Extra Credit Problems
Fourth 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:  February 12, 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 d > 0.  Let A be the area bounded by y = xn, y = 0, and x = 1 + d, where n > 1.
     Let V be the volume of the solid generated by rotating A about the x-axis.
      a.  Find A and V as functions of n and d.  Then show that both A and V® ¥ as n® ¥
           ford > 0.
      b.  Show that

     (Problem taken from Problems in Calculus, 1985 J. Weston Walch, Publisher)

2.  On a particular suspension bridge, the support cable would be closely approximated by the
     graph, drawn to scale, of the catenary curve

    where the origin is placed at the point midway across the bridge and 30 feet beneath the
    lowest point on the cable.  The bridge is 120 feet long and the endpoints of the cable are fixed
    50 feet bove the bridge.  Find the length of the cable to the nearest tenth of a foot.

    (Problem taken from Problems in Calculus, 1985 J. Weston Walch, Publisher)
 

3. 

   (Problem taken from Problems in Calculus, 1985 J. Weston Walch, Publisher)

4. 
 

5.  If ,
        
     where f is a continuous function, find f(4).
 

6.  A closed region R of the plane has y = 1 + sin (px/2) as its upper boundary, y = x/2 as its
     lower boundary, and the y-axis as its left-hand boundary.  Set up, but do not evaluate, an
     integral expression in terms of the single variable x for each of the following:
        a.  the area A of R.
        b.  the volume V of the solid figure obtained by revolving R about the x-axis.
        c.  the total perimeter P of R.

7.  Let f and g be functions that are differentiable for all real numbers x and that have the following
     properties:
             i.  f '(x) = f(x) - g(x)
            ii.  g'(x) = g(x) - f(x)
           iii. f(0) = 5
           iv. g(0) = 1

     a.  Prove that f(x) + g(x) = 6 for all x.
     b.  Find f(x) and g(x).  Show your work.

8.  At time t, t > 0, the volume of a sphere is increasing at a rate proportional to the reciprocal of
     its radius.  At t = 0, the radius of the sphere is 1 and at t = 15, the radius is 2.
        a.  Find the radius of the sphere as a function of t.
        b.  At what time t will the volume of the sphere by 27 times its volume at t = 0?

9.  A circular paper disc has a diameter of 8 inches.  A sector with a central angle of x radians is
     cut out, and the sides of the disk minus the sector are taped together to form a conical
     drinking cup.  Find the angle x which results in a cone of maximum volume.

10.  Consider the quadrant I area bounded by y = xn where n > 1, the x-axis, and the tangent line
       to the graph of y = xn at the point (1,1).  Find the value of n so that the enclosed area is a
       maximum.