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% Document Info
\author{rockcollector2}
\pdfinfo{
  /Title (quantitative-methods-final-exam.pdf)
  /Creator (Cheatography)
  /Author (rockcollector2)
  /Subject (Quantitative Methods Final Exam Cheat Sheet)
}

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\noindent
\begin{multicols}{3}
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    {\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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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Quantitative Methods Final Exam Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{rockcollector2} via \textcolor{DarkBackground}{\uline{cheatography.com/22080/cs/4782/}}}
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\noindent
\begin{multicols}{3}
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  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}rockcollector2 \\
  \uline{cheatography.com/rockcollector2} \\
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  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 10th August, 2015.\\
   Updated 10th May, 2016.\\
   Page {\thepage} of \pageref{LastPage}.
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  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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% www.emerson.emory.edu/services/latex/latex_169.html
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\begin{multicols*}{2}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Statistics \& Probability Notes}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Standard Deviation}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439226211_Screen Shot 2015-08-10 at 1.02.50 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Quartiles}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{IQR = Q3 = Q1 \newline % Row Count 1 (+ 1)
Outliers are beyond: Q1 - 1.5 * IQR, Q3 + 1.5 + IQR% Row Count 3 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Calc Entry for Basic Stats}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{1-VAR Stats% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Correlation (R\textasciicircum{}2 near 1 is better fit)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439226815_Screen Shot 2015-08-10 at 1.13.11 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{\textgreater{}COMBINATORIAL PROBABILITY\textless{}}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{4-Digit PIN with repetition = 10\textasciicircum{}4 \newline % Row Count 1 (+ 1)
4-Digit PIN without repetition = 10!/6! = 5040% Row Count 2 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Permutations}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{How many ways can 6 people be ranked? Ranking n objects leads to n! \newline % Row Count 2 (+ 2)
How many ways can 6 people be ranked into 3 places: \newline % Row Count 4 (+ 2)
n{\emph{(n-1)}}...*(n-r+1) = n!/(n-r)! or 6 nPr 3% Row Count 5 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Combinations}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{Combinations = choosing a certain number of objects from a given set (no order). \newline % Row Count 2 (+ 2)
N choose R or N!/(N-R)!R! or nCr% Row Count 3 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Probability}}  \tn
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\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Example}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{3 cities from 15 are chosen randomly for a visit. A: 4 cost \$800, B: 5 cost \$300, C: 6 cost \$100. What is the probability that the tour will cost \$1000 or less? \newline % Row Count 4 (+ 4)
- All from C 6 nCr 3 \newline % Row Count 5 (+ 1)
- Two from C, one from B or A (6nCr2)(5nCr1) + (6nCr2)(4nCr1) \newline % Row Count 7 (+ 2)
- One from C, two from B (6nCr1)*((5nCr2) \newline % Row Count 8 (+ 1)
- None from C, three from B (5nCr3) \newline % Row Count 9 (+ 1)
Compute and sum: 20+75+60+60+30=245; Divide by the total 15nCr3 = 455 \newline % Row Count 11 (+ 2)
.538 or 53.8\%% Row Count 12 (+ 1)
} \tn 
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\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{\textgreater{}RANDOM VARIABLES\textless{}}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{Example of Random Variable \newline % Row Count 1 (+ 1)
X is the number of heads in 10 flips of a coin. P(X=4)? \newline % Row Count 3 (+ 2)
(10nCr4)/2\textasciicircum{}10 = .205 \newline % Row Count 4 (+ 1)
X is sum of two dice. \newline % Row Count 5 (+ 1)
P(X=5) = 4/36 \newline % Row Count 6 (+ 1)
P(9\textless{}=X\textless{}=11) = P(X=9)+P(X=10)+P(X=11)= 4/36+3/36+2/36% Row Count 8 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Binomial Random Variables}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{Binomial Random Variable represents the number of successes in n trials with probability p of success. The probability of 4 successes in 10 trials (p=0.5) is (10 nCr4\_/2\textasciicircum{}10 = 0.205 \newline % Row Count 4 (+ 4)
Calc: binompdf(10,.5,8) = (n,p,r) r is number of successes \newline % Row Count 6 (+ 2)
Math: (n r) p\textasciicircum{}r(1-p)\textasciicircum{}n-r% Row Count 7 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Binomial Random Variables}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{Binomial with n=15 and p = .4 \newline % Row Count 1 (+ 1)
Compute:  \newline % Row Count 2 (+ 1)
P(X=3)=binompdf(15,.4,3) \newline % Row Count 3 (+ 1)
P(X\textless{}=3)=binomcdf(15,.4,3) \newline % Row Count 4 (+ 1)
P(X\textless{}3)=binomcdf(15,.4,2) \newline % Row Count 5 (+ 1)
P(X\textgreater{}3)=1-binomcdf(15,.4,3) \newline % Row Count 6 (+ 1)
P(X\textgreater{}=3)=1-binomcdf(15,.4,2) \newline % Row Count 7 (+ 1)
P(4\textless{}x\textless{}=8)=binomcdf(15,.4,8)-binomcdf(15,.4,4) \newline % Row Count 8 (+ 1)
P(1.3\textless{}X\textless{}1.7)=0% Row Count 9 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Continuous \& Normal Random Variable}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{For C RV, P(X=3) or any other value is zero. \newline % Row Count 1 (+ 1)
Probability is area under curve. \newline % Row Count 2 (+ 1)
Normal RV is bell curve. \newline % Row Count 3 (+ 1)
Total area under bell is 1. \newline % Row Count 4 (+ 1)
P(X\textless{}a) is the are under the curve up to x=a. \newline % Row Count 5 (+ 1)
1 Std. Dev. P(-1\textless{}Z\textless{}1)= .683 \newline % Row Count 6 (+ 1)
2 Std. Dev. P(-2\textless{}Z\textless{}2)=.954 \newline % Row Count 7 (+ 1)
3 Std. Dev. P(-3\textless{}Z\textless{}3)=.997% Row Count 8 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Pure Numbers}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{X is a random normal variable with mean -3 and standard deviation 0.7 \newline % Row Count 2 (+ 2)
P(-4\textless{}X\textless{}-3) = normalcdf(-4,-3,-3,.7) \newline % Row Count 3 (+ 1)
P(X\textgreater{}-2) = normalcdf(-2,1E99,-3,.7) \newline % Row Count 4 (+ 1)
P(X\textless{}=-3.5)=normalcdf(-1e((,-3.5,-3,.7) \newline % Row Count 5 (+ 1)
P(X=-3) =0 \newline % Row Count 6 (+ 1)
P(|X-(-3)|\textgreater{}.7)=P(X\textless{}-3.7)+P(X\textgreater{}-2.3) = normalcdf(-1E99,-3.7,-3,.7)+normalcdf(-2.3,1E99,-3,.7) \newline % Row Count 8 (+ 2)
A car model gets 24 mpg on the car sticker. The maker knows that this is normally distributed with a std dev of 3 mpg. What is the proportion of cars that get less than 20 mpg? P(X,20)=normalcdf(-1E99,20,24,3)=0.91% Row Count 13 (+ 5)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Conditional Probability}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{P(A|B) is the probability of A given that B happened. \newline % Row Count 2 (+ 2)
P(A|B) = P(AintersectB) /P(B) \newline % Row Count 3 (+ 1)
If P(A|B) = P(A) then independent. \newline % Row Count 4 (+ 1)
P(AintersectB) = P(A)P(B) \newline % Row Count 5 (+ 1)
If mutually exclusive P(A|B)=0% Row Count 6 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Bayes's Theorem}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439232234_Screen Shot 2015-08-10 at 2.43.04 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Central Limit Theorem}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{As n becomes large, the sample mean will be distributed according to the normal distribution with parameters u and standard deviation - std dev/sqrt n \newline % Row Count 4 (+ 4)
*As n gets large, the spread in the sample mean distribution narrows. This means that the sample mean is more likely to be near the true mean.% Row Count 7 (+ 3)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{\textgreater{}INFERENTIAL STATISTICS\textless{}}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{Z-test approximated by normal distribution. If sample size is large or variance is known. \newline % Row Count 2 (+ 2)
T-score/test is used when: \newline % Row Count 3 (+ 1)
- sample size is below 30 \newline % Row Count 4 (+ 1)
- population standard deviation is unknown (estimated from your sample data) \newline % Row Count 6 (+ 2)
otherwise use z-score/test. \newline % Row Count 7 (+ 1)
Generally use 95\% confidence level. \newline % Row Count 8 (+ 1)
Z-Stat represents how many std. deviations away from the mean the sample mean is. \newline % Row Count 10 (+ 2)
Std. dev is  std dev/sqrt n \newline % Row Count 11 (+ 1)
Null hypotheses assumes that whatever you are trying to prove did not happen. \newline % Row Count 13 (+ 2)
p-value of 0.03 means there is a 3\% chance of finding a difference as large as or larger than the one in your study given the null hypothesis is true. \newline % Row Count 17 (+ 4)
If 0.05 or less you typically do not accept the null hypothesis. \newline % Row Count 19 (+ 2)
Type 1 error: rejecting the null hypothesis when true \newline % Row Count 21 (+ 2)
Type 2 error: accepting the null hypothesis when false% Row Count 23 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Two Sample T Test}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439234999_Screen Shot 2015-08-10 at 3.29.27 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Taylor Polynomials}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439237338_Screen Shot 2015-08-10 at 4.08.21 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Arc Length}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439237271_Screen Shot 2015-08-10 at 4.06.49 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Integration by Parts}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439237142_Screen Shot 2015-08-10 at 4.03.30 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Integration by Parts}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439237116_Screen Shot 2015-08-10 at 4.03.21 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Integration by Parts}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439237086_Screen Shot 2015-08-10 at 4.02.57 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Integration by Parts}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439237057_Screen Shot 2015-08-10 at 4.03.07 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{\textgreater{}Integrals, Series\textless{}}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439236939_Screen Shot 2015-08-10 at 4.01.54 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Newton's Method}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{1. Pick a, initial guess. \newline % Row Count 1 (+ 1)
2. Compute tangent line approximation: y = f(a)+f'(a)(x-a) \newline % Row Count 3 (+ 2)
3. Solve y=0 and get x = (f'(a)a-f(a))/f'(a) \newline % Row Count 4 (+ 1)
4. Use x for the next guess. Repeat.% Row Count 5 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Max/Min Word Problems}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{10 meters of string. maximum area dimensions? \newline % Row Count 1 (+ 1)
Perimeter: P(l,w) = 2l\_2w P=10 \newline % Row Count 2 (+ 1)
Area: A9l,w)=lw \newline % Row Count 3 (+ 1)
a(l)=l(5-1) \newline % Row Count 4 (+ 1)
max is at l=2.5% Row Count 5 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Critical Points}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{Min: f goes from decreasing to increasing \newline % Row Count 1 (+ 1)
f' goes from negative to positive. \newline % Row Count 2 (+ 1)
Max: f goes from increasing to decreasing. \newline % Row Count 3 (+ 1)
f' goes from positive to negative. \newline % Row Count 4 (+ 1)
Flat: f continues to change in the same way. \newline % Row Count 5 (+ 1)
f' does not change sign. \newline % Row Count 6 (+ 1)
f" gives concavity. \newline % Row Count 7 (+ 1)
Concave up means second derivative is positive which means first derivative is increasing \newline % Row Count 9 (+ 2)
Concave down: f"\textless{}0, f' decreasing.% Row Count 10 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Basic Derivative Rules}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439236213_Screen Shot 2015-08-10 at 3.49.04 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Basic Derivative Rules}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439236187_Screen Shot 2015-08-10 at 3.48.39 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Basic Derivative Rules}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439236144_Screen Shot 2015-08-10 at 3.48.09 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Limits}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{A function f(x) converges to a limit L at x=a if, for any given error tolerance, we can specify a range of x such that for any x in that range, f(x) is near L, near being given the tolerance.% Row Count 4 (+ 4)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Fundamental Theorem of Calculus}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/rockcollector2_1439235708_Screen Shot 2015-08-10 at 3.41.25 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Computing Areas \& Integrals}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{The definite integral of f from a to b is the area underneath the curve from a to b. \newline % Row Count 2 (+ 2)
Where f is negative, the area contributed is a negative area. \newline % Row Count 4 (+ 2)
Use fnInt% Row Count 5 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Derivatives and Tangent Lines}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{The derivative of a function f at x is the slope of the tangent at x. If all of the slopes are assembled you get f'(x) or df/dx. \newline % Row Count 3 (+ 3)
If we know f'(a) and f(a), the the tangent line at x=a is y=f'(a)(x-a)+f(a) \newline % Row Count 5 (+ 2)
Slope is f'(a) and line passes through (a,f(a)) \newline % Row Count 6 (+ 1)
Approximate slope use: nDeriv% Row Count 7 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Calculus Notes}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


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

\end{document}