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 9, 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.  Show that the ellipse 2x2 + y2 = 6 and the parabola y2 = 4x intersect at right angles and
     sketch the curves, showing their points of intersection.

2.  P1( x1 , y1 ) and P2( x2 , y2 ) are two points on the curve y = ax2 + bx + c (a ¹ 0).  A line is
    drawn parallel to the chord P1P2and tangent to the curve at the point P( x0 , y0 ).  Prove that
    x0 = (1/2)(x1 + x2).

3.  Find the equations of the straight lines through the point (-1,0) which are tangent to the curve
     whose equation is y = x2 - 3x.
     Sketch the curve and the straight line tangents.
     Prove analytically that there is no tangent to the curve through the point (2,0).

4.  A right circular cylinder of radius 12 inches is cut by a plane which passes through a diameter
     of the base and makes an angle of 60o with the base.  Set up an integral which will determine
     the volume of the wedge cut off.  Evaluate the integral.

5.  In the isosceles triangle ABC, the length of each of the sides AB and AC is 10 units and the
     length of BC is 12 units.
     Find the location of the point D on the altitude from A to BC for which the sum of the
     distances from D to the three vertices of the triangle is a minium.

6.  A cylindrical tank whose radius is 10 feet and whose height is 25 feet is full of water.  A leak
     occurs in the bottom of the tank and water escapes at a rate proportional to the square root
     of the depth of the water remaining in the tank.  At the end of the first hour the depth of the
     remaining water is 6 feet.  Obtain a formula for the depth of the water remaining after t hours.

7.  The area to the right of the line x = 1 and inside the ellipse x2 + 2y2 = 9 is revolved around
      the  y-axis, thus generating a solid.  Find its volume.

8.  Sketch the graph of the function f(x) = ln [x + (x/2)]2 showing the following:
     a.  the domain of definition
     b.  all extreme points
     c.  all inflection points
     d.  intervals where the graph is increasing.
     e.  intervals where the graph is concave upward.
     f.  intercepts

9.  Given a function f(x) defined for all real x, and such that f(x + h) - f(x) < 6h2 for all real h
     and x.  Show that f(x) is a constant.

10.  Let .  Is the following statement true or false?

      (1/2 ln x £ f(x) £ ln x.

      Justify your answer.

(Problems taken from the 1957 AP Calculus Exam)