[spec] rates

diff --git a/methods/stuff.md b/methods/stuff.md
index 78c98978c72eaaaeb392b86aa58abcfa3afcf77f..2055b63a7d86e0fb419e1f9b851304547ead4d5b 100644 (file)
--- a/methods/stuff.md
+++ b/methods/stuff.md
@@ -1,5 +1,5 @@
 ---
-geometry: margin=2cm
+geometry: margin=1.5cm
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 graphics: yes
 tables: yes
@@ -41,7 +41,6 @@ If $a<1, \quad \log_{b} a < 0$ (flip inequality operator)
 
 $e^x$ - natural exponential function
 
-
 $$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
 
 ## Logarithm laws
@@ -65,10 +64,6 @@ $$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$$
 
 $$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
 
-## Literal equations
-
-_Literal equation_ - no numerical solutions
-
 ## Exponential and logarithmic modelling
 
 $$A = A_0 e^{kt}$$
@@ -80,29 +75,36 @@ $k$ is a constant
 For continuous growth, $k > 0$  
 For continuous decay, $k < 0$
 
-## Graphing expomnential functions
+## Graphing exponential functions
 
 $$f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1$$
 
-- **$y$-intercept** at $(0, A \cdot a^{-kb}+c)$
+- **$y$-intercept** at $(0, A \cdot a^{-kb}+c)$ as $x \rightarrow \infty$
 - **horizontal asymptote** at $y=c$
 - **domain** is $\mathbb{R}$
 - **range** is $(c, \infty)$
 - dilation of factor $A$ from $x$-axis
 - dilation of factor $1 \over k$ from $y$-axis
 
+![](graphics/exponential-graphs.png){#id .class width=30%} 
+
 ## Graphing logarithmic functions
 
-$log_e x$ is the inverse of $e^x$ (reflection across $y=x$)
+$\log_e x$ 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 $\mathbb{R}^+$
+- **range** is $\mathbb{R}$
 - **vertical asymptote** at $x=b$
 - $y$-intercept exists if $b<0$
 - dilation of factor $A$ from $x$-axis
 - dilation of factor $1 \over k$ from $y$-axis
 
+![](graphics/log-graphs.png){#id .class width=30%} 
+
+## Finding equations
+
+Solve simultaneous equations on CAS: ![](graphics/cas-simultaneous.png){#id .class width=75px}