\[\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 \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
\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])}
\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
\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
\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}