\usepackage{multirow}
\usepackage{newclude}
\usepackage{pgfplots}
+\usepackage{polynom}
\usepackage{pst-plot}
\usepackage{standalone}
\usepackage{subfiles}
\newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm}
\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=important, fontupper=\sffamily\bfseries}
+\newtcolorbox{theorembox}[1]{colback=green!10!white, colframe=blue!20!white, coltitle=black, fontupper=\sffamily, fonttitle=\sffamily, #1}
\begin{document}
\begin{enumerate} \tightlist
\item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\)
- \item Find determinant of first matrix: \(\Delta = ps-qr\)
- \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
- or let \(\Delta \ne 0\) for one unique solution.
- \item Solve determinant equation to find variable \\
+ \item Find \(\det(\text{first matrix}) = ps-qr\)
+ \item Let \(\det = 0\) for \(\{0,\infty\}\) solutions
+ or \(\det \ne 0\) for 1 solution
+ \item Solve to find variable \\ \\
\textbf{For infinite/no solutions:}
\item Substitute variable into both original equations
- \item Rearrange equations so that LHS of each is the same
- \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
- \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
+ \item Rearrange so that LHS of each is the same
+ \item \(\begin{aligned}[t]
+ \infty \text{ solns: } & \text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x \\
+ 0 \text{ solns: } & \text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x
+ \end{aligned}\)
\end{enumerate}
- \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
+ \begin{cas}
+ Action \(\rightarrow\) Matrix \(\rightarrow\) Calculation \(\rightarrow\) \texttt{det}
+ \end{cas}
\subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
\section{Polynomials}
+ \subsection*{Factor theorem}
+
+ \begin{theorembox}{title=General form \(\beta x + \alpha\)}
+ If \(\beta x + \alpha\) is a factor of \(P(x)\), \\
+ \-\hspace{1em}then \(P(-\dfrac{\alpha}{\beta})=0\).
+ \end{theorembox}
+
+ \begin{theorembox}{title=Simple form \(x-a\)}
+ If \((x-a)\) is a factor of \(P(x)\), remainder \(R=0\). \\
+ \-\hspace{1em}\(\implies P(a)=0\)
+ \end{theorembox}
+
+ \subsection*{Remainder theorem}
+
+ \begin{theorembox}{}
+ When \(P(x)\) is divided by \(\beta x + \alpha\), the remainder is \(-\dfrac{\alpha}{\beta}\).
+ \end{theorembox}
+
+ \subsection*{Rational root theorem}
+ Let \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\) be a polynomial of degree \(n\) with \(a_i \in \mathbb{Z} \forall a\). Let \(\alpha, \beta \in \mathbb{Z}\) such that their highest common factor is 1 (i.e. relatively prime).
+
+ If \(\beta x + \alpha\) is a factor of \(P(x)\), then \(\beta\) divides \(a_n\) and \(\alpha\) divides \(a_0\) .
+
+ \subsubsection*{Discriminant}
+ \[\begin{cases}
+ b^2-4ac > 0 & \text{two solutions} \\
+ b^2-4ac = 0 & \text{one solution} \\
+ b^2-4ac < 0 & \text{no solutions}
+ \end{cases}\]
+ \begin{warning}
+ Flip inequality sign when multiplying by -1
+ \end{warning}
+
+ \subsection*{Long division}
+
+ \[ \polylongdiv{x^2+2x+4}{x-1} \]
+
+ \begin{cas}
+ Action \(\rightarrow\) Transformation \(\rightarrow\) \texttt{propFrac}
+ \end{cas}
+
\subsection*{Linear equations}
\subsubsection*{Forms}
Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
\subsection*{Quadratics}
+
\setlength{\abovedisplayskip}{1pt}
\setlength{\belowdisplayskip}{1pt}
+
+ \textbf{Linear factorisation}
\[ x^2 + bx + c = (x+m)(x+n) \]
\hfill where \(mn=c, \> m+n=b\)
\usepackage[dvipsnames, table]{xcolor}
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+\usepackage{array}
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+\usepackage{hhline}
\usepackage{import}
\usepackage{keystroke}
\usepackage{listings}
\usepackage{multirow}
\usepackage{pgfplots}
\usepackage{pst-plot}
+\usepackage{rotating}
\usepackage{subfiles}
\usepackage{tabularx}
\usepackage{tcolorbox}
scopes
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+\newcommand\given[1][]{\:#1\vert\:}
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-\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=important, fontupper=\sffamily\bfseries}
-\newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm}
+\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=darkgray, fontupper=\sffamily\bfseries}
+\newtcolorbox{cas}{colframe=cas!75!black, fonttitle=\sffamily\bfseries, title=On CAS, left*=3mm}
\begin{document}
\((b \cdot c)^n = b^n \cdot c^n\)\\
\({a^m \div a^n} = {a^{m-n}}\)
- \subsection*{Derivative rules}
-
- \renewcommand{\arraystretch}{1.4}
- \begin{tabularx}{\columnwidth}{rX}
- \hline
- \(f(x)\) & \(f^\prime(x)\)\\
- \hline
- \(\sin x\) & \(\cos x\)\\
- \(\sin ax\) & \(a\cos ax\)\\
- \(\cos x\) & \(-\sin x\)\\
- \(\cos ax\) & \(-a \sin ax\)\\
- \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
- \(e^x\) & \(e^x\)\\
- \(e^{ax}\) & \(ae^{ax}\)\\
- \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
- \(\log_e x\) & \(\dfrac{1}{x}\)\\
- \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
- \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
- \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
- \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
- \(\cos^{-1} x\) & \(\dfrac{-1}{sqrt{1-x^2}}\)\\
- \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
- \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) (reciprocal)\\
- \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx} (product rule)\)\\
- \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) (quotient rule)\\
- \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
- \hline
- \end{tabularx}
-
\subsection*{Reciprocal derivatives}
\[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
\subsection*{Differentiating \(x=f(y)\)}
- \begin{align*}
- \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*}
+ Find \(\dfrac{dx}{dy}\), then \(\dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)}\)
\subsection*{Second derivative}
\begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
\begin{tabularx}{\textwidth}{rXXX}
\hline
\rowcolor{shade2}
- & \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
+ & \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\[1.5em]
\hline
\(\dfrac{dy}{dx}>0\) &
- \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-3, xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x))}; \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}&
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-3, xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x)}; \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}&
\makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0.1, xmax=4, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))}; \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}&
\makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1.5, xmax=1.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\
\hline
\(\dfrac{dy}{dx}<0\) &
- \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {(1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}&
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}&
\makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0, xmax=1.5, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(2-x*x)^(1/2)}; \addplot[red] {-x+2}; \end{axis}\end{tikzpicture} \\Falling (concave down)}&
\makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=1.5, xmax=4.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {-x+3.1415}; \end{axis}\end{tikzpicture} \\Falling inflection point}\\
\hline
\(\dfrac{dy}{dx}=0\)&
- \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}& \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x))}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}&
- \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Stationary inflection point}\\
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}& \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x)}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}&
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Stationary inflection point}\\
\hline
\end{tabularx}
\end{table*}
\[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
- \noindent \colorbox{cas}{\textbf{On CAS:}}\\
- Action \(\rightarrow\) Calculation \(\rightarrow\) \texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}\\
- Returns \(y^\prime= \dots\).
-
- \subsection*{Integration}
-
- \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
-
- \subsection*{Integral laws}
-
- \renewcommand{\arraystretch}{1.4}
- \begin{tabularx}{\columnwidth}{rX}
- \hline
- \(f(x)\) & \(\int f(x) \cdot dx\) \\
- \hline
- \(k\) (constant) & \(kx + c\)\\
- \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
- \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
- \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
- \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
- \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
- \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
- \(e^k\) & \(e^kx + c\)\\
- \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
- \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
- \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
- \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
- \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
- \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
- \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
- \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) (substitution)\\
- \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
- \hline
- \end{tabularx}
-
- Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant \(\forall x \in (-a, a)\)
-
- \subsection*{Definite integrals}
+ \begin{cas}
+ Action \(\rightarrow\) Calculation \\
+ \hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)} \hfill(returns \(y^\prime= \dots\))
+ \end{cas}
+
+ \subsection*{Slope fields}
+
+ \begin{tikzpicture}[declare function={diff(\x,\y) = \x+\y;}]
+ \begin{axis}[axis equal, ymin=-4, ymax=4, xmin=-4, xmax=4, ticks=none, enlargelimits=true, ]
+ \addplot[thick, orange, domain=-4:2] {e^(x)-x-1};
+ \pgfplotsinvokeforeach{-4,...,4}{%
+ \draw[gray] ( {#1 -0.1}, {4 - diff(#1, 4) *0.1}) -- ( {#1 +0.1}, {4 + diff(#1, 4) *0.1});
+ \draw[gray] ( {#1 -0.1}, {3 - diff(#1, 3) *0.1}) -- ( {#1 +0.1}, {3 + diff(#1, 3) *0.1});
+ \draw[gray] ( {#1 -0.1}, {2 - diff(#1, 2) *0.1}) -- ( {#1 +0.1}, {2 + diff(#1, 2) *0.1});
+ \draw[gray] ( {#1 -0.1}, {1 - diff(#1, 1) *0.1}) -- ( {#1 +0.1}, {1 + diff(#1, 1) *0.1});
+ \draw[gray] ( {#1 -0.1}, {0 - diff(#1, 0) *0.1}) -- ( {#1 +0.1}, {0 + diff(#1, 0) *0.1});
+ \draw[gray] ( {#1 -0.1}, {-1 - diff(#1, -1) *0.1}) -- ( {#1 +0.1}, {-1 + diff(#1, -1) *0.1});
+ \draw[gray] ( {#1 -0.1}, {-2 - diff(#1, -2) *0.1}) -- ( {#1 +0.1}, {-2 + diff(#1, -2) *0.1});
+ \draw[gray] ( {#1 -0.1}, {-3 - diff(#1, -3) *0.1}) -- ( {#1 +0.1}, {-3 + diff(#1, -3) *0.1});
+ \draw[gray] ( {#1 -0.1}, {-4 - diff(#1, -4) *0.1}) -- ( {#1 +0.1}, {-4 + diff(#1, -4) *0.1});
+ }
+ \end{axis}
+ \end{tikzpicture}
+
+ \subsection*{Parametric equations}
+
+ For each point on \(\left( f(t), g(t) \right)\):
+
+ \begin{align*}
+ \dfrac{dy}{dt} &= \dfrac{dy}{dx} \cdot \dfrac{dx}{dt} \\
+ \therefore \dfrac{dy}{dx} &= \dfrac{\left(\dfrac{dy}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ provided } \dfrac{dx}{dt} \ne 0 \\
+ \text{Also...} \\
+ \dfrac{d^2y}{dx^2} &= \dfrac{\left(\dfrac{dy^\prime}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ where } y^\prime = \dfrac{dy}{dx}
+ \end{align*}
+
+ \subsection*{Integration}
+
+ \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
+
+ \subsubsection*{Definite integrals}
\[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
\subsubsection*{Properties}
- \[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\]
-
- \[\int^a_a f(x) \> dx = 0\]
-
- \[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\]
-
- \[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\]
-
- \[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\]
+ \begin{align*}
+ \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 \\
+ \end{align*}
\subsection*{Integration by substitution}
\[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
- \noindent Note \(f(u)\) must be 1:1 \(\implies\) one \(x\) for each \(y\)
+ \begin{warning}
+ \(\boldsymbol{f(u)}\) must be 1:1 \(\boldsymbol{\implies}\) one \(\boldsymbol{x}\) for each \(\boldsymbol{y}\)
+ \end{warning}
\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\\
\subsection*{Partial fractions}
- \colorbox{cas}{On CAS:}\\
- \indent Action \(\rightarrow\) Transformation \(\rightarrow\)
- \texttt{expand/combine}\\
- \indent Interactive \(\rightarrow\) Transformation \(\rightarrow\)
- Expand \(\rightarrow\) Partial
+ To factorise \(f(x) = \frac{\delta}{\alpha \cdot \beta}\):
+ \begin{align*}
+ \dfrac{\delta}{\alpha \cdot \beta \cdot \gamma} &= \dfrac{A}{\alpha} + \dfrac{B}{\beta} + \dfrac{C}{\gamma} \tag{1} \\
+ \text{Multiply by } & (\alpha \cdot \beta \cdot \gamma) \text{:} \\
+ \delta &= \beta\gamma A + \alpha\gamma B +\alpha\beta C \tag{2} \\
+ \text{Substitute } x &= \{\alpha, \beta, \gamma\} \text{ into (2) to find denominators}
+ \end{align*}
+
+ \subsubsection*{Repeated linear factors}
+
+ \[ \dfrac{p(x)}{(x-a)^n} = \dfrac{A_1}{(x-a)} + \dfrac{A_2}{(x-a)^2} + \dots + \dfrac{A_n}{(x-a)^n} \]
+
+ \subsubsection*{Irreducible quadratic factors}
+
+ \[ \text{e.g. } \dfrac{3x-4}{(2x-3)(x^2+5)} = \dfrac{A}{2x-3} + \dfrac{Bx+C}{x^2+5} \]
+
+ \begin{cas}
+ Action \(\rightarrow\) Transformation:\\
+ \hspace{1em} \texttt{expand(..., x)}
+
+ To reverse, use \texttt{combine(...)}
+ \end{cas}
\subsection*{Graphing integrals on CAS}
- \colorbox{cas}{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\)
- \(\int\) (\(\rightarrow\) Definite)\\
- Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
+ \begin{cas}
+ \textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\
+ Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
+ \end{cas}
\subsection*{Applications of antidifferentiation}
Approximate as sum of infinitesimally-thick cylinders
- \subsubsection*{Rotation about \(x\)-axis}
+ \subsubsection*{Rotation about \(\boldsymbol{x}\)-axis}
- \begin{align*}
- V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
- &= \pi \int^b_a (f(x))^2 \> dx
- \end{align*}
+ \[ V = \pi\int^{x=b}_{x=a} f(x)^2 \> dx \]
- \subsubsection*{Rotation about \(y\)-axis}
+ \subsubsection*{Rotation about \(\boldsymbol{y}\)-axis}
\begin{align*}
- V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
- &= \pi \int^b_a (f(y))^2 \> dy
+ V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\
+ &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy
\end{align*}
- \subsubsection*{Regions not bound by \(y=0\)}
+ \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)}
\[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
\hfill where \(f(x) > g(x)\)
\[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
- \noindent \colorbox{cas}{On CAS:}\\
- \indent Evaluate formula,\\
- \indent or Interactive \(\rightarrow\) Calculation
- \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
+ \begin{cas}
+ \begin{enumerate}[label=\alph*), leftmargin=5mm]
+ \item Evaluate formula
+ \item Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
+ \end{enumerate}
+ \end{cas}
\subsection*{Rates}
\[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
-
+ \include{calculus-rules}
+
\section{Kinematics \& Mechanics}
\subsection*{Constant acceleration}
Note hypotheses are always expressed in terms of population parameters
\end{warning}
- \subsection*{Null hypothesis \(H_0\)}
+ \subsection*{Null hypothesis \(\textbf{H}_0\)}
Sample drawn from population has same mean as control population, and any difference can be explained by sample variations.
- \subsection*{Alternative hypothesis \(H_1\)}
+ \subsection*{Alternative hypothesis \(\textbf{H}_1\)}
Amount of variation from control is significant, despite standard sample variations.
\subsection*{\(p\)-value}
+ Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true.
+ For one-tail tests:
\begin{align*}
- p &= \Pr(\overline{X} \lessgtr \mu(H_1)) \\
- &= 2 \cdot \Pr(\overline{X} <> \mu(H_1) | \mu = 8)
+ p\text{-value} &= \Pr\left( \> \overline{X} \lessgtr \mu(\textbf{H}_1) \> \given \> \mu = \mu(\textbf{H}_0)\> \right) \\
+ &= \Pr\left( Z \lessgtr \dfrac{\left( \mu(\textbf{H}_1) - \mu(\textbf{H}_0) \right) \cdot \sqrt{n} }{\operatorname{sd}(X)} \right) \\
+ &\text{then use \texttt{normCdf} with std. norm.}
\end{align*}
- Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true.
-
\vspace{0.5em}
\begin{tabularx}{23em}{|l|X|}
\hline
\rowcolor{cas}
\(\boldsymbol{p}\) & \textbf{Conclusion} \\
\hline
- \(> 0.05\) & insufficient evidence against \(H_0\) \\
- \(< 0.05\) (5\%) & good evidence against \(H_0\) \\
- \(< 0.01\) (1\%) & strong evidence against \(H_0\) \\
- \(< 0.001\) (0.1\%) & very strong evidence against \(H_0\) \\
+ \(> 0.05\) & insufficient evidence against \(\textbf{H}_0\) \\
+ \(< 0.05\) (5\%) & good evidence against \(\textbf{H}_0\) \\
+ \(< 0.01\) (1\%) & strong evidence against \(\textbf{H}_0\) \\
+ \(< 0.001\) (0.1\%) & very strong evidence against \(\textbf{H}_0\) \\
\hline
\end{tabularx}
- \subsection*{Statistical significance}
+ \subsection*{Significance level \(\alpha\)}
- Significance level is denoted by \(\alpha\).
+ The condition for rejecting the null hypothesis.
\-\hspace{1em} If \(p<\alpha\), null hypothesis is \textbf{rejected} \\
\-\hspace{1em} If \(p>\alpha\), null hypothesis is \textbf{accepted}
Menu \(\rightarrow\) Statistics \(\rightarrow\) Calc \(\rightarrow\) Test. \\
Select \textit{One-Sample Z-Test} and \textit{Variable}, then input:
\begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=2cm, font=\normalfont]
- \item[\(\mu\) cond:] same operator as \(H_1\)
+ \item[\(\mu\) cond:] same operator as \(\textbf{H}_1\)
\item[\(\mu_0\):] expected sample mean (null hypothesis)
\item[\(\sigma\):] standard deviation (null hypothesis)
\item[\(\overline{x}\):] sample mean
\end{cas}
\subsection*{One-tail and two-tail tests}
+
+ \[ p\text{-value (two-tail)} = 2 \times p\text{-value (one-tail)} \]
\subsubsection*{One tail}
\begin{itemize}
\item \(\mu\) has changed in one direction
- \item State ``\(H_1: \mu \lessgtr \) known population mean''
+ \item State ``\(\textbf{H}_1: \mu \lessgtr \) known population mean''
\end{itemize}
\subsubsection*{Two tail}
\begin{itemize}
\item Direction of \(\Delta \mu\) is ambiguous
- \item State ``\(H_1: \mu \ne\) known population mean''
+ \item State ``\(\textbf{H}_1: \mu \ne\) known population mean''
\end{itemize}
- For two tail tests:
\begin{align*}
p\text{-value} &= \Pr(|\overline{X} - \mu| \ge |\overline{x}_0 - \mu|) \\
- &= \left( |Z| \ge \left|\dfrac{\overline{x}_0 - \mu}{\sigma \div \sqrt{n}} \right| \right)
+ &= \left( |Z| \ge \left|\dfrac{\overline{x}_0 - \mu}{\sigma \div \sqrt{n}} \right| \right) \\
\end{align*}
+ where
+ \begin{description}[nosep, labelindent=0.5cm]
+ \item [\(\mu\)] is the population mean under \(\textbf{H}_0\)
+ \item [\(\overline{x}_0\)] is the observed sample mean
+ \item [\(\sigma\)] is the population s.d.
+ \item [\(n\)] is the sample size
+ \end{description}
+
\subsection*{Modulus notation for two tail}
\(\Pr(|\overline{X} - \mu| \ge a) \implies\) ``the probability that the distance between \(\overline{\mu}\) and \(\mu\) is \(\ge a\)''
\subsection*{Errors}
\begin{description}[labelwidth=2.5cm, labelindent=0.5cm]
- \item [Type I error] \(H_0\) is rejected when it is \textbf{true}
- \item [Type II error] \(H_0\) is \textbf{not} rejected when it is \textbf{false}
+ \item [Type I error] \(\textbf{H}_0\) is rejected when it is \textbf{true}
+ \item [Type II error] \(\textbf{H}_0\) is \textbf{not} rejected when it is \textbf{false}
\end{description}
+ \begin{tabularx}{\columnwidth}{|X|l|l|}
+ \rowcolor{cas}\hline
+ \cellcolor{white}&\multicolumn{2}{c|}{\textbf{Actual result}} \\
+ \hline
+ \cellcolor{cas}\(\boldsymbol{z}\)\textbf{-test} & \cellcolor{light-gray}\(\textbf{H}_0\) true & \cellcolor{light-gray}\(\textbf{H}_0\) false \\
+ \hline
+ \cellcolor{light-gray}Reject \(\textbf{H}_0\) & Type I error & Correct \\
+ \hline
+ \cellcolor{light-gray}Do not reject \(\textbf{H}_0\) & Correct& Type II error \\
+ \hline
+ \end{tabularx}
+
% \subsection*{Using c.i. to find \(p\)}
% need more here