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% Document Info
\author{promise123}
\pdfinfo{
  /Title (final.pdf)
  /Creator (Cheatography)
  /Author (promise123)
  /Subject (Final Cheat Sheet)
}

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\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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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Final Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{promise123} via \textcolor{DarkBackground}{\uline{cheatography.com/27564/cs/8023/}}}
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\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}promise123 \\
  \uline{cheatography.com/promise123} \\
  \end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 13th May, 2016.\\
   Page {\thepage} of \pageref{LastPage}.
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\end{multicols}}




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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{final exam}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{{\bf{Data types}}: Numeric data,Singles,Doubles,› Integers,Character data,Logical data,Arrays of arrays (cell arrays and structure arrays), symbolic data \newline % Row Count 4 (+ 4)
{\bf{manipulate expression by hand}}: STEM courses evaluate u on how think,Never trust anything  comes out of a computer.{\bf{Familiar with CAS:}}Scientific expressions can become unwieldy to work by hand,Human mistakesCareless,Absent-minded,Easily distracted,Badhandwriting. \newline % Row Count 10 (+ 6)
 {\bf{symbolic engine is}} {\emph{MuPad}} by SciFace Software previously {\emph{MAPLE}}; MuPad was discontinued as a stand-aloneprogram, and now is only available in MATLAB. \newline % Row Count 14 (+ 4)
\{\{fa-flag\}\}: {\bf{declaring MATLAB symbols}}:`y = sym('x') ` give y a value of x ;`syms x` give x a value of x; `p = sym('a{\emph{(1-e\textasciicircum{}2)') Output: p = -a}}(e\textasciicircum{}2 - 1)` {\bf{{\emph{ expression:}}}}can define a symbolcontaining symbols that are otherwiseunavailable in the Workspace; `my\_p = sym('p == a{\emph{(1-e\textasciicircum{}2)')› Output:p == -a}}(e\textasciicircum{}2 - 1)` entire {\bf{{\emph{equation}}}} \newline % Row Count 21 (+ 7)
\{\{fa-square\}\}{\bf{`simplify`}}:simplifyexpressions or equations using MuPad's rules(use `pretty` with `simplify`:easier to read); `{\bf{expand}}:` multiplies out all of the parts of the expression or equation;`{\bf{factor}}`: factors the expression or equation;`{\bf{collect}}`: collects like terms;`{\bf{numden}}`:find the numerator and denominator of an expression NOT equations; `{\bf{solve}}`:(symbolicroot-finding) set the expression equal to zero and solve it, solve systems of linear or non\{\{fa-exclamation-circle\}\} cannot solve higher-order systems with linear algebra, results are assignedin {\emph{alphabetical order}};`{\bf{subs}}`: substituting\# or other ;`{\bf{ symfun}}`: symbolic function; can usethe result to evaluate different inputs\{\{fa-flag\}\}:`syms x y` \newline % Row Count 36 (+ 15)
} \tn 
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\mymulticolumn{1}{x{3.833cm}}{`f = x\textasciicircum{}2 + x` \newline % Row Count 1 (+ 1)
`g = symfun(f,y)`-{}-{}-{}-{}-{}-{}-`g(y) = x\textasciicircum{}2 + x`; `{\bf{ezplot}}`:plotting for symbolic expression, need a defult range\{\{fa-flag\}\}:`f = sym('x\textasciicircum{}2') `  `ezplot(f, {[}-10,10{]})` \newline % Row Count 5 (+ 4)
\{\{fa-square\}\}:{\bf{Derivative}}: instantaneous time rate of change of a slope; an analogous word is differential`{\bf{diff(f)}}`calculates the symbolic first derivative of a symbolic functionwith respect to the default independent variable;`{\bf{diff(f,symvar)}}` calculates the symbolic first derivative of a symbolicfunction with respect to the default independent variable `{\bf{symvar}}`\{\{fa-exclamation-circle\}\} (symvar has tobe in single quotes if the variable does not already exist as a symbolic \newline % Row Count 15 (+ 10)
variable); `{\bf{diff(f,n)}}` calculates the symbolic nth derivative of the symbolic \newline % Row Count 17 (+ 2)
function with respect to the default independent variable;diff(f,symvar,n) or `{\bf{diff(f,n,symvar)}}`calculates thesymbolic nth derivative of the symbolic function f with respect to the symvar-{}-{}-{}-;  {\bf{Integral}}: the integral represents the area under a curve and above`{\bf{int(f)}}`calculates the symbolic single integral of a symbolic function with respect to the default independent variable;`{\bf{int(f,symvar)}}`;`{\bf{int(f,a,b)}}` evaluates theresults of the integral over the symbolic or numeric range;`{\bf{ int(f,symvar,a,b)}}`;{\bf{Differential Equation (DE):}} An equation containing an unknown function and itsderivatives; `{\bf{dsolve}}`:calculate solutions to differential equations; `{\bf{D}}`specify derivitive if you need to specify a nth order derivative, specify n afterthe symbol D ex: `{\bf{  D4y}}`;`{\bf{ dsolve(equation)}}`;`{\bf{dsolve(equation,symvar) }}`;   `{\bf{dsolve(equation,condition1,condition2, …,conditionN, symvar) }}`.  \{\{fa-exclamation-circle\}\}diff can also be used to calculate the differences then finding slope between points. \newline % Row Count 38 (+ 21)
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\mymulticolumn{1}{x{3.833cm}}{{\bf{Converting Symbolic Expressions to Anonymous Functions}} \newline % Row Count 2 (+ 2)
a. Only available starting in versions of MATLAB starting with version 2007B (this \newline % Row Count 4 (+ 2)
is one of the features that was incorporated with the adoption of MuPad) \newline % Row Count 6 (+ 2)
b. To create anonymous symbolic functions, use the`{\bf{ matlabFunction}}`\{\{fa-flag\}`\}syms x` \newline % Row Count 8 (+ 2)
`y = x\textasciicircum{}2 - x + 1` \newline % Row Count 9 (+ 1)
`dy = diff(y)` \newline % Row Count 10 (+ 1)
`f = matlabFunction(dy)` \newline % Row Count 11 (+ 1)
`f(1)` \newline % Row Count 12 (+ 1)
\{\{fa-square\}\}:{\bf{› Interpolation}}:  consists of "method{[}s{]} of constructing \newline % Row Count 14 (+ 2)
new data points within the range of a discrete set of known data points. `{\bf{{\emph{interp1}}}}`;`{\bf{{\emph{yi = interp1(x,Y,xi)}}}}`Interpolates to find yi, the \newline % Row Count 17 (+ 3)
interpolated function values at the points in the vector or array xi.x contains your known data points (whose functions values are Y),which must be a vector, though xi can be a scalar, vector, ormultidimensional array. yi will always be the same size as xi;`{\bf{{\emph{yi = interp1(Y,xi)}}}}`x = 1:N, where N is length(Y) (for a vector)or size(Y,1) (for a matrix);`{\bf{{\emph{yi = interp1(x,Y,xi,method)}}}}`;`{\bf{{\emph{yi = interp1(x,Y,xi,method,'extrap')}}}}`. \newline % Row Count 26 (+ 9)
 {\bf{Extrapolation}} consists of "the process of estimating,beyond the original observation range, the value of a variable on the basis of its relationship with another variable.\{\{fa-exclamation-circle\}\}Some interpolation are excellent,yielding useful results BUT extrapolation can be a fool's errand. {\bf{Linear Interpolation}}:points (1, 3) and (-2, 5) Use linear interpolationto find estimate the y-value at the point x = −0.5?  \newline % Row Count 35 (+ 9)
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\mymulticolumn{1}{x{3.833cm}}{› ( slope intercept form)𝑦 = 𝑚𝑥 + 𝑏-{}-{}-{}-{}-{}-{}--𝑚=5-3/-2-1=2/-3=𝑦=-2/3𝑥+𝑏-{}-{}-{}-{}-{}-{}--to find b: Substitute in one \newline % Row Count 3 (+ 3)
of points 𝑦 = 𝑚𝑥 + 𝑏= 5=−2/3(−2) +𝑏→𝑏=3  2/3 -{}-{}-{}-{}-{}-{}-{}-{}-{}-the line is 𝑦 =-2/3 𝑥 + 3  2/3-{}-{}-{}-{}-{}- evaluate it at 𝑥 = −0.5 to find 𝑦(−0.5): 𝑦 −0.5 = −2/3 (−0.5) +3 \newline % Row Count 8 (+ 5)
𝑦(−0.5)=4 \newline % Row Count 9 (+ 1)
Approch(pointslopeform) 2:𝑦 − 𝑦1 = 𝑚 (𝑥 − 𝑥1) \newline % Row Count 11 (+ 2)
find the slope m=-2/3 \newline % Row Count 12 (+ 1)
\seqsplit{𝑦−3=−2/3(𝑥−1)→𝑦−3=−2/3} 𝑥 +2/3-{}-{}-{}-{}-{}-{}-𝑦=−2/3𝑥+2/3+3 →𝑦=−2/3𝑥+3 2/3 \newline % Row Count 15 (+ 3)
the line is 𝑦 =-2/3 𝑥 + 3  2/3 evaluate it at 𝑥 = −0.5 to find 𝑦(−0.5): 𝑦 −0.5 = −2/3 (−0.5) +3 \newline % Row Count 18 (+ 3)
𝑦(−0.5)=4 \newline % Row Count 19 (+ 1)
\{\{fa-exclamation-circle\}\}:{\bf{Linear Interpolation}}easy to do, BUT NOT best go-to solution if need accuracy.{\bf{Spline Interpolation}}:A spline is to use a different polynomial between each pair of discrete points.{\bf{Cubic splines}} correct for this flaw by ensuring that at the data points, the adjacent splines have the same 0th, 1st and 2nd derivatives; {\bf{Curve fitting}} is "the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints `{\bf{{\emph{ polyval}}}}`, `{\bf{{\emph{  polyfit}}}}`; \{\{fa-exclamation-circle\}\} / operator use it to solve least squared problems(less error).\{\{fa-exclamation-circle\}\}Goal is the minimize the {\bf{residuals}}: the difference between the actual and predicted values at a given point (i.e. the error).take its derivative and look to see where it is zero (this gives us the {\bf{extrema}} – the extreme points of the function) to find the minimum of function;  \{\{fa-exclamation-circle\}\}numerical differentiation tends to amplify noise.{\bf{Taylor series}} is a series expansion of a function 𝑓(𝑥) about a given point 𝑎.special case, known as {\bf{Maclaurin series}}, of the Taylor series exists in which 𝑎 = 0.\{\{fa-exclamation-circle\}\}The "Big O" notation (asymptotic notation) indicates higher order terms (H.O.T.s). {\bf{Central Difference:}} `{\bf{{\emph{gradient}}}}`; {\bf{Root Finding}}: zero a function; `{\bf{{\emph{fzero}}}}`. \newline % Row Count 48 (+ 29)
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\mymulticolumn{1}{x{3.833cm}}{\{\{fa-square\}\}: {\bf{finding area under curve}}:› Rectangles,Trapezoids,Parabolas.{\bf{Riemann Sums }}:\{\{fa-exclamation-circle\}\}If you average the left and right Riemannsum, you get the trapezoidal sum.{\bf{Left Riemann Sum}}:fits rectangles underneath curve using left of interval as location for hight of rectangle; Overestimate if f decreasing \& vice.versa.{\bf{Right Riemann Sum}}: like left Riemann Sum but in right instead; Overestimate if f is increasing and vice versa.{\bf{Middle Riemann Sum}}: Approximates the function by its value at themiddle point of the subinterval, yieldingmultiple rectangles with a base of Δx and the average height between the left and right.\{\{fa-exclamation-circle\}\} This better than R \& L Riemann sum.{\bf{Trapezoidal Rule}}:Approximates the function by fitting trapezoids underneath the curve.{\bf{Simpson's Rule}}:› Approximates the function by fitting parabolas under the curve. \{\{fa-exclamation-circle\}\} : must use an even number of intervals; › Pros and cons of using this versus trapezoids – more computationally expensive, but a better fit at times.\{\{fa-exclamation-circle\}\} Left-point and right-point sums were just wrong\{\{fa-exclamation-circle\}\}mid-point, trapezoidal and Simpson's) all got the correct answer. The choice between these depends on what the data looks like and what computational expense you can tolerate.{\bf{solve differentialequations}}:Euler's Method( forward Euler method), Runge-Kutta methods. {\bf{"state-space}}: breakyour system down into a system of simultaneous first order differential equations.\{\{fa-exclamation-circle\}\}The number of first order equations will be equal to the sum of number of independent variable(s) times the order.% Row Count 34 (+ 34)
} \tn 
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\seqsplit{Symbolic} \seqsplit{Command} & Description & Symbolic Command Description Numeric Analog \tn 
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ezplot & 2D plot & plot \tn 
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ezmesh & Wireframe mesh & mesh \tn 
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\seqsplit{ezmeshc} & Contour plot under wireframe mesh & meshc \tn 
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ezsurf & Surface plot & surf \tn 
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% Row 5
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\seqsplit{ezsurfc} & Contour plot under surface plot & surfc \tn 
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\seqsplit{ezcontour} & Contour plot & contour \tn 
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\seqsplit{ezcontourf} & Filled contour plot & contourf \tn 
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\seqsplit{ezplot3} & 3D plot & plot3 \tn 
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\seqsplit{ezpolar} & Polar plot & polar \tn 
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% That's all folks
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