Extra Credit Problems
Fifth Six Weeks

Instructions:
 1.  Problems may be submitted any time during the six weeks, but they MUST be turned by the due date.
        (Due Date:  April 4, 2008)
  2.  Your work must accompany each problem and the problem must be stated or printed out.
        No credit will be given for just an answer.
  3.  Parts of problems are worth anywhere from 1 - 3 points depending on level of difficulty.
       Credit is given only for correct work and answers. The whole part must be correct to
       receive credit.
  4.  You may do as many or as few problems as you desire. Try to keep them in order.
Problems:
  1. Let f be the function defined by f(x) = 12x2/3 - 4x.
    2. a.  Analytically, find the intervals on which f is increasing.
    b.  Analytically, find the x- and y-coordinates of all relative maximum points
    c.  Analytically, find the x- and y-coordinates of all relative minimum points.
    d.  Analytically, find the intervals on which f is concave downward.
    e.  Using the information found in parts (a), (b), (c), and (d), sketch the graph of f.
     
  2. Let f be a function defined by

    3. piecewise function

    a.  For what value of k will f be continuous at x = 2?  Justify your answer.
    b.  Using the value of k found in part (a), determine whether f is differentiable at x = 2.
         Use  the definition of the derivative to justify your answer.
    c.  Let k = 4.  Determine whether t is differentiable at x = 2.  Justify your answer.

  3. Let S be the series

    4. a.  Analytically, find the value to which S converges when t = 1.
    b.  Determine the values of t for which S converges.  Justify your answer.
    c.  Find all values of t that make the sum of the series S greater than 10.  

  4. A particle moves along the x-axis so that at time t its position is given by
    5. x(t) = sin (pt2) for -1 < x < 1.
    a.  Analytically, find the velocity at time t.
    b.  Analytically, find the acceleration at time t.
    c.  For what values of t does the particle change direction?
    d.  Find all values of t for which the particle is moving to the left.

  5. a.  A solid is constructed so that it has a circular base of radius r centimeters and every
    6.      plane section perpendicular to a certain diameter of the base is a square, with a side of
         the square being a chord of the circle.  Find the volume of the solid.
    b.  If the solid described in part (a) expands so that the radius of the base increases at a
         constant rate of 1/2 centimeters per minute, how fast is the volume changing when the
         radius is 4 centimeters?
     
  6. Let f be a differentiable function defined for all x > 0 such that
    7.            (i)   f(1) = 0
              (ii)   f ' (1) = 1, and
             (iii)  , for all x > 0
    a.  Find f ' (2)
    b.  Suppose f ' is differentiable.  Prove that there is a number c, 2 < c < 4, such that
         f ''(c) = -1/8.
    c.  Prove that f(2x) = f(2) + f(x) for all x > 0.
     
  7. For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v 3.  It is believed that migrating fish try to minimize the total energy required to swim a fixed distance.  If the fish are swimming against a current u (u < v), then the time required to swim a distance L is

    8. and the total energy E required to swim the distance is given by

      where a is the proportionality constant.
    a.  Determine the value of v that minimizes W.
    b.  Sketch the graph of E.
                                            Note:  This result has been verified experimentally.
    (Taken from Calculus, Stewart, 1996, 3rd Edition, Brooks-Cole)
     
  8. A rescue pilot is flying a plane at a constant altitude of h feet above sea level at a speed of s feet per second.  A food package is to be dropped to a life raft sighted at the angle of depression, f.  Derive the equation of the path of the package relative to a stationary observer assuming only the force of gravity, neglecting air resistance.  As the pilot approaches the raft, f increases.  Show that in order to have the package drop on the raft, the pilot must wait until f satisfies the condition f = tan-1(4 \/h /s)   and then immediately release the package.