1 % Created 2010-11-11 Thu 02:03

     2 \documentclass[presentation]{beamer}

     3 \usepackage[latin1]{inputenc}

     4 \usepackage[T1]{fontenc}

     5 \usepackage{fixltx2e}

     6 \usepackage{graphicx}

     7 \usepackage{longtable}

     8 \usepackage{float}

     9 \usepackage{wrapfig}

    10 \usepackage{soul}

    11 \usepackage{textcomp}

    12 \usepackage{marvosym}

    13 \usepackage{wasysym}

    14 \usepackage{latexsym}

    15 \usepackage{amssymb}

    16 \usepackage{hyperref}

    17 \tolerance=1000

    18 \usepackage[english]{babel} \usepackage{ae,aecompl}

    19 \usepackage{mathpazo,courier,euler} \usepackage[scaled=.95]{helvet}

    20 \usepackage{listings}

    21 \lstset{language=Python, basicstyle=\ttfamily\bfseries,

    22 commentstyle=\color{red}\itshape, stringstyle=\color{darkgreen},

    23 showstringspaces=false, keywordstyle=\color{blue}\bfseries}

    24 \providecommand{\alert}[1]{\textbf{#1}}

    25 

    26 \title{Getting started with symbolics}

    27 \author{FOSSEE}

    28 \date{}

    29 

    30 \usetheme{Warsaw}\usecolortheme{default}\useoutertheme{infolines}\setbeamercovered{transparent}

    31 \begin{document}

    32 

    33 \maketitle

    34 

    35 

    36 

    37 

    38 

    39 

    40 

    41 

    42 

    43 \begin{frame}

    44 \frametitle{Outline}

    45 \label{sec-1}

    46 

    47 \begin{itemize}

    48 \item Defining symbolic expressions in sage.

    49 \item Using built-in constants and functions.

    50 \item Performing Integration, differentiation using sage.

    51 \item Defining matrices.

    52 \item Defining Symbolic functions.

    53 \item Simplifying and solving symbolic expressions and functions.

    54 \end{itemize}

    55 \end{frame}

    56 \begin{frame}

    57 \frametitle{Question 1}

    58 \label{sec-2}

    59 

    60 \begin{itemize}

    61 \item Define the following expression as symbolic

    62     expression in sage.

    63 

    64 \begin{itemize}

    65 \item x$^2$+y$^2$

    66 \item y$^2$-4ax

    67 \end{itemize}

    68 

    69 \end{itemize}

    70 

    71   

    72 \end{frame}

    73 \begin{frame}[fragile]

    74 \frametitle{Solution 1}

    75 \label{sec-3}

    76 

    77 \lstset{language=Python}

    78 \begin{lstlisting}

    79 var('x,y')

    80 x^2+y^2

    81 

    82 var('a,x,y')

    83 y^2-4*a*x

    84 \end{lstlisting}

    85 \end{frame}

    86 \begin{frame}

    87 \frametitle{Question 2}

    88 \label{sec-4}

    89 

    90 

    91 \begin{itemize}

    92 \item Find the values of the following constants upto 6 digits  precision

    93 

    94 \begin{itemize}

    95 \item pi$^2$

    96 \item euler$_{\mathrm{gamma}}$$^2$

    97 \end{itemize}

    98 

    99 \end{itemize}

   100 

   101 \begin{itemize}

   102 \item Find the value of the following.

   103 

   104 \begin{itemize}

   105 \item sin(pi/4)

   106 \item ln(23)

   107 \end{itemize}

   108 

   109 \end{itemize}

   110 \end{frame}

   111 \begin{frame}[fragile]

   112 \frametitle{Solution 2}

   113 \label{sec-5}

   114 

   115 \lstset{language=Python}

   116 \begin{lstlisting}

   117 n(pi^2,digits=6)

   118 n(sin(pi/4))

   119 n(log(23,e))

   120 \end{lstlisting}

   121 \end{frame}

   122 \begin{frame}

   123 \frametitle{Question 3}

   124 \label{sec-6}

   125 

   126 \begin{itemize}

   127 \item Define the piecewise function. 

   128    f(x)=3x+2 

   129    when x is in the closed interval 0 to 4.

   130    f(x)=4x$^2$

   131    between 4 to 6.

   132 \item Sum  of 1/(n$^2$-1) where n ranges from 1 to infinity.

   133 \end{itemize}

   134 \end{frame}

   135 \begin{frame}[fragile]

   136 \frametitle{Solution 3}

   137 \label{sec-7}

   138 

   139 \lstset{language=Python}

   140 \begin{lstlisting}

   141 var('x') 

   142 h(x)=3*x+2 

   143 g(x)= 4*x^2

   144 f=Piecewise([[(0,4),h(x)],[(4,6),g(x)]],x)

   145 f

   146 \end{lstlisting}

   147 

   148 \lstset{language=Python}

   149 \begin{lstlisting}

   150 var('n')

   151 f=1/(n^2-1) 

   152 sum(f(n), n, 1, oo)

   153 \end{lstlisting}

   154 \end{frame}

   155 \begin{frame}

   156 \frametitle{Question 4}

   157 \label{sec-8}

   158 

   159 \begin{itemize}

   160 \item Differentiate the following.

   161 

   162 \begin{itemize}

   163 \item sin(x$^3$)+log(3x), to the second order

   164 \item x$^5$*log(x$^7$), to the fourth order

   165 \end{itemize}

   166 

   167 \item Integrate the given expression

   168 

   169 \begin{itemize}

   170 \item x*sin(x$^2$)

   171 \end{itemize}

   172 

   173 \item Find x

   174 

   175 \begin{itemize}

   176 \item cos(x$^2$)-log(x)=0

   177 \item Does the equation have a root between 1,2.

   178 \end{itemize}

   179 

   180 \end{itemize}

   181 \end{frame}

   182 \begin{frame}[fragile]

   183 \frametitle{Solution 4}

   184 \label{sec-9}

   185 

   186 \lstset{language=Python}

   187 \begin{lstlisting}

   188 var('x')

   189 f(x)= x^5*log(x^7) 

   190 diff(f(x),x,5)

   191 

   192 var('x')

   193 integral(x*sin(x^2),x) 

   194 

   195 var('x')

   196 f=cos(x^2)-log(x)

   197 find_root(f(x)==0,1,2)

   198 \end{lstlisting}

   199 \end{frame}

   200 \begin{frame}

   201 \frametitle{Question 5}

   202 \label{sec-10}

   203 

   204 \begin{itemize}

   205 \item Find the determinant and inverse of :

   206 

   207       A=[[x,0,1][y,1,0][z,0,y]]

   208 \end{itemize}

   209 \end{frame}

   210 \begin{frame}[fragile]

   211 \frametitle{Solution 5}

   212 \label{sec-11}

   213 

   214 \lstset{language=Python}

   215 \begin{lstlisting}

   216 var('x,y,z')

   217 A=matrix([[x,0,1],[y,1,0],[z,0,y]])

   218 A.det()

   219 A.inverse()

   220 \end{lstlisting}

   221 \end{frame}

   222 \begin{frame}

   223 \frametitle{Summary}

   224 \label{sec-12}

   225 

   226 \begin{itemize}

   227 \item We learnt about defining symbolic expression and functions.

   228 \item Using built-in constants and functions.

   229 \item Using <Tab> to see the documentation of a function.

   230 \item Simple calculus operations .

   231 \item Substituting values in expression using substitute function.

   232 \item Creating symbolic matrices and performing operation on them .

   233 \end{itemize}

   234 \end{frame}

   235 \begin{frame}

   236 \frametitle{Thank you!}

   237 \label{sec-13}

   238 

   239   \begin{block}{}

   240   \begin{center}

   241   This spoken tutorial has been produced by the

   242   \textcolor{blue}{FOSSEE} team, which is funded by the 

   243   \end{center}

   244   \begin{center}

   245     \textcolor{blue}{National Mission on Education through \\

   246       Information \& Communication Technology \\ 

   247       MHRD, Govt. of India}.

   248   \end{center}  

   249   \end{block}

   250 \end{frame}

   251 

   252 \end{document}