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% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
\usepackage{amsmath}            % Symbols
\usepackage{MnSymbol}           % Symbols
\usepackage{wasysym}            % Symbols
%\usepackage[english,german,french,spanish,italian]{babel}              % Languages

% Document Info
\author{ejj1999}
\pdfinfo{
  /Title (calculus-2.pdf)
  /Creator (Cheatography)
  /Author (ejj1999)
  /Subject (Calculus 2 Cheat Sheet)
}

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% Commands
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% Header and Footer
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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Calculus 2 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{ejj1999} via \textcolor{DarkBackground}{\uline{cheatography.com/66363/cs/16562/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}ejj1999 \\
  \uline{cheatography.com/ejj1999} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 14th May, 2019.\\
   Updated 5th August, 2018.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
  \SetRowColor{white}
  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{p{0.50347 cm} x{2.01388 cm} x{2.05965 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Taylor Series}}  \tn
% Row 0
\SetRowColor{LightBackground}
1/1-x & 1+x+x\textasciicircum{}2\textasciicircum{}+x\textasciicircum{}3\textasciicircum{}+... & ∑ x\textasciicircum{}n\textasciicircum{} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\seqsplit{sin(x)} & x\textasciicircum{}1\textasciicircum{}-x\textasciicircum{}3\textasciicircum{}/3!+x\textasciicircum{}5\textasciicircum{}/5!-+... & ∑ (-1)\textasciicircum{}n\textasciicircum{}x\textasciicircum{}2n+1\textasciicircum{}/(2n+1)! \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
e\textasciicircum{}x\textasciicircum{} & 1+x+x\textasciicircum{}2\textasciicircum{}/2!+x\textasciicircum{}3\textasciicircum{}/3!+... & ∑ x\textasciicircum{}n\textasciicircum{}/n! \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
\seqsplit{cos(x)} & 1-x\textasciicircum{}2\textasciicircum{}/2!+x\textasciicircum{}4\textasciicircum{}/4!-+... & ∑ (-1)\textasciicircum{}n\textasciicircum{}x\textasciicircum{}2n\textasciicircum{}/(2n)! \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{5.377cm}}{centered around 0 \newline (1/1-x only valid for -1\textless{}x\textless{}1.)}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.89126 cm} x{3.08574 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Trig Sub's}}  \tn
% Row 0
\SetRowColor{LightBackground}
√(x\textasciicircum{}2\textasciicircum{}+a\textasciicircum{}2\textasciicircum{}) & x=atan(θ) \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
√(a\textasciicircum{}2\textasciicircum{}-x\textasciicircum{}2\textasciicircum{}) & x-asin(θ) \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
√(x\textasciicircum{}2\textasciicircum{}-a\textasciicircum{}2\textasciicircum{}) & x=asec(θ) \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
b-ax\textasciicircum{}2\textasciicircum{} & x= √b / √a  sin(θ) \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
ax\textasciicircum{}2\textasciicircum{}+b & x= √b / √a tan(θ) \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
ax\textasciicircum{}2\textasciicircum{}-b & x= √b / √a sec(θ) \tn 
% Row Count 6 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.55618 cm} x{1.19002 cm} x{1.8308 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Convergence|Divergence test}}  \tn
% Row 0
\SetRowColor{LightBackground}
N\textasciicircum{}th\textasciicircum{} term test for divergence & lim(n\textgreater{}∞) an & ≠0 ∑an diverges \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
P-Test & converge p\textgreater{}1 & diverge p≤1 \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Limit Comparison & L= lim(n\textgreater{}∞) (an/bn) & L≠0 series both \seqsplit{diverge|converge} \tn 
% Row Count 8 (+ 3)
% Row 3
\SetRowColor{white}
Ratio test & r= lim(n\textgreater{}∞) |an+1/an| & r\textless{}1 converge r\textgreater{}1 diverge \tn 
% Row Count 11 (+ 3)
% Row 4
\SetRowColor{LightBackground}
Alternating series test & lim(n\textgreater{}∞) an & =0 ∑ (-1)\textasciicircum{}n\textasciicircum{}an converges \tn 
% Row Count 13 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.04057 cm} x{2.93643 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Common Integrals}}  \tn
% Row 0
\SetRowColor{LightBackground}
∫sin(x)dx & -cos(x)+C \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
∫cos(x)dx & sin(x)+C \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
∫tan(x)dx & -ln(cos(x))+C \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
∫sec(x)dx & ln(sec(x)+tan(x))+C \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
∫csc(x)dx & -ln(csc(x)+cot(x))+C \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
∫cot(x)dx & ln(sin(x))+C \tn 
% Row Count 6 (+ 1)
% Row 6
\SetRowColor{LightBackground}
∫sec\textasciicircum{}2\textasciicircum{}(x)dx & tan(x)+C \tn 
% Row Count 7 (+ 1)
% Row 7
\SetRowColor{white}
∫e\textasciicircum{}f(x)\textasciicircum{}dx & e\textasciicircum{}f(x)\textasciicircum{}/f\textasciicircum{}'\textasciicircum{}(x)+C \tn 
% Row Count 8 (+ 1)
% Row 8
\SetRowColor{LightBackground}
∫(1/x)dx & ln(x)+C \tn 
% Row Count 9 (+ 1)
% Row 9
\SetRowColor{white}
∫(1/x\textasciicircum{}n\textasciicircum{})dx & (x\textasciicircum{}n+1\textasciicircum{}/n+1)+C \tn 
% Row Count 10 (+ 1)
% Row 10
\SetRowColor{LightBackground}
∫dx/√(a-x\textasciicircum{}2\textasciicircum{}) & arcsin(x/√(a))+C \tn 
% Row Count 12 (+ 2)
% Row 11
\SetRowColor{white}
∫dx/x\textasciicircum{}2\textasciicircum{}+a & \seqsplit{(1/√a)arctan(x/√a)+C} \tn 
% Row Count 14 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.63781 cm} x{2.33919 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Important Derivatives}}  \tn
% Row 0
\SetRowColor{LightBackground}
d/dx arctan f(x) & f\textasciicircum{}'\textasciicircum{}(x)/x\textasciicircum{}2\textasciicircum{}+1 \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
d/dx sec(θ) & sec(θ)tan(θ) \tn 
% Row Count 2 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.28942 cm} x{2.68758 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Power Series}}  \tn
% Row 0
\SetRowColor{LightBackground}
general form & ∑ an(x-a)\textasciicircum{}n\textasciicircum{} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{an = sequence of coeff.} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
center & x=a \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
radius of convergence & R=lim(n\textgreater{}∞) |an/an+1| \tn 
% Row Count 5 (+ 2)
% Row 4
\SetRowColor{LightBackground}
endpoints & x=a+R and x=a-R in series \tn 
% Row Count 7 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.93643 cm} x{2.04057 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Parametric Curves}}  \tn
% Row 0
\SetRowColor{LightBackground}
Horizontal Tangents (x) & when dy/dx=0 t=? \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.33919 cm} x{2.63781 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Equations for Parabola}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{y=a(x-h)\textasciicircum{}2\textasciicircum{}+k} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Directrix & y=k-(1/4a) \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
Focus & (h,k+1/4a) \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{x=a(y-k)\textasciicircum{}2\textasciicircum{}+h} \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
Directrix & x=h-(1/4a) \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
Focus & (h+1/4a,k) \tn 
% Row Count 6 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.63781 cm} x{2.33919 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Equations for Ellipses}}  \tn
% Row 0
\SetRowColor{LightBackground}
(x-h)\textasciicircum{}2\textasciicircum{}/a\textasciicircum{}2\textasciicircum{} + (y-k)\textasciicircum{}2\textasciicircum{}/b\textasciicircum{}2\textasciicircum{} =1 & c=√(|a\textasciicircum{}2\textasciicircum{}-b\textasciicircum{}2\textasciicircum{}|) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
eccentricity & c/(max a|b) \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
foci (on major axis) & when x= center and y= center \tn 
% Row Count 5 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{y= horizontal axis \newline x= vertical axis}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{3.08574 cm} x{1.89126 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Trig Identities}}  \tn
% Row 0
\SetRowColor{LightBackground}
sec\textasciicircum{}2\textasciicircum{}(θ) & tan\textasciicircum{}2\textasciicircum{}(θ)+1 \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
sin\textasciicircum{}2\textasciicircum{}(θ) & 1-cos\textasciicircum{}2\textasciicircum{}(θ) \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
tan\textasciicircum{}2\textasciicircum{}(θ) & sec\textasciicircum{}2\textasciicircum{}(θ)-1 \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
cos\textasciicircum{}2\textasciicircum{}(θ) & {[}1+cos(2θ){]}/2 \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
sin\textasciicircum{}2\textasciicircum{}(θ) & {[}1-cos(2θ){]}/2 \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
double angle cos\textasciicircum{}2\textasciicircum{}(θ) & (1+cos(2θ)/2 \tn 
% Row Count 6 (+ 1)
% Row 6
\SetRowColor{LightBackground}
double angle sin\textasciicircum{}2\textasciicircum{}(θ) & (1-cos(2θ)/2 \tn 
% Row Count 7 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.33919 cm} x{2.63781 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Polar Coordinates \& Area}}  \tn
% Row 0
\SetRowColor{LightBackground}
Area & ∫1/2 (f(x))\textasciicircum{}2\textasciicircum{} dx \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
One petal of r=sin(nθ) & interval {[}0,π/n{]} \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
One petal of r=cos(nθ) & {[}-π/2n,π/2n{]} \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
Polar \textgreater{} Cartesian & x=rcos(θ) y=rsin(θ) \tn 
% Row Count 6 (+ 1)
% Row 4
\SetRowColor{LightBackground}
Cartesian \textgreater{} Polar & tan(θ)=y/x x\textasciicircum{}2\textasciicircum{}+y\textasciicircum{}2\textasciicircum{}=r\textasciicircum{}2\textasciicircum{} \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}