From 150b93e7009d4e771992eff84c036502b93ac4a2 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Mon, 13 May 2019 16:49:10 +1000 Subject: [PATCH] [methods] tidy up & render notes for SAC --- methods/calculus-ref.md | 64 ++++++++++++++++++++-------------- methods/calculus-ref.pdf | Bin 159642 -> 149615 bytes methods/inverse-functions.pdf | Bin 0 -> 120898 bytes methods/polynomials-ref.pdf | Bin 185882 -> 34202 bytes methods/polynomials.md | 29 +++++++++------ methods/transformations.md | 39 ++++++++++++--------- 6 files changed, 78 insertions(+), 54 deletions(-) create mode 100644 methods/inverse-functions.pdf 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 c783d70a8662c777a856850175dc4270ac3d0ddb..06fb92880bc68c5d098cbadd65499cc2c61f7e45 100644 GIT binary patch delta 121908 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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: -- 2.43.2