Extra Credit Problems
Sixth 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:  May 16, 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.  Find the maximum volume of a cylinder inscribed in a sphere with radius 10.

2.  Analyze and graph y = x2 + 2x + 1/x, identifying all local maximum and minimum points and
     all asymptotes.

3.  In how many zeros does the number 7535 end?

4.  Show that (1 + x)1/3 » 1 + (1/3)x when |x| is small.  Use the result to approximate
   

5.  Show that cos q = (1/2)(e iq + e - iq).

6.  Circle A is fixed and centered at (-a,0) and circle B has the same radius as circle A. Initially
     circles A and B are tangent at point O(0,0). Point P which initially was at the origin is fixed on
     circle B. Circle B rolls counterclockwise around circle A without slippage. Write in polar
     coordinates the equation of the locus of point P(r,0).
    (Problem taken from Problems in Analytic Geometry published by J. Weston Walch,
     1984)

7.  A ship is following a counterclockwise circular course along the curve whose polar equation
     is r = 6 cos 0 with r in miles. A radar station at the origin, O, first places the ship in the
     position (3, \/3, p/6). Eight minutes later the ship is observed at a distance of 3 \/2 miles from
     point O. Assuming constant speed is maintained, how long after the second observation is the
     ship expected to arrive at point O?
 

8.  Do the problems in the limits lessons in the precalculus web pages. Click on the bulleted titles,
     go to the end of the lesson, and click on problems.  There will problems to try in most of the
     lessons.
 
 

9.  A circle with radius 1 rolls along the x-axis at one unit per second.  At t = 0, P is at the
     origin.  The path traced by P as the circle rolls is called a cycloid.  Give a pair of parametric
     equations for the cycloid.  Go to the cycloid generator to see what it looks like.
 

10. Evaluate this limit: