\usepackage[dvipsnames, table]{xcolor}
\usepackage{amsmath}
\usepackage{amssymb}
+\usepackage{array}
\usepackage{blindtext}
\usepackage{dblfloatfix}
\usepackage{enumitem}
\usepackage[a4paper,margin=2cm]{geometry}
\usepackage{graphicx}
\usepackage{harpoon}
+\usepackage{hhline}
\usepackage{import}
\usepackage{keystroke}
\usepackage{listings}
\usepackage{multirow}
\usepackage{pgfplots}
\usepackage{pst-plot}
+\usepackage{rotating}
\usepackage{subfiles}
\usepackage{tabularx}
\usepackage{tcolorbox}
scopes
}
+\newcommand\given[1][]{\:#1\vert\:}
\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
\usepgflibrary{arrows.meta}
\definecolor{cas}{HTML}{e6f0fe}
\definecolor{important}{HTML}{fc9871}
\definecolor{dark-gray}{gray}{0.2}
+\definecolor{light-gray}{HTML}{cccccc}
+\definecolor{peach}{HTML}{e6beb2}
+\definecolor{lblue}{HTML}{e5e9f0}
\newcommand{\tg}{\mathop{\mathrm{tg}}}
\newcommand{\cotg}{\mathop{\mathrm{cotg}}}
\newcommand{\arctg}{\mathop{\mathrm{arctg}}}
\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
-\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}