Extra Credit Problems
First Six Weeks

Instructions:
   1.  Problems may be submitted any time during the six weeks, but they must be turned in at
        least a week prior to the last day of the six weeks. (Due Date:  September 17, 2007)
  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.  Problems are worth anywhere from 1 - 3 points depending on length and level of difficulty.
       Credit is given only for correct work and answers. The whole problem 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. A pendulum hung from the ceiling makes a complete back-and-forth swing each 6 seconds. As
    the pendulum swings, its distance, d cm, from one wall of the room depends on the number of
    seconds, t, since it was set in motion. At t = 1.3 seconds, d is its maximum of 110 cm from the
    wall. The lower bound of d is 50 cm.  Assume that d is a sinusoidal function of t.
        a. Write an equation expressing d as a function of t.
        b. Write an equation for the derivative function.
        c. How fast is the pendulum moving when t = 5?  t = 11?  How do you explain the
            relationship between these two answers?
        d. When t = 20, is the pendulum going toward the wall or away from it?  Explain.
        e.  What is the fastest the pendulum goes?  Where is the pendulum when it is going its fastest?
        f.  What is the first positive value of t at which the pendulum is moving 0 cm/sec?  Where is
            the pendulum at this time?
    (Problem from Calculus by Foerster, Printing 5, 8/94)

2.  Given the curve x2 - xy + y2 = 9.
     a. Write a general expression for the slope of the curve.
     b. Find the coordinates of the points on the curve where the tangents are vertical.
     c. At the point (0,3) find the rate of change in the slope of the curve with respect to x.

3.  Prove that at no point on the graph of y = x2/(x - 1) is there a tangent line whose angle of
     inclination is 45o.
     (Taken from Problems in Calculus, J. Weston Walch, Publisher, 1985)

4.  Given the relation x2y + x - y2 = 0, find the coordinates of all points on its graph where the
     tangent line is horizontal.
     (Taken from Problems in Calculus, J. Weston Walch, Publisher, 1985)