---
-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
$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
-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. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
**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 $=0$
Stationary where $m=0$.
Find derivative, solve for ${dy \over dx} = 0$
-{#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 $B$
## 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}
---
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:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$
$$y=a(bx-h)^3 + c$$
-- $m=0$ 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 $c$
geometry: margin=2cm
columns: 2
author: Andrew Lorimer
+header-includes:
+- \usepackage{graphicx}
+- \usepackage{tabularx}
---
# Transformation
### $x^n$ where $n \in \mathbb{Z}^+$
-| $n$ is even: | $n$ is odd: |
-| ------------ | ----------- |
-|{#id .class width=20%} | {#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: |
-| ------------ | ----------- |
-|{#id .class width=20%} | {#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: |
-| ------------ | ----------- |
-|{#id .class width=20%} | {#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 $n \in \mathbb{Z}^+$
## 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: