Objective

---------

1. How do you find the limit of the function ``x/sin(x)`` as ``x`` tends to

   ``0`` from the negative side.

   Answer: lim(x/sin(x), x=0, dir="below")

#. Find the third differential of the function ``exp(sin(x)*cos(x^2))``

   Answer: diff(exp(sin(x)*cos(x^2), x, 3)

#. Solve the system of linear equations::

     x-2y+3z = 7

     2x+3y-z = 5

     x+2y+4z = 9

   Answer::

     A = Matrix([[1, -2, 3], 

                 [2, 3, -1], 

                 [1, 2, 4]])

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

     solve_right(A, b)

#. How do you get the factorized form of ``x^4 - 4x^2 + x^3 + 2x + 7`` 

   Answer::

      factor( x^4 + x^3 - 4*x^2 + 2*x + 7 )

#. list all the primes between 2009 and 2900

   Answer: prime_range(2009, 2901)

#. Which function is used to check primality

   a. isPrime

   #. isprime

   #. is_prime

   #. prime

   Answer: is_prime

#. How do you list all the combinations of ``[1, 2, 3, 4]``

   Answer::

     c1 = Combinations([1, 2, 3, 4])

     c1.list()

#. How do you list all the permutations of ``[1, 3, 2, 3]``

    Answer::

      p1 = Permutations([1, 3, 2, 3])

      p2.list()

Programming

-----------

1. Obtain the sum of primes between 1 million and 2 million. 

   Answer::

     prime_sum = 0

     for i in range(1000001, 2000000, 2):

         if is_prime(i):

         prime_sum += i

     prime_sum

   OR

   ::

     sum(prime_range(1000000, 2000000))

2. ``graphs.WorldMap()`` gives the world map in the form of a

   graph. ::

       G = graphs.WorldMap()

       G.vertices()

   Suppose, I wish to go from India to France by Road, find out the

   least number of Visas that I'll have to obtain. 

   Answer::

      G.distance("India", "France")