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-\usepackage{tikz-3dplot} +\usepackage{tikz-3dplot} +\usepackage{pgfplots} \usetikzlibrary{calc} \usetikzlibrary{angles} +\usetikzlibrary{datavisualization.formats.functions} +\usetikzlibrary{decorations.markings} \usepgflibrary{arrows.meta} \usepackage{fancyhdr} \pagestyle{fancy} @@ -217,14 +221,14 @@ \begin{itemize} \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)} \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)} - \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)} + \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)} \end{itemize} \subsubsection*{Circles} \begin{itemize} \item \(|z-z_1|^2=c^2|z_2+2|^2\) - \item \(|z-(a+bi)|=c\) + \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\) \end{itemize} \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\) @@ -273,9 +277,7 @@ \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow}; \end{scope} \node[black, right] at (2.5,1.5) {\(y\vec{j}\)}; - \end{tikzpicture}\end{center} - \subsection*{Column notation} \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\] @@ -301,7 +303,7 @@ \begin{itemize} \item Draw each vector head to tail then join lines \item Addition is commutative (parallelogram) - \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v})\) + \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\) \end{itemize} \subsection*{Magnitude} @@ -361,7 +363,7 @@ For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\ \subsection*{Angle between vectors} -\[\cos \theta = {{\boldsymbol{a} \cdot \frac{\boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|}} = {\frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}}\] +\[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\] \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])} @@ -399,10 +401,11 @@ between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and \end{tikzpicture} \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)} + \begin{align*} - \boldsymbol{u}&={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}\\ - &=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)\\ - &=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}} + \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\ + & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\ + & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}} \end{align*} \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)} @@ -458,7 +461,6 @@ between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\ &=(1-m)\boldsymbol{a}+m{b} \end{align*} - \begin{align*} \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\ \text{where } \lambda + \mu &= 1\\ @@ -466,6 +468,42 @@ between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and \end{align*} +\subsubsection*{Parallelograms} + +\begin{center}\begin{tikzpicture} + \coordinate (O) at (0,0) node [below left] {\(O\)}; + \coordinate (A) at (4,0); + \coordinate (B) at (6,2); + \coordinate (C) at (2,2); + \coordinate (D) at (6,0); + + \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)}; + \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)}; + \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)}; + \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O); + + \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)}; + \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)}; + \begin{scope} + \path[clip] (C) -- (A) -- (O); + \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm); + \node at ($(0,0)+(20:8mm)$) {\(\theta\)}; + \end{scope} + \draw [gray, thick, dotted] (B) -- (D) node [pos=0.5, right, black, font=\footnotesize] {\(|\boldsymbol{c}|\sin\theta\)} (A) -- (D) node [pos=0.5, below, black, font=\footnotesize] {\(|\boldsymbol{c}|\cos\theta\)}; + \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B}; +\end{tikzpicture}\end{center} + +\begin{itemize} + \item + Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other + \item + If diagonals are equal length, it is a rectangle + \item + \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\) + \item + Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\) +\end{itemize} + \subsubsection*{Useful vector properties} \begin{itemize} @@ -476,27 +514,23 @@ between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at least one point in common, then they lie on the same straight line \item - Two vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are - perpendicular if \(\boldsymbol{a} \cdot \boldsymbol{b}=0\) + \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\) \item \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\) \end{itemize} \subsection*{Linear dependence} -Vectors \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly -dependent if they are non-parallel and: - -\[k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c} = 0\] -\[\therefore \boldsymbol{c} = m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}\] +\(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and: +\begin{align*} + 0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\ + \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)} +\end{align*} -\(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly +\noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly independent if no vector in the set is expressible as a linear combination of other vectors in set, or if they are parallel. -Vector \(\boldsymbol{w}\) is a linear combination of vectors -\(\boldsymbol{v_1}, \boldsymbol{v_2}, \boldsymbol{v_3}\) - \subsection*{Three-dimensional vectors} Right-hand rule for axes: \(z\) is up or out of page. @@ -541,55 +575,45 @@ at (\ax,\ay,\az){(\ax, \ay, \az)}; Parametric equation of line through point \((x_0, y_0, z_0)\) and 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} +\[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\] \section{Circular functions} -Period of \(a\sin(bx)\) is \(\frac{{2\pi}{b}\) +\(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\) -Period of \(a\tan(nx)\) is \(\frac{\pi}{n}\)\\ -Asymptotes at \(x=\frac{2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\) +\noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\ +\indent\indent\indent 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 - \(\theta = {\frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\) + \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\) \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} + +\begin{center}\includegraphics[width=0.7\columnwidth]{graphics/sec.png}\end{center} \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\] \begin{itemize} -\tightlist + \item \textbf{Domain} - \(= \mathbb{R} \setminus \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}\) + \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\) \item \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\) \item @@ -597,21 +621,17 @@ Asymptotes at \(x=\frac{2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\) \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\) \item \textbf{Asymptotes} at - \(\theta = {{(2n + 1) \pi} \over 2} \> \vert \> n \in \mathbb{Z}\) + \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\) \end{itemize} \subsubsection*{Cotangent} -\begin{figure} -\centering -\includegraphics{graphics/cot.png} -\caption{} -\end{figure} +\begin{center}\includegraphics[width=0.7\columnwidth]{graphics/cot.png}\end{center} \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\] \begin{itemize} -\tightlist + \item \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\) \item @@ -622,30 +642,30 @@ Asymptotes at \(x=\frac{2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\) \subsubsection*{Symmetry properties} -\begin{equation}\begin{split} +\[\begin{split} \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\ \operatorname{sec} (-x) & = \operatorname{sec} x \\ \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\ \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\ \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\ \operatorname{cot} (-x) & = - \operatorname{cot} x -\end{split}\end{equation} +\end{split}\] \subsubsection*{Complementary properties} -\begin{equation}\begin{split} +\[\begin{split} \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\ \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\ \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\ \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x -\end{split}\end{equation} +\end{split}\] \subsubsection*{Pythagorean identities} -\begin{equation}\begin{split} +\[\begin{split} 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\ 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0 -\end{split}\end{equation} +\end{split}\] \subsection*{Compound angle formulas} @@ -655,11 +675,11 @@ Asymptotes at \(x=\frac{2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\) \subsection*{Double angle formulas} -\begin{equation}\begin{split} +\[\begin{split} \cos 2x &= \cos^2 x - \sin^2 x \\ & = 1 - 2\sin^2 x \\ & = 2 \cos^2 x -1 -\end{split}\end{equation} +\end{split}\] \[\sin 2x = 2 \sin x \cos x\] @@ -668,21 +688,17 @@ Asymptotes at \(x=\frac{2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\) \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 \(y=x\) - +Must be 1:1 to find inverse (reflection in \(y=x\)).\\ Domain is restricted to make functions 1:1. -\subsubsection*{\(\arcsin\)} - -\[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y, \quad \text{where } \sin y = x \text{ and } y \in [{-\pi \over 2}, {\pi \over 2}]\] +\[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\] +\hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\) -\subsubsection*{\(\arccos\)} +\[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\] +\hfill where \(\cos y = x, \> y \in [0, \pi]\) -\[\cos^{-1} \rightarrow \mathbb{R}, \quad \cos^{-1} x = y, \quad \text{where } \cos y = x \text{ and } y \in [0, \pi]\] - -\subsubsection*{\(\arctan\)} - -\[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y, \quad \text{where } \tan y = x \text{ and } y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\] +\[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\] +\hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\) \section{Differential calculus} @@ -690,25 +706,13 @@ Domain is restricted to make functions 1:1. \subsection*{Limits} \[\lim_{x \rightarrow a}f(x)\] +\(L^-,\quad L^+\) \qquad limit from below/above\\ +\(\lim_{x \to a} f(x)\) \quad limit of a point\\ -\(L^-\) - 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\\ +\noindent For solving \(x\rightarrow\infty\), put all \(x\) terms 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} +\subsubsection*{Limit theorems} \begin{enumerate} \item @@ -719,23 +723,18 @@ Limits can be solved using normal techniques (if div 0, factorise) \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\) \item \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant -\ite +\item \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\) \item -A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\). + \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\) \end{enumerate} -\subsection{Gradients of secants and tangents} +\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} +\subsection*{First principles derivative} \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\] @@ -745,76 +744,76 @@ As \(Q \rightarrow P, \delta x \rightarrow 0\). Chord becomes tangent \(\log_b x^n = n \log_b x\)\\ \(\log_b y^{x^n} = x^n \log_b y\) -\subsubsection*{Index identities}} +\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} +\renewcommand{\arraystretch}{1.4} +\begin{tabularx}{\columnwidth}{rX} + \hline +\(f(x)\) & \(f^\prime(x)\)\\ +\hline +\(\sin x\) & \(\cos x\)\\ +\(\sin ax\) & \(a\cos ax\)\\ +\(\cos x\) & \(-\sin x\)\\ +\(\cos ax\) & \(-a \sin ax\)\\ +\(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\ +\(e^x\) & \(e^x\)\\ +\(e^{ax}\) & \(ae^{ax}\)\\ +\(ax^{nx}\) & \(an \cdot e^{nx}\)\\ + \(\log_e x\) & \(\dfrac{1}{x}\)\\ + \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\ + \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\ +\(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\ + \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\ + \(\cos^{-1} x\) & \(\dfrac{-1}{sqrt{1-x^2}}\)\\ + \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\ + \hline +\end{tabularx} \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}} - -\[f(x) \longrightarrow f^\prime (x) \longrightarrow f^{\prime\prime}(x)\] + \text{Find }& \frac{dx}{dy}\\ + \text{Then, }\frac{dx}{dy} &= \frac{1}{\frac{dy}{dx}} \\ + \implies {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}\\ + \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}} +\end{align*} -\[\therefore y \longrightarrow {\frac{dy}{dx}} \longrightarrow {d({\frac{dy}{dx}}) \over dx} \longrightarrow {d^2 y \over dx^2}\] +\subsection*{Second derivative} +\begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\ +\implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*} -Order of polynomial \(n\)th derivative decrements each time the -derivative is taken +\noindent 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. +\emph{Stationary point} - i.e. +\(f^\prime(x)=0\)\\ +\emph{Point of inflection} - max \(|\)gradient\(|\) (i.e. \(f^{\prime\prime} = 0\)) - +%\begin{table*}[ht] +%\centering +% \begin{tabularx}{\textwidth}{XXXX} +%\hline +% \rowcolor{shade2} +% & \(\dfrac{d^2 y}{dx^2} > 0\) & \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\ +%\hline +% \(\frac{dy}{dx}>0\) & \begin{tikzpicture} \draw[domain=1:2,smooth,variable=\x,blue] plot ({\x},{(1/10)*\x*\x*\x}) plot ({\x},{0.675*\x-0.677}); \end{tikzpicture} & cell 3\\ +%cell 1 & cell 2 & cell 3\\ +%\hline +%\end{tabularx} +%\end{table*} \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) @@ -828,154 +827,70 @@ derivative is taken 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} +\begin{table*}[ht] + \centering + \includegraphics[width=0.7\textwidth]{graphics/second-derivatives.png} +\end{table*} \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. +\noindent 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: +for all \(x\) and \(y\), then: \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\] +\noindent \colorbox{cas}{\textbf{On CAS:}}\\ +Action \(\rightarrow\) Calculation \(\rightarrow\) \texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}\\ +Returns \(y^\prime= \dots\). + \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}} +\subsection*{Integral laws} + +\renewcommand{\arraystretch}{1.4} +\begin{tabularx}{\columnwidth}{rX} +\hline + \(f(x)\) & \(\int f(x) \cdot dx\) \\ + \hline + \(k\) (constant) & \(kx + c\)\\ + \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\ + \(a x^{-n}\) &\(a \cdot \log_e x + c\)\\ + \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\ + \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\)\\ + \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\ + \(e^k\) & \(e^kx + c\)\\ + \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\ + \(\cos kx\) & \(\frac{1}{k} \sin (kx) + c\)\\ + \(\sec^2 kx\) & \(\frac{1}{k} \tan(kx) + c\)\\ + \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ + \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ + \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\ + \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\ + \(g^\prime(x)\cdot f^\prime(g(x)\) & \(f(g(x))\) (chain rule)\\ + \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\ + \hline +\end{tabularx} + +Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant \(\forall x \in (-a, a)\) + +\subsection*{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: + 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} +\subsubsection*{Properties} \[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\] @@ -987,44 +902,43 @@ all \(x \in (-a, a)\). \[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\] -\subsubsection{Integration by substitution} +\subsection*{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 +\noindent Note \(f(u)\) must be 1:1 \(\implies\) one \(x\) for each \(y\) +\begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\ + \text{let } u&=x+4\\ + \implies& {\frac{du}{dx}} = 1\\ + \implies& x = u - 4\\ + \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\ + &\text{(solve as normal integral)} +\end{align*} \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} +\subsubsection*{Trigonometric integration} \[\sin^m x \cos^n x \cdot dx\] -\textbf{\(m\) is odd:}\\ +\paragraph{\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:}\\ +\paragraph{\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 \(u=\sin x\) +Substitute \(u=\sin x\) -\textbf{\(m\) and \(n\) are even:}\\ -Use identities: +\paragraph{\textbf{\(m\) and \(n\) are even:}} +use identities... \begin{itemize} -\tightlist + \item \(\sin^2x={1 \over 2}(1-\cos 2x)\) \item @@ -1035,22 +949,22 @@ Use identities: \subsection*{Partial fractions} -On CAS: Action \(\rightarrow\) Transformation \(\rightarrow\) +\colorbox{cas}{On CAS:}\\ +\indent Action \(\rightarrow\) Transformation \(\rightarrow\) \texttt{expand/combine}\\ -or Interactive \(\rightarrow\) Transformation \(\rightarrow\) -\texttt{expand} \(\rightarrow\) Partial +\indent Interactive \(\rightarrow\) Transformation \(\rightarrow\) +Expand \(\rightarrow\) Partial \subsection*{Graphing integrals on CAS} -In main: Interactive \(\rightarrow\) Calculation \(\rightarrow\) +\colorbox{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.) +Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..} -\subsection{Applications of antidifferentiation} +\subsection*{Applications of antidifferentiation} \begin{itemize} -\tightlist + \item \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of stationary points on \(y=F(x)\) @@ -1066,28 +980,28 @@ 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}} +\subsection*{Solids of revolution} Approximate as sum of infinitesimally-thick cylinders -\subsubsection{Rotation about \(x\)-axis} +\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} +\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\(y=0\)} +\subsubsection*{Regions not bound by \(y=0\)} -\[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]\\ -where \(f(x) > g(x)\) +\[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\] +\hfill where \(f(x) > g(x)\) \subsection*{Length of a curve} @@ -1095,16 +1009,14 @@ where \(f(x) > g(x)\) \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\] -Evaluate on CAS. Or use Interactive \(\rightarrow\) Calculation -\(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}. +\noindent \colorbox{cas}{On CAS:}\\ +\indent Evaluate formula,\\ +\indent or 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} +\subsubsection*{Gradient at a point on parametric curve} \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0\] @@ -1117,7 +1029,7 @@ Evaluate on CAS. Or use Interactive \(\rightarrow\) Calculation \subsubsection*{Addition of ordinates} \begin{itemize} -\tightlist + \item when two graphs have the same ordinate, \(y\)-coordinate is double the ordinate @@ -1129,38 +1041,34 @@ Evaluate on CAS. Or use Interactive \(\rightarrow\) Calculation other ordinate \end{itemize} -\subsection{Fundamental theorem of calculus} +\subsection*{Fundamental theorem of calculus} If \(f\) is continuous on \([a, b]\), then \[\int^b_a f(x) \> dx = F(b) - F(a)\] +\hfill where \(F = \int f \> dx\) -where \(F\) is any antiderivative of \(f\) - -\subsection*{Differential equations}} - -One or more derivatives +\subsection*{Differential equations} -\textbf{Order} - highest power inside derivative\\ +\noindent\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 +e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad 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} +\subsubsection*{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 +\(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). 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}}\] +\[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\] \subsubsection*{Separation of variables} @@ -1168,17 +1076,7 @@ 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} +\subsubsection*{Euler's method for solving DEs} \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\] -- 2.47.1

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