From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Mon, 13 May 2019 06:49:10 +0000 (+1000)
Subject: [methods] tidy up & render notes for SAC
X-Git-Tag: yr12~141
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/150b93e7009d4e771992eff84c036502b93ac4a2?ds=sidebyside

[methods] tidy up & render notes for SAC
---

diff --git a/methods/calculus-ref.md b/methods/calculus-ref.md
index 0ec4f7b..215845d 100644
--- a/methods/calculus-ref.md
+++ b/methods/calculus-ref.md
@@ -1,13 +1,16 @@
 ---
-geometry: margin=2cm
+geometry: margin=1cm
 columns: 2
 graphics: yes
 tables: yes
 author: Andrew Lorimer
+header-includes:
+- \usepackage{tabularx}
 ---
 
 
 \pagenumbering{gobble}
+\renewcommand{\arraystretch}{1.4}
 
 
 # Methods - Calculus
@@ -16,18 +19,14 @@ author: Andrew Lorimer
 
 $$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
 
-On CAS: (Action|Interactive) $\rightarrow$ Calculation $\rightarrow$ Diff $\rightarrow$ $f(x)$ or $y=\dots$
+On CAS: Action $\rightarrow$ Calculation $\rightarrow$ Diff $\rightarrow$ ($f(x)$ | $y$) $=\dots$
 
 ## Instantaneous rate of change
 
 **Secant** - line passing through two points on a curve  
 **Chord** - line segment joining two points on a curve
 
-Estimated by using two given points on each side of the concerned point.
-
-## Limits & continuity
-
-### Limit theorems
+## Limit theorems
 
 1. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
 2. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
@@ -52,14 +51,10 @@ Not differentiable at:
 **Normal line** - $\perp$ tangent ($m_{{tan}} \cdot m_{\operatorname{norm}} = -1$)  
 **Secant** $={{f(x+h)-f(x)} \over h}$
 
-$$\tan \theta = m = f^\prime (x)$$
-
-where $\theta$ is the angle that tangent line makes with +ve direction of $x$-axis
-
 ## Strictly increasing
 
-- $f$ is **strictly increasing** where $f(x_2) > f(x_1)$ and $x_2 > x_1$
-- $f$ is **strictly decreasing** where $f(x_2) < f(x_1)$ and $x_2 > x_1$
+- **strictly increasing** where $f(x_2) > f(x_1)$ and $x_2 > x_1$
+- **strictly decreasing** where $f(x_2) < f(x_1)$ and $x_2 > x_1$
 - If $f^\prime (x) > 0$ for all $x$ in interval, then $f$ is **strictly increasing**
 - If $f^\prime(x) < 0$ for all $x$ in interval, then $f$ is **strictly decreasing**
 - Endpoints are included, even where gradient $=0$
@@ -74,13 +69,17 @@ where $\theta$ is the angle that tangent line makes with +ve direction of $x$-ax
 Stationary where $m=0$.  
 Find derivative, solve for ${dy \over dx} = 0$
 
-![](graphics/stationary-points.png){#id .class width=50%}
+\begin{center}
+  \includegraphics[height=3cm]{graphics/stationary-points.png}
+\end{center}
+
+**Local maximum at point $A$**
 
-**Local maximum at point $A$**  
 - $f^\prime (x) > 0$ left of $A$
 - $f^\prime (x) < 0$ right of $A$
 
-**Local minimum at point $B$**  
+**Local minimum at point $B$**
+
 - $f^\prime (x) < 0$ left of $B$
 - $f^\prime (x) > 0$ right of $B$
 
@@ -88,14 +87,25 @@ Find derivative, solve for ${dy \over dx} = 0$
 
 ## Function derivatives
 
-
-| $f(x)$ | $f^\prime(x)$ |
-| ------ | ------------- |
-| $x^n$  | $nx^{n-1}$ |
-| $kx^n$ | $knx^{n-1}$ |
-| $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ |
-| $c$    | $0$ |
-| ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ |
-| $uv$ | $u{dv \over dx} + v{du \over dx}$ |
-| $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ |
-
+\begin{tabularx}{\columnwidth}{rl}
+  
+  \hline \(f(x)\) & \(f^\prime(x)\) \\ \hline
+
+  \(kx^n\) & \(knx^{n-1}\)\tabularnewline
+  \(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline
+  \(c\) & \(0\)\tabularnewline
+  \({u \over v}\) &
+  \({{(v{du \over dx} - u{dv \over dx}}) \div v^2}\)\tabularnewline
+  \(uv\) & \(u{dv \over dx} + v{du \over dx}\)\tabularnewline
+  \(f \circ g\) & \({dy \over du} \cdot {du \over dx}\)\tabularnewline
+  \(\sin ax\) & \(a\cos ax\)\tabularnewline
+  \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline
+  \(\cos ax\) & \(-a \sin ax\)\tabularnewline
+  \(\cos(f(x))\) & \(f^\prime(x)(-\sin(f(x)))\) \\
+  \(e^{ax}\) & \(ae^{ax}\)\tabularnewline
+  \(\log_e {ax}\) & \(1 \over x\)\tabularnewline
+  \(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline
+  
+  \hline
+
+\end{tabularx}
diff --git a/methods/calculus-ref.pdf b/methods/calculus-ref.pdf
index c783d70..06fb928 100644
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diff --git a/methods/inverse-functions.pdf b/methods/inverse-functions.pdf
new file mode 100644
index 0000000..5e5f1f8
Binary files /dev/null and b/methods/inverse-functions.pdf differ
diff --git a/methods/polynomials-ref.pdf b/methods/polynomials-ref.pdf
index 3aaef4e..98b6422 100644
Binary files a/methods/polynomials-ref.pdf and b/methods/polynomials-ref.pdf differ
diff --git a/methods/polynomials.md b/methods/polynomials.md
index afd4eab..c826306 100644
--- a/methods/polynomials.md
+++ b/methods/polynomials.md
@@ -1,20 +1,29 @@
 ---
 geometry: margin=1.5cm
+columns: 2
+header-includes:
+- \usepackage{tabularx}
 ---
 
+\pagenumbering{gobble}
+\renewcommand{\arraystretch}{1.4}
+
 # Polynomials
 
-## Factorising
+## Quadratics
+
+\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
+\begin{tabularx}{\columnwidth}{|R|l|}
+  Quadratics & $x^2 + bx + c = (x+m)(x+n)$ \\
+  & where $mn=c, \> m+n=b$ \\
+  Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\
+  Perfect squares & $a^2 \pm 2ab + b^2 = (a \pm b^2)$ \\
+  Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\
+  Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\
+\end{tabularx}
 
-#### Quadratics
-**Quadratics:** $x^2 + bx + c = (x+m)(x+n)$ where $mn=c$, $m+n=b$  
-**Difference of squares:** $a^2 - b^2 = (a - b)(a + b)$  
-**Perfect squares:** $a^2 \pm 2ab + b^2 = (a \pm b^2)$  
-**Completing the square (monic):** $x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$  
-**Completing the square (non-monic):** $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$  
-**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ (if $\Delta$ is a perfect square, rational roots)
+## Cubics
 
-#### Cubics
 **Difference of cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$  
 **Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$  
 **Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$  
@@ -38,7 +47,7 @@ Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
 
 $$y=a(bx-h)^3 + c$$
 
-- $m=0$ at *stationary point of inflection* (i.e. ({h \over b}, k)$)
+- $m=0$ at *stationary point of inflection* (i.e. (${h \over b}, k)$)
 - in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
 - in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
 - in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$
diff --git a/methods/transformations.md b/methods/transformations.md
index 153441b..0378cae 100644
--- a/methods/transformations.md
+++ b/methods/transformations.md
@@ -2,6 +2,9 @@
 geometry: margin=2cm
 columns: 2
 author: Andrew Lorimer
+header-includes:
+- \usepackage{graphicx}
+- \usepackage{tabularx}
 ---
 
 # Transformation
@@ -64,22 +67,24 @@ Function $x^{\pm {p \over q}}$ is odd if $q$ is odd
 
 ### $x^n$ where $n \in \mathbb{Z}^+$
 
-| $n$ is even: | $n$ is odd: |
-| ------------ | ----------- |
-|![](graphics/parabola.png){#id .class width=20%} | ![](graphics/cubic.png){#id .class width=20%} |
+\begin{tabularx}{\textwidth}{|c|c|}
+  \(n\) is even & \(n\) is odd\\
+  {\includegraphics[height=1cm]{graphics/parabola.png}} & {\includegraphics[height=1cm]{graphics/cubic.png}}
+\end{tabularx}
 
 ### $x^n$ where $n \in \mathbb{Z}^-$
 
-| $n$ is even: | $n$ is odd: |
-| ------------ | ----------- |
-|![](graphics/truncus.png){#id .class width=20%} | ![](graphics/hyperbola.png){#id .class width=20%} |
+\begin{tabularx}{\textwidth}{|c|c|}
+  \(n\) is even & \(n\) is odd\\
+  {\includegraphics[height=1cm]{graphics/truncus.png}} & {\includegraphics[height=1cm]{graphics/hyperbola.png}}
+\end{tabularx}
 
 ### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$
 
-| $n$ is even: | $n$ is odd: |
-| ------------ | ----------- |
-|![](graphics/square-root-graph.png){#id .class width=20%} | ![](graphics/cube-root-graph.png){#id .class width=20%} |
-
+\begin{tabularx}{\textwidth}{|c|c|}
+  \(n\) is even & \(n\) is odd\\
+  {\includegraphics[height=1cm]{graphics/square-root-graph.png}} & {\includegraphics[height=1cm]{graphics/cube-root-graph.png}}
+\end{tabularx}
 
 ### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
 
@@ -102,18 +107,18 @@ $$x^{p \over q} = \sqrt[q]{x^p}$$
 
 ## Combinations of functions (piecewise/hybrid)
 
-$$\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
+$$\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
 
 Open circle - point included  
 Closed circle - point not included  
 
 ### Sum, difference, product of functions
-| | | |
-|---|-----|-----|
-|sum|$f+g$|domain $= \text{dom}(f) \cap \text{dom}(g)$|
-|difference|$f-g$ or $g-f$|domain $=\text{dom}(f) \cap \text{dom}(g)$|
-|product|$f \times g$|domain $=\text{dom}(f) \cap \text{dom}(g)$|
-
+\begin{tabularx}{\columnwidth}{X|X}
+  sum & $f+g$ & domain $= \text{dom}(f) \cap \text{dom}(g)$ \\
+  difference & $f-g$ or $g-f$ & domain $=\text{dom}(f) \cap \text{dom}(g)$ \\
+  product & $f \times g$ & domain $=\text{dom}(f) \cap \text{dom}(g)$
+\end{tabularx}
+  
 Addition of linear piecewise graphs - add $y$-values at key points
 
 Product functions: