Extra Credit Problems
Second Quarter

  Instructions:
  1.  Problems may be submitted any time during the quarter, but they MUST be turned in by the due date.
        (Due Date:  December 3, 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.   A builder is purchasing a rectangular plot of land with frontage on a road for the purpose of
      constructing a rectangular warehouse. Its floor area must be 300,000 square feet. Local
      building codes require that the building be set back 40 feet from the road and that there be
      empty buffer strips of land 25 feet wide on the sides and 20 feet wide in the back. Find the
      overall dimensions of the parcel of land and building which will minimize the total area of the
      land parcel that the builder must purchase.
      (Problem taken from Problems in Calculus published by J. Weston Walch--1985)

2.  Determine the maximum area of a rectangle with one side on the x-axis and the opposite
     corners touching the parabola y = -x2 + 9.

3.  Find the lengths of the sides of the isosceles triangle with perimeter 12 and maximum area.

4. One ship, A, is sailing due south at 16 knots and a second ship, B, initially 32 nautical miles
    south of A, is sailing due east at 12 knots.
    a. At what rate are they approaching or separating at the end of one hour?
    b. When do they cease to approach one another and how far apart are they at this time? What
         is the significance of this distance? 
    (Taken from Problems in Calculus, J. Weston Walch, Publisher, 1985)

These problems were written to be done without a calculator.  Try your hand at them.

5.  Given the function f(x) = x2ln x, 0 < x < 1:
      a.  Find the coordinates of any points where the graph of f(x) has a horizontal tangent line.
      b.  Find the coordinates of any points of inflection on the graph of f(x).
      c.  Find
              lim         f(x)
              x-> 0+
                                            and
             lim         f '(x)
             x-> 0+

    d.  Sketch the graph of f using the information obtained in parts (a), (b), and (c).
         State the coordinates of all relative and/or absolute maximum or minimum
         points.  Clearly show the  concavity of the graph and its behavior near (0,0).
              Suggestion: Use a large scale for the y-axis.
             (Problem taken from Problems in Calculus published by J. Weston Walch--1985)
     

6.  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)
7.  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.
    (taken from AP Calculus AB Test 1976 -- AB2)
8.     a.  A point moves on the hyperbola 3x2 - y2 = 23 so that its y-coordinate is increasing at
         a constant rate of 4 units per second. How fast is the x-coordinate changing when
         x = 4?
    b.  For what values of k will the line 2x + 9y + k = 0 be normal to the hyperbola
         3x2 - y2 = 23?
    (taken from AP Calculus AB Test 1976 -- AB4)
9.  Given the function defined by f(x) = esin x for all x such that -p < x < 2p.
    a. Find the x- and y-coordinates of all maximum and minimum points on the given
        interval.  Justify your answers.
    b. Sketch the graph of the function, showing your analyses.
    c. Write an equation for the axis of symmetry of the graph.
    (taken from AP Calculus AB Test 1976 -- AB5)

10.  Consider the curve defined by f(x) = x4 - 4x2
    a. Find the coordinates of all relative aximum and minimum points. Justify your answers.
    b. Find the coordinates of all points of inflection. Justify your answers.
    c. Sketch the graph of the function.
    d. Find the area bounded by the curve, the x-axis, and the lines x = 0 and x = 1.
    (Problem taken from Arco's AP Exams in Mathematics; Calculus AB and
     Calculus BC, published in 1990)
11.  Given the curve defined by 2x2 + xy - y = 8.
    a. Find dy/dx and d2y/dx2.
    b. Find the equation of the tangent line to the curve at the point where x = 0.
    c. Find the coordinates of those points on the curve where the tangent lines are vertical.
    (Problem taken from Arco's AP Exams in Mathematics; Calculus AB and Calculus BC, published in 1990)
12.  The vertices of a triangle are (0,0), (x, cos x), and (sin3x, 0), where 0 < x < p / 2.
    a. If A(x) represents the area of the triangle, write a formula for A(x).
    b. Find the value of x for which A(x) is a maximum. Justify your answer.
    c. What is the maximum area of the triangle?
    (Problem taken from Arco's AP Exams in Mathematics; Calculus AB and Calculus BC, published in 1990)