Function sampling
=================

In these exercises we want to sample a function `f` in a given
interval `[a, b]` and with `d` increments. `a`, `b` and `d` are such
that :math:`a < b` and :math:`d < b-a`. Sampling `f` consists in
computing points of this function with abscissas beginning with `x=a`,
with uniform increments of `d`, and such all of them are less than or
equal to `b`. Values `a`, `b` and `d` can be real.

Save the following functions into file ``sampling.py``.

#. Write the function ``positives(a, b, d, k)`` that returns the
   average of the function `f` values (ordinates of the sampled
   points) in the given interval which are positive. Function `f` is
   defined as: :math:`f(x, k) = e^{\frac{x}{k}} - k sin(x)`, :math:`k
   >0`. If there is no positive value, this function must return
   0.0. Examples:

   .. literalinclude:: positives.txt
      :language: python3
      :lines: 3-
             
   .. note::

      More tests can be found in file :download:`positives.txt
      <positives.txt>`

#. Write the function ``localmax(a, b, d, k)`` that returns a list of
   tuples where each tuple is the coordinates pair `(x, y)` of those points
   that are local maxima of the function :math:`f(x, k) =
   e^{\frac{x}{k}} - k sin(x)`, :math:`k >0` in the given interval. A
   point (x, y) is a local maxima if its ordinate is greater than the
   ordinate of the previous sampled point and of that of the next
   one. Examples:

   .. literalinclude:: localmax.txt
      :language: python3
      :lines: 3-
             
   .. note::

      More tests can be found in file :download:`localmax.txt <localmax.txt>`

#. Write the function ``fposneg (a, b, k, n)`` that returns the number
   of sampled positive or zero values and the number of
   sampled negative values of function :math:`f(x, k) =
   e^{\frac{x}{k}} - k sin(x)`, :math:`k >0` in the given interval. In
   this case ``n`` stands for the number of points to be sampled
   (including the interval extreme points). Examples:

   .. literalinclude:: fposneg.txt
      :language: python3
      :lines: 3-
             
   .. note::

      More tests can be found in file :download:`fposneg.txt <fposneg.txt>`

.. rubric:: Solution
               
Solutions can be found in file :download:`sampling.py <solutions/sampling.py>`