Exercise 2
==========

#. Complete the 8-bit, unsigned addition given below. Show all the steps
   in both hexadecimal (left) and binary (right).

   .. math::

         \begin{aligned}
         &\textsf{hex} \quad & &\textsf{binary}\\
         &\texttt{AF}   & &\square\square\square\square\square\square\square\square\\
         +&\texttt{72}   &+&\square\square\square\square\square\square\square\square\\ \hline
         &\!\!\square\square  &&\square\square\square\square\square\square\square\square
         \end{aligned}

   Is there an overflow?

#. Suppose we want to find an iterative algorithm for raising the number
   :math:`\mathsf{a}` to a rational power---i.e., for computing
   :math:`\mathsf{a^{p/q}}` with :math:`\mathsf{p}` and
   :math:`\mathsf{q}` integer.  Convince yourself that the sequence
   defined by

   .. math::

      \begin{aligned}
      \mathsf{x_0} & := \mathsf{1}\\
      \mathsf{x_{n+1}} & := \mathsf{\Bigl(1-\frac{1}{q}\Bigr)x_n} + \mathsf{\frac{a^p}{q}x_n^{1-q}}
      \end{aligned}

   has :math:`\mathsf{a^{p/q}}` as its limit. I suggest you proceed in
   this way: suppose that :math:`\mathsf{x_n}` is a good guess to the
   answer; construct a better guess :math:`\mathsf{x_{n+1}}
   = \mathsf{x_n} + \mathsf{\delta x} \approx \mathsf{a^{p/q}}`; solve
   for :math:`\mathsf{\delta x}` to lowest order in a series expansion.

#. Find the value of :math:`\mathsf{b}` such that the functions

   .. math::

      \begin{aligned}
      \mathsf{t(x)} &= \mathsf{x} - \mathsf{\frac{1}{3}x^3} + \mathsf{\frac{2}{15}x^5}\\
      \textsf{and} \ \ \mathsf{r(x)} & = \mathsf{\frac{15x+x^3}{15+b x^2}}
      \end{aligned}

agree to sixth order. Plot the functions :math:`\mathsf{t(x)}`,
:math:`\mathsf{r(x)}`, and :math:`\mathsf{tanh}\,\mathsf{x}` in the
range :math:`\mathsf{0} \le \mathsf{x} \le \mathsf{2}`.  Also, plot the
differences :math:`\mathsf{t(x)}-\textsf{tanh}\,\mathsf{x}` and
:math:`\mathsf{r(x)} - \textsf{tanh}\,\mathsf{x}` with the log scale on
the y-axis.  How do they compare?  What do :math:`\mathsf{t(x)}` and
:math:`\mathsf{r(x)}` represent?