From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Tue, 5 Nov 2019 05:30:19 +0000 (+1100)
Subject: [spec] additions to complex graphs and exp identities
X-Git-Tag: yr12~1
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/4de6207254326a62227ef93cc0f3b6ccf30f69e2?hp=c26a7be1bbee390dfb0a98d61a52d8dd18ebf7cf

[spec] additions to complex graphs and exp identities
---

diff --git a/spec/calculus-rules.tex b/spec/calculus-rules.tex
index 2b6b712..f60a82f 100644
--- a/spec/calculus-rules.tex
+++ b/spec/calculus-rules.tex
@@ -34,10 +34,10 @@
   \flushbottom
   \subsubsection*{Index identities}
   \begin{align*}
-    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}}
+    a^{x+y} &= a^x \cdot a^y \\
+    a^{x-y} &= a^x \div a^y \\
+    (a^x)^y &= a^{x \cdot y} \\
+    (a \cdot b)^x  &=  a^x \cdot b^x
   \end{align*}
 }
   
@@ -82,7 +82,7 @@ Note \(\sin^{-1} \left(\dfrac{x}{a}\right) + \cos^{-1} \left(\dfrac{x}{a}\right)
   \begin{align*}
     \log_b (xy) &= \log_b x + \log_b y \\
     \log_b\left(\frac{x}{y}\right) &= \log_b(x) - \log_b(y) \\
-    \log_b x^n &= n \log_b x \\
-    \log_b y^{x^n} &= x^n \log_b y
+    \log_b y^{x^n} &= x^n \log_b y \\
+    \log_b x^n &= n \log_b x
   \end{align*}
 }
diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf
index 41d3d5c..8ff3726 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 1358143..97385d2 100644
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -296,12 +296,31 @@
 
     \subsection*{Sketching complex graphs}
 
-    \subsubsection*{Linear}
+    \subsubsection*{Rays/lines \qquad \(\operatorname{Arg}( z\pm b)=\theta\)}
+
+    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
+      \draw [->] (-0.75,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
+      \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
+      \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
+      \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
+      \begin{scope}
+        \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
+        \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
+      \end{scope}
+      \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
+      \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
+      \node [brown, mydot] at (-0.25,0) {};
+      \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
+      \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
+      \node [below, font=\footnotesize] at (1,0) {\(1\)};
+    \end{tikzpicture}\end{center}
 
     \begin{itemize}
+      \item \(\operatorname{Arg}(z \pm b) = \theta\) (ray)
       \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
       \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
-      \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
+      \item \(|z - (a+bi)|=|z - (c+di)| \\ \implies \frac{2(c-a)x + a^2 + b^2 - c^2 - d^2}{2(b-d)}\)
+      \item \(\operatorname{Re}(z) \pm \operatorname{Im}(z) = c\)
     \end{itemize}
 
     \subsubsection*{Circles}
@@ -309,9 +328,10 @@
     \begin{itemize}
       \item \(|z-z_1|^2=c^2|z_2+2|^2\)
       \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
+      \item \(z \overline{z} = r^2\)
     \end{itemize}
 
-    \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
+    \subsubsection*{Regions \qquad \(\operatorname{Arg}(z) \lessgtr \theta\)}
 
     \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
       \draw [->] (0,0) -- (1,0) node [right]  {$\operatorname{Re}(z)$};
@@ -328,24 +348,6 @@
       \node [blue, mydot] {};
     \end{tikzpicture}\end{center}
 
-    \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
-
-    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
-      \draw [->] (-0.75,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
-      \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
-      \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
-      \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
-      \begin{scope}
-        \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
-        \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
-      \end{scope}
-      \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
-      \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
-      \node [brown, mydot] at (-0.25,0) {};
-      \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
-      \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
-      \node [below, font=\footnotesize] at (1,0) {\(1\)};
-    \end{tikzpicture}\end{center}
 
     \section{Vectors}
     \begin{center}\begin{tikzpicture}
@@ -1205,12 +1207,12 @@
 
                   \begin{align*}
                     V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\
-                    &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy
+                    &= 2\pi \int^{x=b}_{x=a} x|f(x)| \> dx
                   \end{align*}
 
-                  \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)}
+                  \subsubsection*{Rotating the area between two graphs}
 
-                  \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
+                  \[V = \pi \int^b_a \left( f(x)^2 - g(x)^2 \right) \> dx\]
                   \hfill where \(f(x) > g(x)\)
 
                   \subsection*{Length of a curve}