---
-geometry: margin=1cm
+geometry: a4paper, margin=2cm
columns: 2
-graphics: yes
-tables: yes
author: Andrew Lorimer
header-includes:
+- \usepackage{fancyhdr}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+- \usepackage{graphicx}
- \usepackage{tabularx}
+- \usepackage[dvipsnames, table]{xcolor}
---
-
+\linespread{3}
\pagenumbering{gobble}
\renewcommand{\arraystretch}{1.4}
+\definecolor{cas}{HTML}{e6f0fe}
-
-# Methods - Calculus
+# Calculus
## Average rate of change
$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
-On CAS: Action $\rightarrow$ Calculation $\rightarrow$ Diff $\rightarrow$ ($f(x)$ | $y$) $=\dots$
+\colorbox{cas}{On CAS:} Action $\rightarrow$ Calculation $\rightarrow$ `diff`
## Instantaneous rate of change
**Normal line** - $\perp$ tangent ($m_{{tan}} \cdot m_{\operatorname{norm}} = -1$)
**Secant** $={{f(x+h)-f(x)} \over h}$
-## Strictly increasing
+## Strictly increasing/decreasing
-- **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**
+For $x_2$ and $x_1$ where $x_2 > x_1$:
+
+- **strictly increasing** where $f(x_2) > f(x_1)$
+or $f^\prime(x)>0$
+- **strictly decreasing** where $f(x_2) < f(x_1)$
+or $f^\prime(x)<0$
- Endpoints are included, even where gradient $=0$
+\columnbreak
+
### Solving on CAS
-**In main**: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line)
-**In graph**: define function. Analysis $\rightarrow$ Sketch $\rightarrow$ (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.
+\colorbox{cas}{\textbf{In main}}: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line)
+\colorbox{cas}{\textbf{In graph}}: define function. Analysis $\rightarrow$ Sketch $\rightarrow$ (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.
## Stationary points
## Function derivatives
-\begin{tabularx}{\columnwidth}{rl}
+\definecolor{shade1}{HTML}{ffffff}
+\definecolor{shade2}{HTML}{F0F9E4}
+\rowcolors{1}{shade1}{shade2}
+\begin{tabularx}{\columnwidth}{rX}
\hline \(f(x)\) & \(f^\prime(x)\) \\ \hline
- \(kx^n\) & \(knx^{n-1}\)\tabularnewline
+ \hspace{6em} \(kx^n\) & \(knx^{n-1}\)\tabularnewline
\(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline
\(c\) & \(0\)\tabularnewline
\({u \over v}\) &
---
-geometry: margin=2cm
+geometry: a4paper, margin=2cm
columns: 2
-graphics: yes
+author: Andrew Lorimer
+header-includes:
+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \usepackage{graphicx}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
---
-# Circular functions
-<!-- ## Radians and degrees -->
+\setstretch{1.2}
+\pagenumbering{gobble}
-<!-- $$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$ -->
+# Circular functions
## Exact values
\includegraphics[width=0.2\textwidth]{./graphics/exact-values-1.png}
\includegraphics[width=0.2\textwidth]{./graphics/exact-values-2.png}
-<!--  -->
-<!--  -->
+$$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$
## $\sin$ and $\cos$ graphs
+---
+geometry: margin=2cm
+author: Andrew Lorimer
+header-includes:
+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+---
+
+\setstretch{1.3}
+\pagenumbering{gobble}
+
# Inverse functions
## Functions
## One to one functions
-- $f(x)$ is *one to one* if $f(a) \ne f(b)$ if $a, b \in \operatorname{dom}(f)$ and $a \ne b$
-- i.e. unique $y$ for each $x$ ($\sin x$ is not 1:1, $x^3$ is)
+- $f(x)$ is *one to one* if $f(a) \ne f(b)$ if $a, b \in \operatorname{dom}(f)$ and $a \ne b$
+$\implies$ unique $y$ for each $x$ ($\sin x$ is not 1:1, $x^3$ is)
- horizontal line test
- if not one to one, it is many to one
-## Inverse functions $f^{-1}$
+## Deriving $f^{-1}$
- if $f(g(x)) = x$, then $g$ is the inverse of $f$
- reflection across $y-x$
- $\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}$
-- inverse $\ne$ inverse *function* (i.e. inverse must pass vertical line test)
-- - $\implies f^{-1}(x)$ exists $\iff f(x)$ is one to one
+- inverse $\ne$ inverse *function* (i.e. inverse must pass vertical line test)
+$\implies f^{-1}(x)$ exists $\iff f(x)$ is one to one
- $f^{-1}(x)=f(x)$ intersections may lie on line $y=x$
-Requirements for showing working for $f^{-1}$:
+### Requirements for showing working for $f^{-1}$
-- start with *"let $y=f(x)$"*
-- must state *"take inverse"* for line where $y$ and $x$ are swapped
-- do all working in terms of $y=\dots$
-- for square root, state $\pm$ solutions then show restricted
-- for inverse *function*, state in function notation
+1. start with *"let $y=f(x)$"*
+2. must state *"take inverse"* for line where $y$ and $x$ are swapped
+3. do all working in terms of $y=\dots$
+4. for square root, state $\pm$ solutions then show restricted
+5. for inverse *function*, state in function notation
---
-geometry: margin=1.5cm
+geometry: a4paper, margin=2cm
columns: 2
+author: Andrew Lorimer
header-includes:
+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+- \usepackage{graphicx}
- \usepackage{tabularx}
+- \usepackage[dvipsnames]{xcolor}
---
+\setstretch{1.3}
+\definecolor{cas}{HTML}{e6f0fe}
\pagenumbering{gobble}
\renewcommand{\arraystretch}{1.4}
## Quadratics
\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
-\begin{tabularx}{\columnwidth}{|R|l|}
- Quadratics & $x^2 + bx + c = (x+m)(x+n)$ \\
- & where $mn=c, \> m+n=b$ \\
+\begin{tabularx}{\columnwidth}{Rl}
+ General form& \parbox[t]{5cm}{$x^2 + bx + c = (x+m)(x+n)$\\ where $mn=c, \> m+n=b$} \\
+ \hline
Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\
- Perfect squares & $a^2 \pm 2ab + b^2 = (a \pm b^2)$ \\
+ \hline
+ Perfect squares & \parbox[c]{5cm}{$a^2 \pm 2ab + b^2 = (a \pm b^2)$} \\
+ \hline
Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\
+ \hline
Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\
\end{tabularx}
**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. (${h \over b}, k)$)
+- 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 $c$
+
## Linear and quadratic graphs
### Forms of linear equations
Parallel lines: $m_1 = m_2$
Perpendicular lines: $m_1 \times m_2 = -1$
-Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
-
-
-## Cubic graphs
-
-$$y=a(bx-h)^3 + c$$
-
-- $m=0$ 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 $c$
+Distance: $|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
## Quartic graphs
$y=a(x-b)^2(x-c)^2$
$y=a(x-b)(x-c)^3$
-## Literal equations
-
-Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters)
-
## Simultaneous equations (linear)
- **Unique solution** - lines intersect at point
- **Infinitely many solutions** - lines are equal
- **No solution** - lines are parallel
-### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases}$ for one, infinite and no solutions
+### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>$ for $\{0,1,\infty\}$ solutions
where all coefficients are known except for one, and $a, b$ are known
- *--- for infinite/no solutions: ---*
5. Substitute variable into both original equations
6. Rearrange equations so that LHS of each is the same
-7. If $\text{RHS}(1) = \text{RHS}(2)$, lines are coincident (infinite solutions)
- If $\text{RHS}(1) \ne \text{RHS}(2)$, lines are parallel (no solutions)
+7. $\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x$ ($\infty$ solns)
+ $\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x$ (0 solns)
-Or use Matrix -> `det` on CAS.
+\colorbox{cas}{On CAS:} Matrix $\rightarrow$ `det`
### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
a_2 x + b_2 y + c_2 z = d_2 \\
---
-geometry: margin=1.5cm
-<!-- columns: 2 -->
-graphics: yes
-tables: yes
+geometry: a4paper, margin=2cm
+columns: 2
author: Andrew Lorimer
-classoption: twocolumn
-header-includes: \pagenumbering{gobble}
+header-includes:
+- \usepackage{fancyhdr}
+- \usepackage{setspace}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+- \usepackage{graphicx}
+- \usepackage{tabularx}
+- \usepackage[dvipsnames]{xcolor}
---
-# Exponential and Index Functions
+\pagenumbering{gobble}
+\setstretch{1.5}
+\definecolor{cas}{HTML}{e6f0fe}
+
+# Exponentials & Logarithms
## Index laws
-\begin{equation}\begin{split}
+\begin{equation*}\begin{split}
a^m \times a^n & = a^{m+n} \\
- a^m \div a^n & = a^{m-n}4 \\
+ a^m \div a^n & = a^{m-n} \\
(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
-
-$$^n\sqrt{x}=x^{1/n}$$
-
-## Logarithms
-
-$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
-
-## Using logs to solve index eq's
-
-Used for equations without common base exponent
-
-Or change base:
-$$\log_b c = {{\log_a c} \over {\log_a b}}$$
-
-If $a<1, \quad \log_{b} a < 0$ (flip inequality operator)
-
-## Exponential functions
-
-$e^x$ - natural exponential function
-
-$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
+ {({a \over b})}^m & = {a^m \over b^m} \\
+ ^n\sqrt{x} &=x^{1/n}
+\end{split}\end{equation*}
## Logarithm laws
-\begin{equation}\begin{split}
+\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}
-
+ \log_a1 = 0 & \text{ and } \log_aa = 1 \\
+ \log_b c &= {{\log_a c} \over {\log_a b}}
+\end{split}\end{equation*}
## Inverse functions
For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is:
-$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$$
+$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax$$
+
+## Exponentials
-## Euler's number
+$$e^x \quad \text{natural exponential function}$$
$$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
-## Exponential and logarithmic modelling
+## Modelling
$$A = A_0 e^{kt}$$
-where
-$A_0$ is initial value
-$t$ is time taken
-$k$ is a constant
-For continuous growth, $k > 0$
-For continuous decay, $k < 0$
+- $A_0$ is initial value
+- $t$ is time taken
+- $k$ is a constant
+- For continuous growth, $k > 0$
+- For continuous decay, $k < 0$
+
+\columnbreak
## Graphing exponential functions
- **horizontal asymptote** at $y=c$
- **domain** is $\mathbb{R}$
- **range** is $(c, \infty)$
-- dilation of factor $A$ from $x$-axis
+- dilation of factor $|A|$ from $x$-axis
- dilation of factor $1 \over k$ from $y$-axis
{#id .class width=30%}
- **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 $|A|$ from $x$-axis
- dilation of factor $1 \over k$ from $y$-axis
{#id .class width=30%}
## Finding equations
-Solve simultaneous equations on CAS: {#id .class width=75px}
+\colorbox{cas}{On CAS:} {#id .class width=75px}
---
-geometry: margin=2cm
+geometry: a4paper, margin=2cm
columns: 2
author: Andrew Lorimer
header-includes:
+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
- \usepackage{graphicx}
- \usepackage{tabularx}
---
-# Transformation
+\setstretch{1.6}
+\pagenumbering{gobble}
-**Order of operations:** DRT - Dilations, Reflections, Translations
+# Transformations
+
+**Order of operations:** DRT
+
+\begin{center}dilations --- reflections --- translations\end{center}
## Transforming $x^n$ to $a(x-h)^n+K$
-- $|a|$ is the dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
+- dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
- if $a<0$, graph is reflected over $x$-axis
-- $k$ - translation of $k$ units parallel to $y$-axis or from $x$-axis
-- $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
+- translation of $k$ units parallel to $y$-axis or from $x$-axis
+- translation of $h$ units parallel to $x$-axis or from $y$-axis
- for $(ax)^n$, dilation factor is $1 \over a$ parallel to $x$-axis or from $y$-axis
- when $0 < |a| < 1$, graph becomes closer to axis
+## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
+
+Applies to exponential, log, trig, $e^x$, polynomials.
+Functions must be written in form $y=Af[n(x+c)]+b$
+
+- dilation by factor $|A|$ from $x$-axis (if $A<0$, reflection across $y$-axis)
+- dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis)
+- translation of $c$ units from $y$-axis ($x$-shift)
+- translation of $b$ units from $x$-axis ($y$-shift)
+
## Dilations
-For the graph of $y = f(x)$, there are two pairs of equivalent processes:
+Two pairs of equivalent processes for $y=f(x)$:
1. - Dilating from $x$-axis: $(x, y) \rightarrow (x, by)$
- Replacing $y$ with $y \over b$ to obtain $y = b f(x)$
For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
+## Matrix transformations
+
+Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
+
## Reflections
- Reflection **in** axis = reflection **over** axis = reflection **across** axis
For $y = f(x)$, these processes are equivalent:
- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
-- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
-
-## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
-
-Applies to exponential, log, trig, power, polynomial functions.
-Functions must be written in form $y=Af[n(x+c)] + b$
-
-$A$ - dilation by factor $A$ from $x$-axis (if $A<0$, reflection across $y$-axis)
-$n$ - dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis)
-$c$ - translation from $y$-axis ($x$-shift)
-$b$ - translation from $x$-axis ($y$-shift)
+- replacing $x$ with $x-h$ and $y$ with $y-k$ to obtain $y-k = f(x-h)$
## Power functions
Function is even if it can be reflected across $y$-axis $\implies f(x)=f(-x)$
Function $x^{\pm {p \over q}}$ is odd if $q$ is odd
-### $x^n$ where $n \in \mathbb{Z}^+$
-
-\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 $n \in \mathbb{Z}^-$
-
-\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 $n \in \mathbb{Z}^+$
-\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}
+\newcolumntype{C}{>{\centering\arraybackslash} m{3cm} }
+\begin{center}
+\begin{tabular}{m{1.2cm}|C|C}
+ & $n$ is even & $n$ is odd \\
+ \hline
+ \parbox[c]{1.2cm}{$x^n,\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/parabola.png}} & {\includegraphics[height=3cm]{graphics/cubic.png}}\\
+ \parbox[c]{1.2cm}{$x^n$,\\ $n \in \mathbb{Z}^-$} & {\includegraphics[height=3cm]{graphics/truncus.png}} & {\includegraphics[height=3cm]{graphics/hyperbola.png}}\\
+ \parbox[c]{1.2cm}{$x^{1 \over n},\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/square-root-graph.png}} & {\includegraphics[height=3cm]{graphics/cube-root-graph.png}}\\
+\end{tabular}
+\end{center}
### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
If $n$ is odd, it is an odd function.
+\columnbreak
+
### $x^{p \over q}$ where $p, q \in \mathbb{Z}^+$
$$x^{p \over q} = \sqrt[q]{x^p}$$
- points $(0, 0)$ and $(1, 1)$ will always lie on graph
- Domain is: $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
-
-## Combinations of functions (piecewise/hybrid)
+## Piecewise functions
$$\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
-Open circle - point included
-Closed circle - point not included
+**Open circle:** point included
+**Closed circle:** point not included
-### Sum, difference, product of functions
-\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
+## Operations on functions
-Product functions:
+For $f \pm g$ and $f \times g$: \quad $\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)$
-- product will equal 0 if one of the functions is equal to 0
-- turning point on one function does not equate to turning point on product
+Addition of linear piecewise graphs: add $y$-values at key points
-## Matrix transformations
+Product functions:
-Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
+- product will equal 0 if $f=0$ or $g=0$
+- $f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0$
## Composite functions