Lecture 45: Fourth Review
=========================

The question on the fourth review focus on 

1. function definitions, differentiation, integration;

2. plotting in two and three dimensions; 

3. solving linear, differential, polynomial equations, linear programming.

The material corresponds to the second review,
which prepared for the second midterm exam.

Calculus
--------

1. Write a function to make polynomials in a system.
   The *k*-th polynomial in the system is

   .. math:: 

      f_k(x_1,x_2, \ldots, x_n) = x_k + \sum_{i=1}^{n-k} x_i x_{k+i},
      \quad k=1,2,\ldots,n.

   For example, for :math:`n=8`, the polynomials are

   ::

       x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 + x6*x7 + x7*x8 + x1
       x1*x3 + x2*x4 + x3*x5 + x4*x6 + x5*x7 + x6*x8 + x2
       x1*x4 + x2*x5 + x3*x6 + x4*x7 + x5*x8 + x3
       x1*x5 + x2*x6 + x3*x7 + x4*x8 + x4
       x1*x6 + x2*x7 + x3*x8 + x5
       x1*x7 + x2*x8 + x6
       x1*x8 + x7
       x8

2. Define a piecewise function ``int_inv_cub`` which
   as function of the end point :math:`b`
   always returns the correct value 
   of :math:`{\displaystyle \int_{-1}^b \frac{1}{x^3} dx}`.

3. The arc length of continuous function :math:`f(x)` over an interval
   :math:`[a,b]` can be defined 
   as :math:`{\displaystyle \int_a^t \sqrt{1+[f'(x)]^2}}`.

   1. Compute the arc length of the positive half
      of the unit circle, i.e.: :math:`f(x) = \sqrt{1-x^2}`
      (answer :math:`= \pi`).

   2. Create a function (call it ``arc_length``) in :math:`t`
      which returns a 10-digit floating-point approximation
      of the arc length of the positive half of the circle,
      for :math:`x \in [0,t]`.

4. Consider the recurrence relation

   .. math::

      h(n) = 5 h(n-1) - 6 h(n-2), \quad {\rm for} \ n \geq 2,
      \quad {\rm with} \ h(0) = 1 \ {\rm and} \ h(1) = -2.

   Answer the following the questions.

   1. The generating function
      :math:`{\displaystyle g(x) = \frac{1-7x}{1-5x + 6x^2}}`
      defines :math:`h(n)` as the coefficient with :math:`x^n` in the Taylor
      expansion of :math:`g(x)`.  Use :math:`g(x)` to define :math:`h` as a  
      function (call it ``t``) of :math:`n` which gives the value
      of :math:`h(n)`.

   2. Write a function to compute :math:`h(n)`, directly using the
      recurrence relation from above.  Make sure your function
      can compute :math:`h(120)`.  Compare with the result of (a).

5. The Legendre polynomials are defined by 

   .. math::

      P_0(x) = 1, \quad P_1(x) = x, 
      \quad P_n(x) = \frac{2n-1}{n} x P_{n-1}(x)
      - \frac{n-1}{n} P_{n-2}(x), \ {\rm for} \ n \geq 2.

   Write a efficient recursive function ``legendre`` to compute
   :math:`P_n(x)`.  The function ``legendre`` takes on input 
   the degree :math:`n` and the variable :math:`x`.

   Compare the output of your ``legendre`` (50, :math:`x` ) 
   with the ``legendre_P`` (50, :math:`x` ).

6. Consider the point :math:`P = (1,1)` on the curve
   defined by :math:`xy - 2 x + 1 = 0`.

   Compute the slope of the tangent line to the curve at :math:`P`
   in two ways: 

   1. with implicit differentiation,

   2. with a Taylor series.

Plotting and Solving Equations
------------------------------

1. Suppose we want to plot the curve :math:`x^4 + x^2 y^2 - y^2 = 0`
   for :math:`x` and :math:`y` both between :math:`-1` and :math:`+1`.

   1. Sampling this curve as given in rectangular coordinates,
      how many samples do we need to take from the curve to 
      obtain a nice plot?

   2. Convert the curve into polar coordinates and plot.
      Give all commands used to obtain the plot.
      How many samples of the curve are needed here?

2. Solve :math:`x^2 a^2 - 2x^2 a - 3 x^2 - x a^2 + 4 x a - 3 x + a^2
   + 2 a - 15` for :math:`x` for all values of the parameter :math:`a`.

   Be as complete as possible in your description of the solution.

3. Find the point with real coordinates on the curve 
   :math:`xy - 2 x + 3 = 0` closest to the origin.

4. Consider the system

   .. math::

      \left\{ 
      \begin{array}{rcl}
      x^2 - 2 y^2 - 1 & = & 0 \\
      x y - 2 x - 3 & = & 0. \\
      \end{array}
      \right.

   How many real solutions does this system have?

5. Consider :math:`y'' + 6 y' + 13 y = 0`, with :math:`y(\pi/2) = -2`
   and :math:`y'(\pi/2) = 8`.

   1. Find an exact solution to this initial value problem
      and use this to create a function :math:`f` which returns
      a numerical 10-digit floating-point approximation
      of the solution.

   2. Solve this initial value problem numerically.
      Compare the solution with the value for :math:`y(2)` and 
      also with :math:`f(2)` obtained in (a).

6. A 5-by-5 variable Toeplitz matrix has the following form:

   ::

       [t0 t1 t2 t3 t4]
       [t8 t0 t1 t2 t3]
       [t7 t8 t0 t1 t2]
       [t6 t7 t8 t0 t1]
       [t5 t6 t7 t8 t0]

   for the symbols in the list ``[t0, t1, t2, t3, t4, t5, t6, t7, t8]``.

   For general dimension :math:`n`, the :math:`(i,j)`-th element
   of the Toepliz matrix :math:`T` is

   .. math::

      T_{(i,j)} = \left\{
      \begin{array}{lcl}
      j - i & {\rm if} & j \geq i \\
      j - i + 2 n - 1 & {\rm if} & j < i.
      \end{array}
      \right.

   Give the command(s) to define a variable Toeplitz matrix,
   for any dimension :math:`n`.

7. Maximize :math:`x_1 + x_2` subject to :math:`-x_1 + 2 x_2 \leq 8`,
   :math:`4 x_1 - 3 x_2 \leq 8`, :math:`2 x_1 + x_2 \leq 14`,
   :math:`x_1 \geq 0`, and :math:`x_2 \geq 0`.

   Write the commands to define this problem and then solve it.
   What are the values of :math:`x_1` and :math:`x_2` at the optimal solution?