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\texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})} \subsubsection*{Properties} @@ -584,6 +586,78 @@ Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\) \end{itemize} + \subsubsection*{Perpendicular bisectors of a triangle} + +\hspace{-1.5cm}\begin{tikzpicture} + [ + scale=3, + >=stealth, + point/.style = {draw, circle, fill = black, inner sep = 1pt}, + dot/.style = {draw, circle, fill = black, inner sep = .2pt}, + thick + ] + + \node at (-1,1) [text width=5cm, rounded corners, fill=lblue, inner sep=1ex] + { + \sffamily The three bisectors meet at the circumcenter \(Z\) where \(|\overrightharp{ZA}| = |\overrightharp{ZB}| = |\overrightharp{ZC}|\). + }; + + % the circle + \def\rad{1} + \node (origin) at (0,0) [point, label = {right: {\(Z\)}}]{}; + \draw [thin] (origin) circle (\rad); + + % triangle nodes: just points on the circle + \node (n1) at +(60:\rad) [point, label = above:\(A\)] {}; + \node (n2) at +(-145:\rad) [point, label = below:\(B\)] {}; + \node (n3) at +(-45:\rad) [point, label = {below right:\(C\)}] {}; + + % triangle edges: connect the vertices, and leave a node at the midpoint + \draw[orange] (n3) -- node (a) [label = {above right:\(D\)}] {} (n1); + \draw[blue] (n3) -- node (b) [label = {below right:\(F\)}] {} (n2); + \draw[red] (n1) -- node (c) [label = {left: \(E\)}] {} (n2); + + % Bisectors + % start at the point lying on the line from (origin) to (a), at + % twice that distance, and then draw a path going to the point on + % the line lying on the line from (a) to the (origin), at 3 times + % that distance. + \draw[orange, dotted] + ($ (origin) ! 2 ! (a) $) + node [right] {\sffamily Bisector \(AC\)} + -- ($(a) ! 3 ! (origin)$ ); + + % similarly for origin and b + \draw[blue, dotted] + ($ (origin) ! 2 ! (b) $) + -- ($(b) ! 3 ! (origin)$ ) + node [right] {\sffamily Bisector \(BC\)}; + + \draw[red, dotted] + ($ (origin) ! 5 ! (c) $) + -- ($(c) ! 7 ! (origin)$ ) + node [right] {\sffamily Bisector \(AB\)}; + + \draw[gray, dashed, thin] (n1) -- (origin) -- (n2); + \draw[gray, dashed, thin] (origin) -- (n3); + + % Right angle symbols + \def\ralen{.5ex} % length of the short segment + \foreach \inter/\first/\last in {a/n3/origin, b/n2/origin, c/n2/origin} + { + \draw [thin] let \p1 = ($(\inter)!\ralen!(\first)$), % point along first path + \p2 = ($(\inter)!\ralen!(\last)$), % point along second path + \p3 = ($(\p1)+(\p2)-(\inter)$) % corner point + in + (\p1) -- (\p3) -- (\p2); % path + } +\end{tikzpicture} + + \begin{theorembox}{title=Perpendicular bisector theorem} + If a point \(P\) lies on the perpendicular bisector of line \(\overrightharp{XY}\), then \(P\) is equidistant from the endpoints of the bisected segment + \[ \text{i.e. } |\overrightharp{PX}| = |\overrightharp{PY}| \] + \end{theorembox} + \subsubsection*{Useful vector properties} \begin{itemize} @@ -830,11 +904,11 @@ arc[start angle=220, end angle=320, radius=2cm] -- cycle; \node {Major Segment}; - \node at (-90:2) {Minor Segment}; + \node at (-90:1.5) {Minor Segment}; \begin{scope}[xshift=4.5cm] - \draw circle (2cm); - \filldraw[fill=lblue] + \draw [fill=lblue] circle (2cm); + \filldraw[fill=white] (320:2cm) node[right] {} -- (0,0) node[above] {} -- (220:2cm) node[left] {} @@ -1185,6 +1259,64 @@ \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\] + \subsection*{Euler's method} + + \[\dfrac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\] + + \[\implies f(x+h) \approx f(x) + hf^\prime(x)\] + + \begin{theorembox}{} + If \(\dfrac{dy}{dx} = g(x)\) with \(x_0 = a\) and \(y_0 = b\), then: + \[\begin{cases} + x_{n+1} = x_n + h \\ + y_{n+1} = y_n + hg(x_n) + \end{cases}\] + \end{theorembox} + + \[ + \dfrac{d^2y}{dx^2} + \begin{cases} + > 0 \implies \text{ underestimate (concave up)} \\ + < 0 \implies \text{ overestimate (concave down)} + \end{cases} + \] + + \begin{center}\begin{tikzpicture} + \begin{axis}[xmin=0, xmax=1.6, ticks=none, enlargelimits=true, samples=100] + \addplot[blue, domain=-0.25:1.5, postaction={decorate,decoration={text along path, text align={align=center, left indent=3cm}, text={|\sffamily|solution curve}}}] {e^(x-3/2)+1/4}; + \addplot[red] {(x+1/2)*e^(-1)+1/4} (1.7,1.0593) node [above, black] {\(\ell\)}; + \addplot[mark=*, black] coordinates {(0.5,0.6179)} node[above left]{\((x_0, y_0)\)}; + \addplot[mark=*, orange] coordinates {(1.4,1.1548)} node[left]{\color{black} \sffamily correct solution}; + \addplot[mark=*, black] coordinates {(1.4,0.94897)} node[above right] {\((x_1,y_1)\)}; + \draw [gray, dashed] (0.5,0) -- (0.5,0.6179) -- (1.6,0.6179); + \draw [gray, dashed] (1.4,0) -- (1.4, 1.1548); + \draw [<->] (0.5,0.48) -- (1.4,0.48) node[midway, fill=white] {\(h\)}; + \draw [gray, dashed] (1.4,0.94897) -- (1.6,0.94897); + \draw [<->] (1.5,0.94897) -- (1.5,0.6179) node[midway, rotate=90, below] {\(hg(x_0)\)}; + \end{axis} + \end{tikzpicture}\end{center} + + \begin{cas} + Menu \(\rightarrow\) Sequence \(\rightarrow\) Recursive + + \textbf{To generate \(\boldsymbol{x}\)-values:} + \begin{itemize} + \item Enter \(a_{n+1}=a_n + h\) where \(h\) is the step size \\ + (input \(a_n\) from menu bar) + \item In \(a_0\), set the initial value \(x_0\) as a constant + \end{itemize} + + \textbf{To generate \(\boldsymbol{y}\)-values:} + \begin{itemize} + \item In \(b_{n+1}\), enter \(\dfrac{dy}{dx}\), replacing \(x\) with \(a_n\) + \item Set \(b_0 = y(x_0)\) as a constant + \end{itemize} + + To view table of values, tap table icon (top left) \\ + To compare approximations with actual values, enter in \(c_{n+1} = a_{n+1} - f(a_{n+1})\) where \(f(x) = \int \dfrac{dy}{dx} \> dx\) + + \end{cas} + \subsection*{Fundamental theorem of calculus} If \(f\) is continuous on \([a, b]\), then @@ -1201,29 +1333,65 @@ \begin{warning} To verify solutions, find \(\frac{dy}{dx}\) from \(y\) and substitute into original \end{warning} - - + + \vspace*{1cm} + \hspace*{-1cm} + + { \tabulinesep=1.2mm + \begin{tabu}{|c|c|} + + \hline + \taburowcolors 2{gray..white} + \textbf{DE} & \textbf{Method} \\ + \hline + + \tabureset + \(\dfrac{dy}{dx} = f(x)\) + & + {\(\begin{aligned} + y &= \int f(x) \> dx \\ + &= F(x) + c \quad \text{where } F^\prime(x) = f(x) + \end{aligned}\)} \\ + + \hline + + \(\dfrac{d^2y}{dx^2} = f(x)\) + & + {\(\begin{aligned} + \dfrac{dy}{dx} &= \int f(x) \> dx \\ + &= F(x) + c \quad \text{where } F^\prime(x) = f(x) \\ + \therefore y &= \iint f(x) \> dx = \int \left( F(x) + c \right) \> dx \\ + &= G(x) + cx + d \\ + & \text{where } G^\prime(x) = F(x) + \end{aligned}\)} \\ + + \hline + + \(\dfrac{dy}{dx} = g(y)\) + & + {\(\begin{aligned} + \dfrac{dx}{dy} &= \dfrac{1}{g(y)} \\ + \therefore x &= \int \dfrac{1}{g(y)} \> dy \\ + &= F(y) + c \\ + & \text{where } F^\prime(y) = \dfrac{1}{g(y)} + \end{aligned}\)} \\ + + \hline + + \(\dfrac{dy}{dx} = f(x) g(y)\) + & + {\(\begin{aligned} + f(x) &= \dfrac{1}{g(y)} \cdot \dfrac{dy}{dx} \\ + \int f(x) \> dx &= \int \dfrac{1}{g(y)} \> dy + \end{aligned}\)} \\ + + \hline + \end{tabu}} \subsubsection*{Mixing problems} \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\] - \subsection*{Euler's method} - - \[\dfrac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\] - - \[\implies f(x+h) \approx f(x) + hf^\prime(x)\] - - \begin{theorembox}{} - If \(\dfrac{dy}{dx} = g(x)\) with \(x_0 = a\) and \(y_0 = b\), then: - \begin{align*} - x_{n+1} &= x_n + h \\ - y_{n+1} &= y_n + hg(x_n) - \end{align*} - \end{theorembox} - - - \include{calculus-rules} \section{Kinematics \& Mechanics} -- 2.47.1