From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Sat, 16 Mar 2019 07:01:49 +0000 (+1100)
Subject: Merge branch 'master' of ssh://charles/tank/andrew/school/notes
X-Git-Tag: yr12~202
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/06f0a815384df34b94cf6ac201e7d5ef67de3b73?hp=bceb0cd45bfb006f84ff715ef58d016f00dfa811

Merge branch 'master' of ssh://charles/tank/andrew/school/notes
---

diff --git a/methods/stuff.md b/methods/stuff.md
index bf8fbb0..78c9897 100644
--- a/methods/stuff.md
+++ b/methods/stuff.md
@@ -12,15 +12,17 @@ header-includes: \pagenumbering{gobble}
 
 ## Index laws
 
-$a^m \times a^n = a^{m+n}$  
-$a^m \div a^n = a^{m-n}4$  
-$(a^m)^n = a^{_mn}$  
-$(ab)^m = a^m b^m$  
-${({a \over b})}^m = {a^m \over b^m}$
+\begin{equation}\begin{split}
+  a^m \times a^n & = a^{m+n} \\
+  a^m \div a^n & = a^{m-n}4 \\
+  (a^m)^n & = a^{_mn} \\
+  (ab)^m & = a^m b^m \\
+  {({a \over b})}^m & = {a^m \over b^m}
+\end{split}\end{equation}
 
 ## Fractional indices
 
-$^n\sqrt{x}=x^{1/n}$
+$$^n\sqrt{x}=x^{1/n}$$
 
 ## Logarithms
 
@@ -44,16 +46,20 @@ $$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
 
 ## Logarithm laws
 
-$\log_a(mn) = \log_am + \log_an$  
-$\log_a({m \over n}) = \log_am - \log_an$  
-$\log_a(m^p) = p\log_am$  
-$\log_a(m^{-1}) = -\log_am$  
-$\log_a1 = 0$ and $\log_aa = 1$
+\begin{equation}\begin{split}
+  \log_a(mn) & = \log_am + \log_an \\
+  \log_a({m \over n}) & = \log_am - \log_a \\
+  \log_a(m^p) & = p\log_am \\
+  \log_a(m^{-1}) & = -\log_am \\
+  \log_a1 = 0 & \text{ and } \log_aa = 1
+\end{split}\end{equation}
 
 
 ## Inverse functions
 
-Inverse of $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$ is $f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$
+For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is:
+
+$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$$
 
 ## Euler's number
 
@@ -76,9 +82,9 @@ For continuous decay, $k < 0$
 
 ## Graphing expomnential functions
 
-$$f(x)=Aa^{k(x-b)} + c, \quad \vert a > 1$$
+$$f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1$$
 
-- **$y$-intercept** at $(0, {{1+c} \over {a^b}})$
+- **$y$-intercept** at $(0, A \cdot a^{-kb}+c)$
 - **horizontal asymptote** at $y=c$
 - **domain** is $\mathbb{R}$
 - **range** is $(c, \infty)$
@@ -92,10 +98,11 @@ $log_e x$ is the inverse of $e^x$ (reflection across $y=x$)
 $$f(x)=A \log_a k(x-b) + c$$
 
 where
+
 - **domain** is $(b, \infty)$
 - **range** is $\mathbb{R}^+$
 - **vertical asymptote** at $x=b$
 - $y$-intercept exists if $b<0$
-- dilation of factor $A$ from $x$-axis (reflection across $x$-axis when $A < 0$)  
-- dilation of factor $1 \over k$ from $y$-axis (reflection across $y$-axis when $k < 0$)  
+- dilation of factor $A$ from $x$-axis
+- dilation of factor $1 \over k$ from $y$-axis
 
diff --git a/methods/stuff.pdf b/methods/stuff.pdf
index c62aeaf..a15caf8 100644
Binary files a/methods/stuff.pdf and b/methods/stuff.pdf differ