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\title{Calculus BC Equations}
\author{The Hudson School, Hoboken, NJ}
% * <mzhang1.student@thehudsonschool.org> 2017-03-30T13:34:20.173Z:
%
% ^.
\date{2017}
%
\newcommand{\abs}[1]{\left| #1 \right|}
%\renewcommand{\vec}[1]{\mathbf{#1}}
%
\usepackage{blindtext}
\begin{document}
\maketitle
\begin{multicols}{2}
%\section*{Trigonometry}
\section*{Limits}
%\subsection*{Projectile Motion}
% \begin{tabularx}{\columnwidth}{c c}
% $x_o+v_{xo}t+\frac{1}{2}at^2$ & $ y_o+v_{yo}t-\frac{1}{2}gt^2$ \\
% $ v_x=v_{xo}t+at$ & $v_y=v_{yo}-gt $\\
% \end{tabularx}
\begin{tabular}{lll}
If $\lim_{x\to a} \frac{f(x)}{g(x)}=\frac{0}{0}, \text{or} \frac{\infty}{\infty}$ \\ then $\lim_{x\to a} \frac{f(x)}{g(x)}=\lim_{x\to a} \frac{f'(x)}{g'(x)}$ & &
$\lim_{x\to 0} \frac{\sin x}{x} = 1$\\
%$\vec{x}=x_o + {v}_{xo}t + \frac{1}{2}at^2$ & & $\vec{v}=v_{o}+at $\\[4ex]
%$\vec{v}_{avg}=\frac{d}{t}$ & $\rightarrow d=vt$& $ \rightarrow t=\frac{d}{v}$ \\[4ex]
%$ \vec{a}_{avg}=\frac{\Delta v}{t}$ & $v_f^2=v_o^2 +2a\Delta x $ & \\
\end{tabular}
%\noindent For projectile motion problems with no air resistance the main Kinematics equation reduces to:
%$$ \langle x_o+v_{xo}t, \; y_o+v_{yo}t-\frac{1}{2}gt^2 \rangle$$
%\section*{Differentiation}
\section*{Applications of Differentiation}
\begin{tabular}{lcl}
$V=\frac{4}{3}\pi r^3 $ &
$V=\frac{1}{3}h\pi r^2$
\end{tabular}
\section*{Integration}
\begin{tabularx}{\columnwidth}{c c}
$\int u \, dv = uv - \int v\,du$ &
$\frac{1}{b-a}\int_a^b{f(x)}\, dx$ \\[5mm]
$\frac{1}{(x-1)(x-2)}=\frac{A}{x-1}+\frac{B}{x-2}$ \\[2em]
$ \int_a^b \sqrt{1+\left( \frac{dy}{dx} \right) ^2 } \, dx $\\
%\includegraphics[width=0.28\textwidth]{Arc_Length_Formula}
\end{tabularx}
\section*{Applications of Integration}
%\blindtext[4]
\begin{tabularx}{\columnwidth}{X X r}
$\pi\int_a^b{(R_2)\,^2-(R_1)^2}dx$ \\ [5mm]
\end{tabularx}
\section*{Infinite Series}
\begin{description}
\item [Monotonic Sequence] A sequence $\{a_n\}$ that is nondecreasing (i.e. $\{1, 1, 2, 3\}$) where $$a_1\leq a_2\leq a_3\leq \cdots \leq a_n \leq \cdots$$ or if terms are nonincreasing like $$a_1\geq a_2\geq a_3\geq \cdots \geq a_n \geq \cdots$$
\item [Bounded Monotonic Sequence] A bounded monotonic sequence converges.
A sequence is bounded if it bounded above by M and below by N such that\\ $ N< a_n < M $, $ \forall \, n \geq 0$.
\item [Infinite Series]
Infinite series are defined as $$S=\sum_{n=1}^\infty a_n$$ where $S_n \text{denotes the } n^{\text{th}}\text{ partial sum}$\\
\item [Convergence:]
For an infinite series $S=\sum a_n$, where $ S_n \text{denotes the } n^{\text{th}}\text{ partial sum} $, if the sequence $\{S_n\}$ converges to $S$ then the series $S=\sum a_n$ converges. The limit $S$ is called the sum of the series.\\
\item [Integral Test:] For an infinite series $S=\sum f(x)$ if the improper integral $\int f(x) = L$ converges then the series converges and if the improper integral $\int f(x)$ does not exist or is infinity, it diverges. It does not give any information about the actual sum of the series.\\
\item [ P series:]
$$N=\sum_{n=1}^\infty \frac{1}{n^p}=\frac{1}{1^p}+\frac{1}{2^p}+\frac{1}{3^p}+\cdots$$
$ P=1$ diverges
$ P>1 $ converges
$ P$ < $1 $ diverges
$ 0>P>1 $ diverges\\
\item [Taylor Polynomials]
If f has n derivatives at c, then the polynomial\\
\[ P_n(x) = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \cdots + \frac{f^{(n)}(c)}{n!} (x-c)^n \]\\
\noindent is defined as the \textbf{nth} degree \textbf{taylor polynomial}.
\item [Taylor Series]
If f is infinitely differentiable, then f is represented exactly by the series, centered at $x=c$ \\
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x-c)^n\]\\
%\includegraphics[width=\linewidth]{Garamond_Series_Chart}
\end{description}
\begin{tabularx}{\columnwidth}{X X r}
\end{tabularx}
\section*{Parametric Equations}
\begin {tabularx}{\columnwidth}{X l}
$x=r\cos{\theta}$ & $y=r\sin{\theta}$\\ [5mm]
Distance Formula:&
$\Delta s=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$\\[5mm]
$x(t)=x_0+v_{x0}+\frac{1}{2}at^2$&
$y(t)=y_0+v_{y0}+\frac{1}{2}at^2$\\[5mm]
$\int_a^b\sqrt{{\left(\frac{dx}{dt}\right)^2}+{\left(\frac{dy}{dt}\right)}^2}$\\
\vspace{2mm}
\textbf{Projectile Motion} \\[5mm]
Maximum Height:&
$H=\frac{v_o^2\sin{\theta}^2}{2g}$\\[5mm]
Horizontal Range:&
$R=\frac{v_o^2\sin{2\theta}}{g}$\\[5mm]
Flight Time:&
$t=\frac{2v_{y0}sin{\theta}}{g}$\\[5mm]
\end{tabularx}
\subsection*{Example}
Eliminating the Parameter:\\
Finding a rectangular equation that represents the graph of a set of parametric equations is called \textit{eliminating the parameter}.\\
\begin{enumerate}
\item Parametric Equations
\begin{tabular}{lcl}
$x=t^2-4$&$y=\frac{t}{2}$
\end{tabular}
\item Solve for t in one equation.\\
$x=(2y)^2-4$
\item Rectangular Equation\\
$x=4y^2-4$
\end{enumerate}
\section*{Polar Equations}
\begin{tabularx}{\columnwidth}{X X r}
$\int_{\alpha}^{\beta}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta$ &
%$\int_{\alpha}^{\beta}\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$\\[3mm]
$\frac{1}{2}\int_\alpha^\beta{r^2} \, d\theta$\\[5mm]
$x=r\cos{\theta}$&
$ x^2+y^2=r^2$\\
$y=r\cos{\theta}$ & $\tan \theta = \frac{y}{x} $\\
\includegraphics[width=0.45\textwidth]{Polar_Graph}
\end{tabularx}
\section*{Vectors}
Angle Between Two Vectors\\
If $\theta$ is the angle between two nonzero vectors $\vec {u} \text{ and } \vec {v}$, then \\
$$\Large \cos \theta = \frac{\vec{u} \cdot \vec{v}}{\Vert \vec {u} \Vert \Vert \vec {v} \Vert}$$\\
Alternatively,\\
\[ \vec{u} \cdot \vec{v}= \Vert \vec{u} \Vert \, \Vert \vec{v} \Vert \cos{\theta} \]\\
This form can be used to calculate the dot product without knowing the component form of the vectors.
\section*{Vector-Valued Functions}
A function of the form:
\[
\vec{r}(t) = f(t) \vec{i} + g(t) \vec{j} +h(t) \vec{k}
\]
\noindent Can also be written as:
\[
\vec{r}(t) = \langle f(t), g(t), h(t) \rangle
\]
\noindent Projectile Motion equations (without air resistance) can be written as Vector Valued Functions:
\begin{align*}
\vec{s}(t) & = \langle x_o + v_{xo}t, y_o+v_{yo}t-\frac{1}{2}gt^2 \rangle \\[1em]
\vec{v}(t) & = \langle v_{xo}, v_{yo}-gt \rangle \\[1em]
\vec{a}(t) & = \langle 0, -g \rangle \end{align*}
\section*{Differential Equations}
%
% Eulers method
%%%%%%%%%%%%%%%
\begin{center}
\begin{itemize}
\item Separable Differentiable Equations
\begin{enumerate}
\item Separate the variables into standard form:
\[ F(y) \, dy=G(x) \, dx \]
\end{enumerate}
\item First Order Differentiable
\begin{enumerate}
\item Rearrange equation into standard form:\\
{\centering $y'+py=q$\\}
\item Integrating factor:\\
{\centering $u(x) = e^{\int pdx}$\\}
\item Multiply both sides:\\
{\centering $uy'+upy=uq$\\
$(uy)'=uq$\\}
\item Integrate:\\
{\centering $uy=\int(uq)dx$\\}
\end{enumerate}
\item Euler's Method Formula
\[x_{n+1} = x_n + h \]
\[y_{n+1} = y_n + h f(x_n, y_n)\]
\end{itemize}
\end{center}
\end{multicols}
\section*{Elementary Power Series}
\begin{align*}
1-(x-1)+(x-1)^2-(x-1)^3+(x-1)^4-\cdots+(-1)^n(x-1)^n+\cdots & \\[1em] % 1/x
1-x+x^2-x^3+x^4-x^5+\cdots+(-1)^n x^n+\cdots & \\[1em] % 1/(1-x)
(x-1)-\frac{(x-1)^2}{2}+\frac{(x-1)^3}{3}-\frac{(x-1)^4}{4}+\cdots+\frac{(-1)^{n-1}(x-1)^n}{n}+\cdots & \\[1em] % ln(X)
1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} +\cdots+ \frac{x^n}{n!} +\cdots & \\[1em] % e^x
x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} -\cdots+ \frac{(-1)^n x^{2n+1}}{(2n+1)!}+\cdots & \\[1em]
1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!}-\cdots+ \frac{(-1)^n x^{2n}}{(2n)!}+\cdots & \\[1em]
x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \cdots+\frac{(-1)^n x^{2n+1}}{2n+1} +\cdots & \\[1em]
x + \frac{x^3}{2\cdot3} + \frac{1\cdot3x^5}{2\cdot4
\cdot5} + \frac{1\cdot3\cdot5x^7}{2\cdot4\cdot6\cdot7} +\cdots+\frac{(2n)! x^{2n+1}}{(2^nn!)^2(2n+1)} +\cdots & \\[1em]
1 + kx + \frac{k(k-1)x^2}{2!} + \frac{k(k-1)(k-2)x^3}{3!} + \frac{k(k-1)(k-2)(k-3)x^4}{4!} +\cdots & \\
\end{align*}
% $1-(x-1)+(x-1)^2-(x-1)^3+(x-1)^4-\cdots+(-1)^n(x-1)^n+\cdots$
% $1-x+x^2-x^3+x^4-x^5+\cdots+(-1)^n x^n+\cdots$
% $(x-1)-\frac{(x-1)^2}{2}+\frac{(x-1)^3}{3}-\frac{(x-1)^4}{4}+\cdots+\frac{(-1)^{n-1}(x-1)^n}{n}+\cdots$
% $ 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} +\cdots+ \frac{x^n}{n!} +\cdots$
% $ x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} -\cdots+ \frac{(-1)^n x^{2n+1}}{(2n+1)!}+\cdots$
% $ 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!}-\cdots+ \frac{(-1)^n x^{2n}}{(2n)!}+\cdots$
% $ x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \cdots+\frac{(-1)^n x^{2n+1}}{2n+1} +\cdots$
% $x + \frac{x^3}{2\cdot3} + \frac{1\cdot3x^5}{2\cdot4
% \cdot5} + \frac{1\cdot3\cdot5x^7}{2\cdot4\cdot6\cdot7} +\cdots+\frac{(2n)! x^{2n+1}}{(2^nn!)^2(2n+1)} +\cdots$
% $1 + kx + \frac{k(k-1)x^2}{2!} + \frac{k(k-1)(k-2)x^3}{3!} + \frac{k(k-1)(k-2)(k-3)x^4}{4!} +\cdots$
\newpage
\begin{multicols}{2}
\newpage
%\section*{Trigonometry}
\section*{Limits}
\begin{itemize}
\item Limits at infinity
\begin{itemize}
\item Three possibilities for horizontal asymptotes
\end{itemize}
\item Removeable vs. non-removeable discontinuity
\begin{itemize}
\item One-sided limits
\end{itemize}
\item L'H\^{o}pital's Rule
\begin{itemize}
\item Conditions for use
\end{itemize}
\end{itemize}
% \begin{description}
% \item [limits] \blindtext
% \end{description}
%\subsection*{Projectile Motion}
% \begin{tabularx}{\columnwidth}{c c}
% $x_o+v_{xo}t+\frac{1}{2}at^2$ & $ y_o+v_{yo}t-\frac{1}{2}gt^2$ \\
% $ v_x=v_{xo}t+at$ & $v_y=v_{yo}-gt $\\
% \end{tabularx}
%\noindent For projectile motion problems with no air resistance the main Kinematics equation reduces to:
\section*{Differentiation}
\begin{itemize}
\item Definition of derivative at a point
\item Derivatives of polynomials, trig, and exponential functions
\item Differentiation rules
% \begin{itemize}
% \item Chain rule
% \item Product rule
% \end{itemize}
\item Equation of a tangent line to a curve
\item Interpreting the signs of the first and second derivative
\begin{itemize}
\item Find the min or max of a function
\end{itemize}
\item Sketching the first and second derivative from a graph
\item Implicit differentiation
\end{itemize}
%\medskip
\section*{Applications of Differentiation}
\begin{itemize}
\item Optimization
\begin{itemize}
\item Distance, area, volume
\end{itemize}
\item Newton's Method
\item Related Rates
\begin{itemize}
\item Distance, area, volume, depth, ladder, and shadows
\end{itemize}
\end{itemize}
\section*{Integration}
\begin{itemize}
\item Reimann Sums
\item Difference between area and definite integral
\item Integration by substitution
\item Integration by parts
\item Partial fractions
\begin{itemize}
\item Distinct linear, repeated linear, quadratic and repeated quadratic factors
\end{itemize}
\item Improper integrals
\end{itemize}
\section*{Applications of Integration}
\begin{itemize}
\item Volumes of revolution
\item Work done by a variable force
\item Average value of a function
\item Arc length of a curve
\item Area between two curves
\end{itemize}
\section*{Infinite Series}
\begin{itemize}
\item P-series
\item Geometric Series
\item Convergence or divergence
\begin{itemize}
\item Integral test, ratio test, comparison test and root test
\end{itemize}
\item Taylor polynomial approximation with desired accuracy
\item Taylor and Maclaurin series for elementary functions
\begin{itemize}
\item Radius of convergence
\item Interval of convergence
\end{itemize}
\end{itemize}
\section*{Parametric Equations}
\begin{itemize}
\item Converting to/from rectangular functions
\item Difference between $ \frac{dy}{dx}$, $\frac{dy}{dt}$ and $\frac{dx}{dt}$
\item Second derivative of a parametric equation
\item Arc length
\item Projectile Motion: Range, hangtime, and max height
\end{itemize}
\section*{Polar Equations}
\begin{itemize}
\item Converting to/from rectangular functions
\item Area and arc length of polar functions
\end{itemize}
\section*{Vectors}
\begin{itemize}
\item Dot product of two vectors
\item Differentiation and integration of vector-valued functions (Initial value problems)
\item Tangential acceleration and centripetal acceleration
\end{itemize}
\section*{Differential Equations}
\begin{itemize}
\item Logistic differential equations and population growth
\item Standard form of first order linear differential equations
\item Solve by integrating factor
\end{itemize}
\end{multicols}
\end{document}
