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#+TITLE:    
#+AUTHOR:    FOSSEE
#+EMAIL:     
#+DATE:    

#+DESCRIPTION: 
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#+LANGUAGE:  en
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* 
#+begin_latex
\begin{center}
\vspace{12pt}
\textcolor{blue}{\huge Using Sage}
\end{center}
\vspace{18pt}
\begin{center}
\vspace{10pt}
\includegraphics[scale=0.95]{../images/fossee-logo.png}\\
\vspace{5pt}
\scriptsize Developed by FOSSEE Team, IIT-Bombay. \\ 
\scriptsize Funded by National Mission on Education through ICT\\
\scriptsize  MHRD,Govt. of India\\
\includegraphics[scale=0.30]{../images/iitb-logo.png}\\
\end{center}
#+end_latex
* Objectives
 At the end of this tutorial, you will be able to,

 - Learn the range of things for which Sage can be used. 
 - Know some of the functions for Calculus.
 - Learn about graph theory and number theory using Sage.
* Pre-requisite
  Spoken tuorial on -
  - Getting started with Sage
* Equation
  Ax = v,
  where A is the matrix,'' matrix([[1,2],[3,4]])''
  v is the vector, ''vector([1,2])''. 
* Summary
In this tutorial, we have learnt to,  
 - Use functions for calculus like --
    - lim()-- to find out the limit of a function
    - diff()-- to find out the differentiation of an expression
    - integrate()-- to integrate over an expression  
    - integral()-- to find out the definite integral of an 
      expression by specifying the limits
    - solve()-- to solve a function, relative to it's postion. 
 - Create Both a simple graph and a directed graph, using the 
    functions ``graph()`` and ``digraph()`` respectively.
 - Use functions for Number theory.For eg: 
    - primes\_range()-- to find out the prime numbers within the 
      specified range
    - factor()-- to find out the factorized form of the number specified
    - Permutations(), Combinations()-- to obtain the required permutation 
      and combinations for the given set of values.  
* Evaluation
 1. How do you find the limit of the function ``x/sin(x)`` as ``x`` tends 
    to ``0`` from the negative side.

 2. List all the primes between 2009 and 2900.

 3. Solve the system of linear equations
     
    x-2y+3z = 7
    2x+3y-z = 5
    x+2y+4z = 9
* Solutions
1. lim(x/sin(x), x=0, dir="below")

2. prime\_range(2009, 2901)

3. A = Matrix([[1, -2, 3], 
               [2, 3, -1], 
               [1, 2, 4]])

   b = vector([7, 5, 9])

   solve\_right(A, b)
* 
#+begin_latex
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  \textcolor{blue}{\Large THANK YOU!} 
  \end{center}
  \end{block}
\begin{block}{}
  \begin{center}
    For more Information, visit our website\\
    \url{http://fossee.in/}
  \end{center}  
  \end{block}
#+end_latex