Extra Credit Problems
Third 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:  December 13, 2007)
  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:

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

  1. Given the function f(x) = x2ln x, 0 < x < 1:

    2.      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)
     
  2. The cooling system in my old truck holds about 10 liters of coolant. Last summer, I flushed
    3. the system by running tap water into a tap-in on the heater hose while the engine was
    running and simultaneously draining the thoroughly mixed fluid from the bottom of the radiator.
    Water flowed in at the same rate that the mixture flowed out -- at about 2 liters per minute.
    The system was initially 50% antifreeze. If we let W be the amount of water in the system
    after t minutes, then it follows that
     

    dW/dt = 2 - 2(W/10)

  a.  Explain why the differential equation above is the correct one.
        b.  Find W as a function of time.
        c.  How long should I have let water run into the system to ensure that 95% of the mixture
             was water?
            (Problem written by Lang Moore -- Duke University)


  1. Let
    2. 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)
  1. Given the two functions f and h such that f(x) = x3 - 3x2 - 4x + 12 and


    2. 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)
  1. Let R be the region bounded by the curves f(x) = 4/x and g(x) = (x - 3)2.
    2. a. Find the area of R.
    b. Find the volume of the solid generated by revolving R about the x-axis.
    (taken from AP Calculus AB Test 1976 -- AB3)
  1. a.  A point moves on the hyperbola 3x2 - y2 = 23 so that its y-coordinate is increasing at
    2.      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)
  1. Given the function defined by f(x) = esin x for all x such that -p< x < 2 p.
    2. 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)

  1. Consider the curve defined by f(x) = x4 - 4x2
    2. 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)
  1. Given the curve defined by 2x2 + xy - y = 8.
    2. 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)
  1. A particle moves along a straight line so that its acceleration at any time t for t > 0 is
    2. a(t) = - 4t.
    a. Find the velocity function, v(t), if v(0) = 32.
    b. Find the distance function, s(t), if s(0) = 10.
    c. Find all times for which the particle is at rest.
    d. Find the average velocity of the particle between t = 2 and t = 5.
    (Problem taken from Arco's AP Exams in Mathematics; Calculus AB and Calculus BC, published in 1990)
  1. The vertices of a triangle are (0,0), (x, cos x), and (sin3x, 0), where 0 < x < p/2.
    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)
  1. Given the function defined by f(x) = 4e - x.
    2. 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)
  1. Given that p and q are constants, q not equal to 0, and that a function g is defined by



    2.  

    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)