrow\) Calculation \\

-                      \hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)} \hfill(returns \(y^\prime= \dots\))
+                      \-\hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}

                   \end{cas}
 
                   \subsection*{Slope fields}
                   \end{cas}
 
                   \subsection*{Slope fields}


@@ -963,12 +974,10 @@
 
                   \subsection*{Parametric equations}
 
 
                   \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 \\
 
                   \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*}
 
                     \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*}
 


@@ -1066,7 +1075,7 @@
 
                   \begin{cas}
                     Action \(\rightarrow\) Transformation:\\
 
                   \begin{cas}
                     Action \(\rightarrow\) Transformation:\\

-                    \hspace{1em} \texttt{expand(..., x)}
+                    \-\hspace{1em} \texttt{expand(..., x)}

 
                     To reverse, use \texttt{combine(...)}
                   \end{cas}
 
                     To reverse, use \texttt{combine(...)}
                   \end{cas}


@@ -1075,7 +1084,7 @@
 
                   \begin{cas}
                     \textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\
 
                   \begin{cas}
                     \textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\

-                    Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
+                    For restrictions, \texttt{Define\ f(x)=...} then \texttt{f(x)\textbar{}x\textgreater{}...}

                   \end{cas}
 
                   \subsection*{Applications of antidifferentiation}
                   \end{cas}
 
                   \subsection*{Applications of antidifferentiation}


@@ -1142,21 +1151,6 @@
 
                   \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
 
 
                   \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
 

-                  \subsubsection*{Addition of ordinates}
-
-                  \begin{itemize}
-
-                    \item
-                      when two graphs have the same ordinate, \(y\)-coordinate is double the
-                      ordinate
-                    \item
-                      when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
-                      (\(x\)-intercept)
-                    \item
-                      when one of the ordinates is 0, the resulting ordinate is equal to the
-                      other ordinate
-                  \end{itemize}
-

                   \subsection*{Fundamental theorem of calculus}
 
                   If \(f\) is continuous on \([a, b]\), then
                   \subsection*{Fundamental theorem of calculus}
 
                   If \(f\) is continuous on \([a, b]\), then


@@ -1170,10 +1164,9 @@
                   \textbf{Degree} - highest power of highest derivative\\
                   e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
 
                   \textbf{Degree} - highest power of highest derivative\\
                   e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
 

-                  \subsubsection*{Verifying solutions}
-
-                  Start with \(y=\dots\), and differentiate. Substitute into original
-                  equation.
+                  \begin{warning}
+                    To verify solutions, find \(\frac{dy}{dx}\) from \(y\) and substitute into original
+                  \end{warning}

 
                   \subsubsection*{Function of the dependent
                   variable}
 
                   \subsubsection*{Function of the dependent
                   variable}


@@ -1213,23 +1206,23 @@
 
       \subsubsection*{Velocity-time graphs}
 
 
       \subsubsection*{Velocity-time graphs}
 

-      \begin{itemize}
-        \item Displacement: \textit{signed} area between graph and \(t\) axis
-        \item Distance travelled: \textit{total} area between graph and \(t\) axis
-      \end{itemize}
+      \begin{description}[nosep, labelindent=0.5cm, leftmargin=0.5\columnwidth]
+        \item[Displacement:] \textit{signed} area
+        \item[Distance travelled:] \textit{total} area
+      \end{description}

 
       \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
 
         \begin{center}
           \renewcommand{\arraystretch}{1}
           \begin{tabular}{ l r }
 
       \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
 
         \begin{center}
           \renewcommand{\arraystretch}{1}
           \begin{tabular}{ l r }

-              \hline & no \\ \hline
-              \(v=u+at\) & \(x\) \\
-              \(v^2 = u^2+2as\) & \(t\) \\
-              \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
-              \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
-              \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
-            \end{tabular}
+            \hline & no \\ \hline
+            \(v=u+at\) & \(x\) \\
+            \(v^2 = u^2+2as\) & \(t\) \\
+            \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
+            \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
+            \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
+          \end{tabular}

         \end{center}
 
         \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
         \end{center}
 
         \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]