Extra Credit Problems
First Quarter

Instructions:
  1.  Problems may be submitted any time during the quarter, but they MUST be turned in by the due date.
        (Due Date:  October 1, 2010)
  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.  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.

2.  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)

3.  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)

4.  Let
    f(x) = 4x3 - 3x - 1
    a.  Find the x-intercepts of the graph of  f.
    b. Write an equation for the tangent line to the graph of  f at x = 2.
    c. Write an equation of the graph that is the reflection across the y-axis of the
         graph of  f.
    (taken from AP Calculus AB Test 1972 -- AB1)
5.  Given the two functions f and h such that f(x) = x3 - 3x2 - 4x + 12 and

    a. Find all zeros of the function f.
    b. Find the value of p so that the function h is continuous at x = 3. Justify your answer.
    c. Using the value of p found in (b), determine whether h is an even function. Justify your
        answer.