Extra Credit Problems
Third Quarter

Instructions:
  1.  Problems may be submitted any time during the quarter, but they MUST be turned in by the due date.
        (Due Date:  February 28, 2011)
  2.  Your work must accompany each problem.  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. 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)

  2.  
    (Problem taken from Problems in Calculus, 1985 J. Weston Walch, Publisher)

  3.  

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

  5. 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.

  6. 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.

  7. 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?

  8. 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.

  9. 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.

  10. Given the function defined by f(x) = 4e - x.
    a. Find the area of the region bounded by f, the x-axis, the y-axis, and the line x = p,
        where p is a positive constant.
    b. Find the volume of the solid generated if the region in (a) is rotated abut the x-axis.
    c. What is the limit of the volume in (b) as p®¥?
    (Problem taken from Arco's AP Exams in Mathematics; Calculus AB and Calculus BC, published in 1990)

  11. Given that p and q are constants, q not equal to 0, and that a function g is defined by
            
    a. Find all values for which g(x) = 0.
    b. Find all points of discontinuity for g.
    c. Find the values for p and q so that g is continuous at x = 0. Justify your answer.
    d. Given that w > 0, find w such that
          
    (Problem taken from Arco's AP Exams in Mathematics; Calculus AB and Calculus BC, published in 1990)


  12. Let f be the function defined by f(x) = 12x2/3 - 4x.
    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.

  13. Let f be a function defined by  
    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.

  14. Let S be the series

    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. 

  15. A particle moves along the x-axis so that at time t its position is given by
    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.

  16. a.  A solid is constructed so that it has a circular base of radius r centimeters and every
         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?

  17. Let f be a differentiable function defined for all x > 0 such that
          (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.