From f37f49cb0221a4928e8aad605201916c7d7524fc Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Mon, 27 May 2019 11:52:44 +1000 Subject: [PATCH] [spec] copy notes for circ. fn's & calculus (untidied) --- spec/spec-collated.pdf | Bin 254419 -> 220084 bytes spec/spec-collated.tex | 649 ++++++++++++++++++++++++++++++++++++++++- 2 files changed, 644 insertions(+), 5 deletions(-) diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 5cbbb9b38be1e00abf121ebba764a1894b076800..881e8cce35bb330f82ec6766966659220694407e 100644 GIT binary patch delta 111018 zcmV)BK*PV&!w`YU9{EP4YG4Q6q?dpSnXWV1`Dz;7)wDk^`S01Q?Es98F*HP(7Xd=7P8oIAbROl59obZ9alF)=tbGn0Ya6a_IdIW;tw;l=?dfA$4X z8`{nEe~}4eDYe#>4{fe+0YP0<=J` zAQumi72uDS0ct=;(BG}Gq0#|#Y{9O7DYT*1Zk|9F5a31N0Ja1{Twh$=Ayyz4z)N<3 zwxS9^-3bKw%UIgP&t3M2Z4zAD_e}A9{5bOZ7crp01bRa-ZQWF4t>G1FVTrFL|PHwJju3(2h zdSw42%*!>)Lad~rj*cLRn=9%c^~r!;K$b7}?#=%9aP1*bPl)e-kTn=$W&KARR_;#h zx)89lJ4jLHZ<`kp>OV3YkQ;!PgM)*gPXGXN27tUQZQ1`wpyTZXfBI9&`G@$W2Y+8D 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zU7t#qkX&=UCPAh|U diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex index 616033b..eca12dc 100644 --- a/spec/spec-collated.tex +++ b/spec/spec-collated.tex @@ -325,7 +325,7 @@ For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\ \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\] \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)} -\[\begin{split}\hat{\boldsymbol{a}} & = {1 \over {|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\] +\[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\] \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)} @@ -361,7 +361,7 @@ For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\ \subsection*{Angle between vectors} -\[\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}\] +\[\cos \theta = {{\boldsymbol{a} \cdot \frac{\boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|}} = {\frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}}\] \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])} @@ -372,7 +372,7 @@ For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\ \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\) which makes angles \(\alpha, \beta, \gamma\) with positive side of \(x, y, z\) axes: -\[\cos \alpha = {a_1 \over |\boldsymbol{a}|}, \quad \cos \beta = {a_2 \over |\boldsymbol{a}|}, \quad \cos \gamma = {a_3 \over |\boldsymbol{a}|}\] +\[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\] \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and @@ -470,8 +470,7 @@ between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and \begin{itemize} \item - If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel, then - \(\boldsymbol{b}=k\boldsymbol{a}\) for some + \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some \(k \in \mathbb{R} \setminus \{0\}\) \item If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at @@ -544,6 +543,646 @@ parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is: \begin{equation}\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\end{equation} +\section{Circular functions} + +Period of \(a\sin(bx)\) is \(\frac{{2\pi}{b}\) + +Period of \(a\tan(nx)\) is \(\frac{\pi}{n}\)\\ +Asymptotes at \(x=\frac{2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\) + +\subsection*{Reciprocal functions} + +\subsubsection*{Cosecant} + +\begin{figure} +\centering +\includegraphics{graphics/csc.png} +\caption{} +\end{figure} + +\[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\] + +\begin{itemize} +\tightlist +\item + \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\) +\item + \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\) +\item + \textbf{Turning points} at + \(\theta = {\frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\) +\item + \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\) +\end{itemize} + +\subsubsection*{Secant} + +\begin{figure} +\centering +\includegraphics{graphics/sec.png} +\caption{} +\end{figure} + +\[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\] + +\begin{itemize} +\tightlist +\item + \textbf{Domain} + \(= \mathbb{R} \setminus \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}\) +\item + \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\) +\item + \textbf{Turning points} at + \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\) +\item + \textbf{Asymptotes} at + \(\theta = {{(2n + 1) \pi} \over 2} \> \vert \> n \in \mathbb{Z}\) +\end{itemize} + +\subsubsection*{Cotangent} + +\begin{figure} +\centering +\includegraphics{graphics/cot.png} +\caption{} +\end{figure} + +\[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\] + +\begin{itemize} +\tightlist +\item + \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\) +\item + \textbf{Range} \(= \mathbb{R}\) +\item + \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\) +\end{itemize} + +\subsubsection*{Symmetry properties} + +\begin{equation}\begin{split} + \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\ + \operatorname{sec} (-x) & = \operatorname{sec} x \\ + \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\ + \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\ + \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\ + \operatorname{cot} (-x) & = - \operatorname{cot} x +\end{split}\end{equation} + +\subsubsection*{Complementary properties} + +\begin{equation}\begin{split} + \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\ + \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\ + \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\ + \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x +\end{split}\end{equation} + +\subsubsection*{Pythagorean identities} + +\begin{equation}\begin{split} + 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\ + 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0 +\end{split}\end{equation} + +\subsection*{Compound angle formulas} + +\[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\] +\[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\] +\[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\] + +\subsection*{Double angle formulas} + +\begin{equation}\begin{split} + \cos 2x &= \cos^2 x - \sin^2 x \\ + & = 1 - 2\sin^2 x \\ + & = 2 \cos^2 x -1 +\end{split}\end{equation} + +\[\sin 2x = 2 \sin x \cos x\] + +\[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\] + +\subsection*{Inverse circular functions} + +Inverse functions: \(f(f^{-1}(x)) = x, \quad f(f^{-1}(x)) = x\)\\ +Must be 1:1 to find inverse (reflection in \(y=x\) + +Domain is restricted to make functions 1:1. + +\subsubsection*{\(\arcsin\)} + +\[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y, \quad \text{where } \sin y = x \text{ and } y \in [{-\pi \over 2}, {\pi \over 2}]\] + +\subsubsection*{\(\arccos\)} + +\[\cos^{-1} \rightarrow \mathbb{R}, \quad \cos^{-1} x = y, \quad \text{where } \cos y = x \text{ and } y \in [0, \pi]\] + +\subsubsection*{\(\arctan\)} + +\[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y, \quad \text{where } \tan y = x \text{ and } y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\] + + +\section{Differential calculus} + +\subsection*{Limits} + +\[\lim_{x \rightarrow a}f(x)\] + +\(L^-\) - limit from below + +\(L^+\) - limit from above + +\(\lim_{x \to a} f(x)\) - limit of a point + +\begin{itemize} +\item + Limit exists if \(L^-=L^+\) +\item + If limit exists, point does not. +\item + For solving \(x\rightarrow\infty\), factorise so that all \(x\) terms are in denominators\\ + e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\] + \item +Limits can be solved using normal techniques (if div 0, factorise) +\end{itemize} + + +\begin{enumerate} +\item + For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\) +\item + \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\) +\item + \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\) + \item +\(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant +\ite + \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\) +\item +A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\). +\end{enumerate} + +\subsection{Gradients of secants and tangents} + +\textbf{Secant (chord)} - line joining two points on curve\\ +\textbf{Tangent} - line that intersects curve at one point + +\(m\left(\overrightharp{PQ}\right){m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x} \text{ for } P(x,y),\quad Q(x+\delta x, y+ \delta y)\) + +As \(Q \rightarrow P, \delta x \rightarrow 0\). Chord becomes tangent +(two infinitesimal points are equal). + +\subsection{First principles derivative} + +\[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\] + +\subsubsection*{Logarithmic identities} + +\(\log_b (xy)=\log_b x + \log_b y\)\\ +\(\log_b x^n = n \log_b x\)\\ +\(\log_b y^{x^n} = x^n \log_b y\) + +\subsubsection*{Index identities}} + +\(b^{m+n}=b^m \cdot b^n\)\\ +\((b^m)^n=b^{m \cdot n}\)\\ +\((b \cdot c)^n = b^n \cdot c^n\)\\ +\({a^m \div a^n} = {a^{m-n}}\) + +\subsubsection{\texorpdfstring{\(e\) as a +logarithm}{e as a logarithm}}\label{e-as-a-logarithm} + +\[\operatorname{if} y=e^x, \quad \operatorname{then} x=\log_e y\] +\[\ln x = \log_e x\] + +\subsection*{Derivative rules} + +\begin{longtable}[]{@{}ll@{}} +\toprule +\(f(x)\) & \(f^\prime(x)\)\tabularnewline +\midrule +\endhead +\(\sin x\) & \(\cos x\)\tabularnewline +\(\sin ax\) & \(a\cos ax\)\tabularnewline +\(\cos x\) & \(-\sin x\)\tabularnewline +\(\cos ax\) & \(-a \sin ax\)\tabularnewline +\(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\tabularnewline +\(e^x\) & \(e^x\)\tabularnewline +\(e^{ax}\) & \(ae^{ax}\)\tabularnewline +\(ax^{nx}\) & \(an \cdot e^{nx}\)\tabularnewline +\(\log_e x\) & \(1 \over x\)\tabularnewline +\(\log_e {ax}\) & \(1 \over x\)\tabularnewline +\(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline +\(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline +\(\sin^{-1} x\) & \(1 \over {\sqrt{1-x^2}}\)\tabularnewline +\(\cos^{-1} x\) & \(-1 \over {sqrt{1-x^2}}\)\tabularnewline +\(\tan^{-1} x\) & \(1 \over {1 + x^2}\)\tabularnewline +\bottomrule +\end{longtable} + +\subsection*{Reciprocal derivatives} + +\[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\] + +\subsection*{Differentiating \(x=f(y)\)} + +Find \(\frac{dx}{dy}\). Then: + +\begin{align*} + {\frac{dx}{dy}} =& {1 \over {\frac{dy}{dx}}} \\ + \implies {\frac{dy}{dx}} &= {1 \over {\frac{dx}{dy}}}\). + +\[{\frac{dy}{dx}} = {1 \over {\frac{dx}{dy}}}\] + +\subsection*{Second derivative}} + +\[f(x) \longrightarrow f^\prime (x) \longrightarrow f^{\prime\prime}(x)\] + +\[\therefore y \longrightarrow {\frac{dy}{dx}} \longrightarrow {d({\frac{dy}{dx}}) \over dx} \longrightarrow {d^2 y \over dx^2}\] + +Order of polynomial \(n\)th derivative decrements each time the +derivative is taken + +\subsubsection*{Points of Inflection} + +\emph{Stationary point} - point of zero gradient (i.e. +\(f^\prime(x)=0\))\\ +\emph{Point of inflection} - point of maximum \(|\)gradient\(|\) (i.e. +\(f^{\prime\prime} = 0\)) + +\begin{itemize} +\tightlist +\item + if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point + \((a, f(a))\) is a local min (curve is concave up) +\item + if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point + \((a, f(a))\) is local max (curve is concave down) +\item + if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of + inflection +\item + if also \(f^\prime(a)=0\), then it is a stationary point of inflection +\end{itemize} + +\begin{figure} +\centering +\includegraphics{graphics/second-derivatives.png} +\caption{} +\end{figure} + +\subsection*{Implicit Differentiation} + +\textbf{On CAS:} Action \(\rightarrow\) Calculation \(\rightarrow\) +\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}. Returns \(y^\prime= \dots\). + +Used for differentiating circles etc. + +If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\), +for all \(x\) nd \(y\), then: + +\[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\] + +\subsection*{Integration} + +\[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\] + +\[\int x^n \cdot dx = {x^{n+1} \over n+1} + c\] + +\begin{itemize} +\tightlist +\item + area enclosed by curves +\item + \(+c\) should be shown on each step without \(\int\) +\end{itemize} + +\subsubsection*{Integral laws} + +\(\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx\)\\ +\(\int k f(x) dx = k \int f(x) dx\) + +\begin{longtable}[]{@{}ll@{}} +\toprule +\begin{minipage}[b]{0.42\columnwidth}\raggedright\strut +\(f(x)\)\strut +\end{minipage} & \begin{minipage}[b]{0.38\columnwidth}\raggedright\strut +\(\int f(x) \cdot dx\)\strut +\end{minipage}\tabularnewline +\midrule +\endhead +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(k\) (constant)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\(kx + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(x^n\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\({x^{n+1} \over {n+1}} + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(a x^{-n}\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\(a \cdot \log_e x + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\({1 \over {ax+b}}\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\({1 \over a} \log_e (ax+b) + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\((ax+b)^n\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\({1 \over {a(n+1)}}(ax+b)^{n-1} + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(e^{kx}\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\({1 \over k} e^{kx} + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(e^k\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\(e^kx + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(\sin kx\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\(-{1 \over k} \cos (kx) + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(\cos kx\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\({1 \over k} \sin (kx) + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(\sec^2 kx\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\({1 \over k} \tan(kx) + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(1 \over \sqrt{a^2-x^2}\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\(\sin^{-1} {x \over a} + c \>\vert\> a>0\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(-1 \over \sqrt{a^2-x^2}\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\(\cos^{-1} {x \over a} + c \>\vert\> a>0\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(a \over {a^2-x^2}\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\(\tan^{-1} {x \over a} + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\({f^\prime (x)} \over {f(x)}\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\(\log_e f(x) + c\)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(g^\prime(x)\cdot f^\prime(g(x)\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\(f(g(x))\) (chain rule)\strut +\end{minipage}\tabularnewline +\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut +\(f(x) \cdot g(x)\)\strut +\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut +\(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\strut +\end{minipage}\tabularnewline +\bottomrule +\end{longtable} + +Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant for +all \(x \in (-a, a)\). + +\subsubsection*{Definite integrals}} + +\[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\] + +\begin{itemize} +\tightlist +\item + Signed area enclosed by: + \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\). +\item + \emph{Integrand} is \(f\). +\item + \(F(x)\) may be any integral, i.e. \(c\) is inconsequential +\end{itemize} + +\paragraph{Properties}\label{properties} + +\[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\] + +\[\int^a_a f(x) \> dx = 0\] + +\[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\] + +\[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\] + +\[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\] + +\subsubsection{Integration by substitution} + +\[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\] + +Note \(f(u)\) must be one-to-one \(\implies\) one \(x\) value for each +\(y\) value + +e.g.~for \(y=\int(2x+1)\sqrt{x+4} \cdot dx\):\\ +let \(u=x+4\)\\ +\(\implies {\frac{du}{dx}} = 1\)\\ +\(\implies x = u - 4\)\\ +then \(y=\int (2(u-4)+1)u^{1 \over 2} \cdot du\)\\ +Solve as a normal integral + +\subsubsection*{Definite integrals by substitution} + +For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and +\(b\) for \(f(u) \cdot du\). + +\subsubsection{Trigonometric integration} + +\[\sin^m x \cos^n x \cdot dx\] + +\textbf{\(m\) is odd:}\\ +\(m=2k+1\) where \(k \in \mathbb{Z}\)\\ +\(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\ +Substitute \(u=\cos x\) + +\textbf{\(n\) is odd:}\\ +\(n=2k+1\) where \(k \in \mathbb{Z}\)\\ +\(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\ +Subbstitute \(u=\sin x\) + +\textbf{\(m\) and \(n\) are even:}\\ +Use identities: + +\begin{itemize} +\tightlist +\item + \(\sin^2x={1 \over 2}(1-\cos 2x)\) +\item + \(\cos^2x={1 \over 2}(1+\cos 2x)\) +\item + \(\sin 2x = 2 \sin x \cos x\) +\end{itemize} + +\subsection*{Partial fractions} + +On CAS: Action \(\rightarrow\) Transformation \(\rightarrow\) +\texttt{expand/combine}\\ +or Interactive \(\rightarrow\) Transformation \(\rightarrow\) +\texttt{expand} \(\rightarrow\) Partial + +\subsection*{Graphing integrals on CAS} + +In main: Interactive \(\rightarrow\) Calculation \(\rightarrow\) +\(\int\) (\(\rightarrow\) Definite)\\ +Restrictions: \texttt{Define\ f(x)=...} \(\rightarrow\) +\texttt{f(x)\textbar{}x\textgreater{}1} (e.g.) + +\subsection{Applications of antidifferentiation} + +\begin{itemize} +\tightlist +\item + \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of + stationary points on \(y=F(x)\) +\item + nature of stationary points is determined by sign of \(y=f(x)\) on + either side of its \(x\)-intercepts +\item + if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree + \(n+1\) +\end{itemize} + +To find stationary points of a function, substitute \(x\) value of given +point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find +original function. + +\subsection*{Solids of revolution}} + +Approximate as sum of infinitesimally-thick cylinders + +\subsubsection{Rotation about \(x\)-axis} + +\begin{align*} + V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\ + &= \pi \int^b_a (f(x))^2 \> dx +\end{align*} + +\subsubsection{Rotation about \(y\)-axis} + +\begin{align*} + V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\ + &= \pi \int^b_a (f(y))^2 \> dy +\end{align*} + +\subsubsection{Regions not bound by\(y=0\)} + +\[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]\\ +where \(f(x) > g(x)\) + +\subsection*{Length of a curve} + +\[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\] + +\[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\] + +Evaluate on CAS. Or use Interactive \(\rightarrow\) Calculation +\(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}. + +\subsection*{Rates} + +\subsubsection*{Related rates} + +\[{\frac{da}{db}} \quad \text{(change in } a \text{ with respect to } b)\] + +\subsubsection{Gradient at a point on parametric curve} + +\[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0\] + +\[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\] + +\subsection*{Rational functions} + +\[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\] + +\subsubsection*{Addition of ordinates} + +\begin{itemize} +\tightlist +\item + when two graphs have the same ordinate, \(y\)-coordinate is double the + ordinate +\item + when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e. + (\(x\)-intercept) +\item + when one of the ordinates is 0, the resulting ordinate is equal to the + other ordinate +\end{itemize} + +\subsection{Fundamental theorem of calculus} + +If \(f\) is continuous on \([a, b]\), then + +\[\int^b_a f(x) \> dx = F(b) - F(a)\] + +where \(F\) is any antiderivative of \(f\) + +\subsection*{Differential equations}} + +One or more derivatives + +\textbf{Order} - highest power inside derivative\\ +\textbf{Degree} - highest power of highest derivative\\ +e.g. \({\left(\frac{dy^2}{d^2} x\right)}^3\): order 2, degree 3 + +\subsubsection*{Verifying solutions} + +Start with \(y=\dots\), and differentiate. Substitute into original +equation. + +\subsubsection{Function of the dependent +variable}\label{function-of-the-dependent-variable} + +If \({\frac{dy}{dx}}=g(y)\), then +\(\frac{{dx}{dy} = 1 \div {\frac{dy}{dx}} = \frac{1}{g(y)}\). Integrate +both sides to solve equation. Only add \(c\) on one side. Express +\(e^c\) as \(A\). + +\subsubsection*{Mixing problems} + +\[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left({\frac{dm}{dt}\)_{\text{out}}\] + +\subsubsection*{Separation of variables} + +If \({\frac{dy}{dx}}=f(x)g(y)\), then: + +\[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\] + +\subsubsection{Using definite integrals to solve DEs} + +Used for situations where solutions to \({\frac{dy}{dx}} = f(x)\) is not +required. + +In some cases, it may not be possible to obtain an exact solution. + +Approximate solutions can be found by numerically evaluating a definite +integral. + +\subsubsection{Using Euler's method to solve a differential equation} + +\[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\] + +\[\implies f(x+h) \approx f(x) + hf^\prime(x)\] \end{multicols} \end{document} -- 2.43.2