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\begin{document}
\raggedright
\footnotesize
\begin{multicols*}{4}
  \setlength{\columnseprule}{0.25pt}
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  \section{Function and Limits}
  \begin{itemize}
    \item $\lim\limits_{x\to \pm \infty}\frac{Ax^\alpha}{Bx^\beta} 
          \begin{cases}
            0 & \text{if} \alpha < \beta\\
            \frac{A}{B} & \text{if} \alpha = \beta\\
            \pm \infty & \text{if} \alpha > \beta\\
          \end{cases}$
    \item $\lim\limits_{x\to c}\frac{sin(g(x))}{g(x)} = 1(\lim\limits_{x \to c}g(x) = 0)$
    \item $\lim\limits_{x\to c}\frac{tan(g(x))}{g(x)} = 1$
    \item $\lim\limits_{x\to 0}\frac{sin(x)}{x} = 1$
    \item $\lim\limits_{x\to 0}\frac{tan(x)}{x} = 1$
    \item $\lim\limits_{n \to \infty} \frac{n!}{n^{n}} = 0$
  \end{itemize}


  \section{Differentiation}
  parametric differentiaton: $\frac{d^2y}{dx^2} = \frac{d}{dx}(\frac{dy}{dx}) = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$

  \begin{tabular}{|>{\color{black}}c | >{\color{black}}c|}
    \hline
    $f(x)$ & $f'(x)$
    \\ \hline
    \rule{0pt}{2.3ex}  % top spacing
    $\tan x$ & $\sec ^2 x$ \\
    $\csc x$ & $-\csc x \cot x$ \\
    $\sec x$ & $\sec x \tan x$ \\
    $\cot x$ & $- \csc ^2 x$
    \\ \hline
    \rule{0pt}{2.3ex}  % top spacing
    $a^{f(x)}$ & $\ln a \cdot f'(x)a^{f(x)}$ \\
    $\log_af(x)$ & $\log_a e \cdot \frac{f'(x)}{f(x)}$
    \\[1ex] \hline
    \rule{0pt}{3ex}  % top spacing
    $\sin^{-1} f(x)$ & $\frac{f'(x)}{\sqrt{1-[f(x)]^2}}, \ \ _{\vert f(x) \vert < 1}$ \\[1.5ex]
    $\cos^{-1} f(x)$ & $-\frac{f'(x)}{\sqrt{1-[f(x)]^2}}, \ \ _{\vert f(x) \vert < 1}$ \\[1.5ex]
    $\tan^{-1} f(x)$ & $\frac{f'(x)}{1 + [f(x)]^2}$ \\[1.5ex]
    $\cot^{-1} f(x)$ & $-\frac{f'(x)}{1 + [f(x)]^2}$ \\[1.5ex]
    $\sec^{-1} f(x)$ & $\frac{f'(x)}{\vert f(x) \vert \sqrt{[f(x)]^2-1}}$ \\[1.5ex]
    $\csc^{-1} f(x)$ & $-\frac{f'(x)}{\vert f(x) \vert \sqrt{[f(x)]^2-1}}$ \\[2ex]
    \hline
  \end{tabular}

  \textbf{Second Derivative} Test: $f'(c) = 0, f''(c) < 0$ then local max, $f''(c) > 0$ local min.

  \textbf{L'Hopital's Rule}: Given $\lim\limits_{x\to c}f(x) $ and $ g(x) = 0 $ or $ \pm \infty$ $ \lim\limits_{x \to c}\frac{f(x)}{g(x)} = \lim\limits_{x \to c}\frac{f'(x)}{g'(x)}$
  \begin{itemize}
    \item Use for $\frac{0}{0}$ or $\frac{\infty}{\infty}$
  \end{itemize}

  \subsection*{Trigo Identities}
  \begin{enumerate}
    \item $\sec^2x - 1 = \tan^2x$
    \item $\csc^2x - 1 = \cot^2x$
    \item $\sin A\cos A = \frac{1}{2}\sin2A$
    \item $\cos^2A = \frac{1}{2}(1+\cos2A)$
    \item $\sin^2A = \frac{1}{2}(1-\cos2A)$
    \item $\sin A\cos B = \frac{1}{2}(\sin(A + B) + \sin(A - B)$
    \item $\cos A\sin B = \frac{1}{2}(\sin(A + B) - \sin(A - B)$
    \item $\cos A\cos B = \frac{1}{2}(\cos(A + B) + \cos(A - B)$
    \item $\sin A\sin B = \frac{1}{2}(\cos(A + B) - \cos(A - B)$
  \end{enumerate}

  \section{Integration}

  \begin{tabular}{|>{\color{black}}c | >{\color{black}}c|}
    \hline
    $f(x)$ & $\int f(x)$\\
    $\tan ax$ & $\frac{1}{a}\ln|\sec(ax)|$\\
    $\cot ax$ & $\frac{1}{a}\ln|\cot(ax)|$\\
    $\sec ax$ & $\frac{1}{a}\ln|\sec(ax) + tan(ax)|$\\
    $\csc ax$ & $\frac{1}{a}\ln|\csc(ax) + cot(ax)|$\\
    \hline
    $\frac{1}{a^2+(x+b)^2}$ & $\frac{1}{a}\tan^{-1}(\frac{x+b}{a})$\\
    $\frac{1}{\sqrt{a^2-(x+b)^2}}$ & $\sin^{-1}(\frac{x+b}{a})$\\
    $\frac{1}{a^2-(x+b)^2}$ & $\frac{1}{2a}\ln|\frac{x+b+a}{x+b-a}|$\\
    $\frac{1}{(x+b)^2-a^2}$ & $\frac{1}{2a}\ln|\frac{x+b-a}{x+b+a}|$\\
    \hline
  \end{tabular}

  \textbf{Substitution} $\int f(g(x)) \cdot g'(x) dx = \int f(u) du, u = g(x)$

  \textbf{By Parts} $\int u v' dx = uv - \int u'v dx$, order: LIATE: Differentiate to integrate

  \subsection{Application of Integration}
  about x axis
  \begin{itemize}
    \item Vol Disk:  $V = \pi \int^b_a f(x)^2 - g(x)^2 dx$
    \item Vol Shell: $V = 2\pi\int^b_a x|f(x)-g(x)|dx$ (absolute!!)
    \item Length of curve: $\int^b_a \sqrt{1+f'(x)^2}dx$
  \end{itemize}

  \section{Vectors}
  unit vector: $\hat{p} = \frac{p}{|p|}$, $\VV{AB} = \VV{OB} - \VV{OA}$
  \begin{center}
    \begin{multicols}{2}
      \begin{tikzpicture}[scale=0.8, every node/.style={transform shape}]
        \coordinate[label=below left:O] (O) at (0,0);
        \coordinate[label=A] (A) at (0.3,1.6);
        \coordinate[label=B] (B) at (1.5, 1.4);
        \coordinate[label=P] (P) at (1, 1.5);
        \draw (O)
        -- node[left] {$\vv{a}$} (A)
        -- node[above] {$\lambda$} (P)
        -- node[above] {$\mu$} (B)
        -- node[right] {$\vv{b}$} (O)
        -- node[left] {$\vv{p}$} (P);
      \end{tikzpicture}
      \\ \textbf{ratio theorem}
      \\* $\vv{p} = \frac{\mu\vv{a} + \lambda\vv{b}}{\lambda + \mu}$
      \newline
      \\ \textbf{midpoint theorem}
      \\* $\vv{p} = \frac{\vv{a} + \vv{b}}{2}$
    \end{multicols}
  \end{center}
  \subsection{Dot Product}
  \begin{itemize}
    \item $\VV{a} \cdot \VV{b} = a_1b_1 + a_2b_2 + a_3b_3 = |a||b|\cos\theta$
    \item $a \perp b \Then a \cdot b = 0$
    \item $a \parallel b \Then a \cdot b = |a||b|$
  \end{itemize}

  \subsection{Cross Product}
  $
  \vv{a} \times \vv{b} =
  \begin{vmatrix} \vv{i} & \vv{j} & \vv{k} \\ a_i & a_2 & a_3 \\ b_i & b_2 & b_3 \end{vmatrix} =
  \begin{pmatrix} (a_2b_3 - a_3b_2) \\ -(a_1b_3 - a_3b_1) \\ (a_1b_2 - a_2b_1) \end{pmatrix}
  $
  \begin{center}
    \begin{multicols}{2}
      $|\vv{a} \times \vv{b}| = |\vv{a}||\vv{b}|\sin\theta$

      $a \perp b \Then a \times b = |a||b|$

      $a \parallel b \Then a \times b = 0$

      Area Parallelogram = $|\vv{a} \times \vv{b}|$
    \end{multicols}
  \end{center}

  \subsection{Projection}
  \begin{multicols}{2}
    \begin{tikzpicture}[scale=0.7, every node/.style={transform shape}]
      \coordinate[label=below left:O] (O) at (0,0);
      \coordinate[label=right:A] (A) at (2, 1);
      \coordinate[label=below:B] (B) at (3, 0);
      \coordinate[label=below:N] (N) at (2, 0);
      \draw (A)
      -- node[above] {$\vv{a}$} (O)
      -- node[below] {$\vv{b}$} (B);
      \draw[shorten >=0pt, dashed] (A) -- (N);
    \end{tikzpicture}
    $\triangle ANO = \frac{1}{2} \abs{\VV{OA} \times \VV{ON}}$
    \columnbreak

    $\text{comp}_{\vv{b}}\vv{a} = |\vv{b}|\cos\theta = \frac{\vv{a}\cdot \vv{b}}{|\vv{a}|}$
    $\text{proj}_{\vv{b}}\vv{a} = \text{comp}_{\vv{b}}\vv{a} \cdot \frac{a}{|a|} = \VV{ON} = \frac{\vv{a}\cdot \vv{b}}{\vv{a}\cdot \vv{a}}\vv{a} = \frac{\vv{a} \cdot \vv{b}}{|\vv{a}|^2}\vv{b}$
  \end{multicols}

  \subsection{Lines}
  \begin{multicols}{2}
    $\vv{r} = \vv{r}_0 + t\vv{v} = \langle x,y,z\rangle$
    $\langle x_0,y_0,z_0\rangle + t\langle a,b,c\rangle$
    $\begin{pmatrix}
       x_0 + at \\
       y_0 + bt \\
       z_0 + ct \\
     \end{pmatrix}$
   \end{multicols}
   \subsection{Planes}
   $\vv{n} = \langle a, b, c \rangle, \vv{r} = \langle x, y, z \rangle,\vv{r}_0\langle x_0, y_0, c_0 \rangle$\\
   Scalar: $\vv{n} \cdot \vv{r} = \vv{n} \cdot \vv{r}_0$\\
   Cartesian: $ax + by + cz = d$
   \subsection{Distance from Point to Plane}
   $\frac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}}$

   \section{Partial Derivatives}
   \subsection{Chain Rule}
   For $z(t) = f(x(t), y(t))$,
   \\* $\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}$

   For $z(s, t) = f(x(s,t), y(s,t))$,
   \\* $\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$
   \\* $\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial s}$

   Arc Length of $r(t)$: $\int^b_a |\vv{r}'(t)|dt$

   \subsection{Implicit Differentiation}
   $\frac{\partial z}{\partial x} =- \frac{F_x}{F_z}$
   $\frac{\partial z}{\partial y} =- \frac{F_y}{F_z}$

   \subsection{Directional Derivative}
   Gradient vector at $f(x,y): \triangledown f = f_x\vv{i} + f_y\vv{j}$

   $D_uf(x, y) = \langle f_x, f_y \rangle \cdot \langle a, b \rangle = \langle f_x, f_y\rangle \cdot \hat{\vv{u}} = \	\triangledown f \cdot \hat{\vv{u}}$ (Unit Vector)

   Tangent Plane: $\langle f_{x}, f_{y} -1 \rangle \cdot \langle x-x_0, y-y_0,z-z_0\rangle = 0$

   \subsection{Critical Points}

   $f_x = 0$ and $f_y = 0$, OR ($f_x$ or $f_y$ does not exist)

   $D = f_{xx}(a,b)f_{yy}(a,b) - (f_{xy}(a,b))^2$
   \def\arraystretch{1.2}
   \begin{tabular}{| c | c | c |}
     \hline $D$ & $f_{xx}(a,b)$ & \textbf{local}
     \\\hline + & + & \text{min}
     \\\hline + & - & \text{max}
     \\\hline - & \text{any} & \text{saddle point}
     \\\hline 0 & \text{any} & \text{no conclusion}
     \\\hline
   \end{tabular}
   \section{Double Integrals}
   \subsection{Type I}
   \begin{multicols}{2}
     \includegraphics[width=\linewidth]{Type I}
     \columnbreak

     $\int^b_a\int^{g_2(x)}_{g_1(x)}f(x,y)dydx$
     \newline
     \newline
     $D = \{(x,y): a \leq x \leq b,$ $g_1(x) \leq y \leq g_2(x)\}$
   \end{multicols}
   \subsection{Type II}
   \begin{multicols}{2}
     \includegraphics[width=\linewidth]{Type II}
     \columnbreak

     $\int^d_c\int^{h_2(y)}_{h_1(y)}f(x,y)dxdy$
     \newline
     \newline
     $D = \{(x,y): c \leq y \leq d,$ $ h_1(y) \leq x \leq h_2(y)\}$
   \end{multicols}

   \subsection{Polar Coordinates}
   \begin{multicols}{2}
     \includegraphics[width=\linewidth]{polar}
     \columnbreak

     $x = r\cos\theta$\\
     $y = r\sin\theta$\\
     $R = \{(r, \theta): 0 \leq a \leq r \leq b,$ $\alpha \leq \theta \leq \beta\}$
     \newline
     \newline
     $\int^\beta_\alpha\int^b_af(r\cos\theta, r\sin\theta)rdrd\theta$
   \end{multicols}

   \subsection{Surface Area}
   $S = \iint_R\sqrt{f_x^2 + f_y^2 + 1} dA$, get in the form of $z = f(x,y)$ first

   \section{ODE}
   \begin{tabular}{|>{\color{black}}c | >{\color{black}}c|}
     \hline
     form & change of variable \\
     \hline
     $\frac{dy}{dx} = f(x)g(y)$ & $\int \frac{1}{g(y)}dy = \int f(x)dx + C$\\
     \hline
     $y'=g(\frac{y}{x})$ & \makecell{Set $v = \frac{y}{x}$ \\ $\Then y' = v + xv' $}\\
     \hline
     \makecell{ $y'=f(ax + by + c)$\\ $\Then y' = \frac{ax+by+c}{\alpha x + \beta y + \gamma}$} & Set $v = ax+by$ \\
     \hline
     $y' + P(x)y = Q(x)$ & \makecell{$R = e^{\int P(x)dx}$ \\ $\Then y \cdot R = \int Q \cdot R dx $}\\
     $y' + P(x)y = Q(x)y^n$ & \makecell{$z = y^{1-n}$ \\ $\Then$ sub in Z \\ solve linear}\\
   \end{tabular}

   \section{Population Models}
   \begin{center}
     $N_{\infty} = \frac{B}{s}$, $\hat{N} = $ Population Now
     \begin{multicols}{2}
       \textbf{Malthus}\\
       $N(t) = \hat{N}e^{kt}$\\
       $k = B - D$

       \columnbreak
       \textbf{Logistic}\\
       $\frac{1}{N} = \frac{1}{N_{\infty}} + (\frac{1}{\hat{N}} - \frac{1}{N_{\infty}})e^{-Bt}$\\
       $N = \frac{N_{\infty}}{1+(\frac{N_{\infty}}{N} - 1)e^{-Bt}}$
     \end{multicols}
   \end{center}

   \subsection{Uranium Decay into Thorium}
   $U(t) = U_{0}e^{-k_ut}$, $k = \frac{\ln2}{\text{halflife}}, \frac{dU}{dt} = -k_{u}U$\\
   Thorium: $T(t) = \frac{K_{u}U_{0}}{K_{t}-K_{u}}(e^{-k_{u}t} - e^{-k_{t}t}), \frac{dT}{dt} = k_{u}U - k_{T}T$
 \end{multicols*}
 \newpage

 \begin{multicols*}{4}
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   \section{Series}
   \subsection{Geometric Series}
   $\sum_{n=1}^{\infty}ar^{n-1}, a \ne 0$ converges to $\frac{a}{1-r}$ when $|r| < 1$, diverges otherwise

   If series $\sum_{n=1}^{\infty} a_{n}$ is convergent, then $\lim\limits_{n \to \infty} a_{n} = 0$

   \subsection{Tests}
   Decreasing function -> differentiate and see the range where $x < 0$
   \begin{center}

     \begin{tabular}{|>{\color{black}}p{0.2\linewidth} | >{\color{black}}p{0.7\linewidth}|}
       \hline
       Test & Method \\\hline
       $n^{th}$ term & $\lim\limits_{n \to \infty} a_{n} \ne 0$ or does not exist, then divergent \\\hline
       Integral & $f(n)=a_{n}$ is continuous, positive, decreasing function $\forall x\geq 1$ and $\int_{1}^{\infty}f(x)dx$ converges else divergent \\\hline
       p-series & $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$convergent $\leftrightarrow p > 1$ \\\hline
       Harmonic Series & $\sum_{n=1}^{\infty} \frac{1}{n}$ divergent \\\hline
       Ratio \tiny{If Factorial} & $0 \geq \lim\limits_{n \to \infty} |\frac{a_{n+1}}{a_{n}}|=L < 1$ abs. convergent, $> 1$ divergent, $= 1$ inconclusive \\\hline
       Root \tiny{If nth power} & $0 \geq \lim\limits_{n \to \infty} \sqrt[n]{a_{n}}=L < 1$ abs. convergent, $> 1$ divergent, $= 1$ inconclusive \\\hline
       Alternating series & $b_{n}$ decreasing, $\lim\limits_{n \to \infty}b_{n} = 0$, then $\sum_{n=1}^{\infty}(-1)^{n-1}b_{n} = b_{1}-b_{2}+b_{3}... $ is convergent \\\hline
       Power Series & $b_{n}$ decreasing, $\lim\limits_{n \to \infty}b_{n} = 0$, then $\sum_{n=1}^{\infty}(-1)^{n-1}b_{n} = b_{1}-b_{2}+b_{3}... $ is convergent \\\hline
       Comparison Test & $\sum a_{n} $ and $ \sum b_{n} $ s.t. $a_{n} \leq b_{n}$ Then if $\sum b_{n}$ convergent, $\sum a_{n}$ convergent. If $\sum a_{n}$ divergent, $\sum b_{n}$ divergent\\\hline
     \end{tabular}
 \end{center}

 \subsection{Power Series}
 $\sum_{n=0}^{\infty} c_{n}(x-a)^{n}$ converges at \textbf{ONE OF}
 \begin{itemize}
   \item $x=a$
   \item For all $x$
   \item converges if $|x-a| < R$ and diverges if $|x-a| > R$ (R is radius of convergence)
 \end{itemize}

 If $\lim_{n \to \infty} \left| \frac{c_{n+1}}{c_{n}} = L \right|$ or $\lim_{n \to \infty} \sqrt[n]{|c_{n}|}=L$, $L \in \mathbb{R}$ or $\infty$, then $R = \frac{1}{L}$

 If power series $\sum_{n=0}^{\infty} c_{n}(x-a)^{n}$ has radius of convergence $R>0$, then function $f$ is differentiable on interval $|x-a| < R$ and

 \begin{itemize}
   \item $f'(x) = \sum_{n=1}^{\infty} nc_{n}(x-a)^{n-1}$, for $|x-a| < R$
   \item $\int f(x) = \sum_{n=0}^{\infty} c_{n}\frac{(x-a)^{n+1}}{n+1}+C$ for $|x-a| < R$
 \end{itemize}


 \subsection{Taylor and Maclaurin Series}
 If f has power series repr @ $f = a$, $f(x) = \sum_{n=0}^{\infty} c_{n}(x-a)^{n}, |x-a| < R, R > 0$, then $c_{n} = \frac{f^{(n)}(a)}{n!}$. \\
 Maclaurin Series: $f(x) = \sum_{n=0}^{\infty} \frac{f^{n}(0)}{n!}x^{n}$
For $-\infty < x < \infty$
\\ \;  % spacing
\setlength\tabcolsep{1.5pt} % default value: 6pt
\begin{tabular}{rl}
           $\sin x$ & $= \sum\limits^\infty_{n = 0} \frac{(-1)^nx^{2n + 1}}{(2n+1)!} $
        \\ $\cos x$ & $= \sum\limits^\infty_{n = 0} \frac{(-1)^nx^{2n}}{(2n)!}$
        \\ $e^x$ & $= \sum\limits^\infty_{n = 0} \frac{x^n}{n!}$
\end{tabular}

For $-1 < x < 1$
\\ \;  % spacing
\setlength\tabcolsep{1.5pt} % default value: 6pt
\begin{tabular}{rl}
           $\frac{1}{1 - x}$ & $= \sum\limits^\infty_{n = 0} x^n $
        \\ $\frac{1}{1 + x}$ & $= \sum\limits^\infty_{n = 0} (-1)^nx^n $
        \\ $\frac{1}{1 + x^2}$ & $= \sum\limits^\infty_{n = 0} (-1)^nx^{2n} $
        \\ $\ln(1 + x)$ & $= \sum\limits^\infty_{n = 1} \frac{(-1)^{n - 1}x^n}{n} $
        \\ $\tan^{-1}x$ & $= \sum\limits^\infty_{n = 0} \frac{(-1)^n}{2n + 1} x^{2n+1}$
        \\ $\frac{1}{(1+x)^2}$ & $= \sum\limits^\infty_{n = 1} (-1)^{n-1}nx^{n-1}$
        \\ $\frac{1}{(1-x)^2}$ & $= \sum\limits^\infty_{n = 1} nx^{n-1}$
        \\ $\frac{1}{(1-x)^3}$ & $= \frac{1}{2} \sum\limits^\infty_{n = 2} n(n - 1)x^{n-2}$
        \\ $(1 + x)^k$ & $= \sum\limits^\infty_{n = 0} \binom{k}{n}x^n$
        \\   & $= 1 + kx + \frac{k(k-1)}{2!}x^2 + \dots$
\end{tabular}

\subsection{Useful Math}

\begin{itemize}
  \item Line: $y-y_{1} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})$
  \item $\int \sqrt{a^{2}-x^{2}}dx = \frac{a^{2}}{2}\sin^{-1}(\frac{x}{a}) + \frac{x}{2}\sqrt{a^{2}-x^{2}}, x = a\sin\theta, dx = a\cos\theta d\theta, $A
  \item $\sqrt{a^{2}+x^{2}}dx = \frac{1}{2}\left(x\sqrt{a^{2}+x^{2}} + a^{2}\ln\left|\frac{x+\sqrt{a^{2}+x^{2}}}{a}\right|\right), x = a\tan\theta, \frac{-\pi}{2} < \theta < \frac{\pi}{2}$
  \item $\int\cos^{2}x = \frac{1}{4} \sin2x + \frac{x}{2} = \frac{1}{2}\cos x \sin x + \frac{1}{2}x$
  \item $\int\sin^{2}x = -\frac{1}{4} \sin2x + \frac{x}{2}$
  \item $(x-y)^{3} = x^{3} - 3x^{2}y + 3xy^{2}-y^{3}$
  \item $(x+y)^{3} = x^{3} + 3x^{2}y + 3xy^{2}+y^{3}$

\end{itemize}

\end{multicols*}
\end{document}
