BC 1-2
Spring 2016
Problem Set #6
Name
Due Date: Tuesday, 4/5. (at beginning of class)
Please show appropriate work – no big calculator leaps – except as indicated. Work should
be shown clearly, using correct mathematical notation. Please show enough work on all
problems (unless specified otherwise) so that others could follow your work and do a similar
problem without help. Collaboration is encouraged, but in the end, the work should be your
own.
1.
(1993 BC #3, NO CALCULATOR allowed). Let  be the function defined by
  x   ln  2  sin  x   for   x  2
a.
Find the absolute maximum value and the absolute minimum value of . Show the
analysis that leads to your conclusions.
b.
Find the x-coordinates of each inflection point on the graph of . Justify your
answer.
BC 1-2
Spring 2016
Problem Set #6
Name
Due Date: Tuesday, 4/5. (at beginning of class)
2. The cost of fuel for a luxury liner is proportional to the square of the speed, v. At a speed of
10 km/hr, fuel costs $3000 per hour. Other fixed costs (e.g., labor) amount to $12,000 per
hour. Assuming that the ship is to make a trip of total distance d, find the speed that
minimizes the cost of the trip. Does this speed depend on the distance?
3. Suppose ABC is a right triangle with right angle at C, Suppose that vertex A is at the
1
origin, vertex B is on the curve y  4e  x ,  x  4 , and vertex C is along the positive x-axis.
2
Find the minimum area of such a triangle.
BC 1-2
Spring 2016
Problem Set #6
Name
Due Date: Tuesday, 4/5. (at beginning of class)
4. Who says you can’t solve Differential Equations?
Let P  P (t )  number of IMSA students who have heard a certain rumor t days after it was
started. Assume that three students know the rumor initially (i.e., P(0)  3 ).
a. Suppose that the rate at which the rumor spreads is proportional to the product of the
number who have heard the rumor with the number who have not heard the rumor. At
this point there are 632 students at IMSA (could be one less depending on the nature of
dP
the rumor). Find an expression for
in terms of P and the proportionality constant, k.
dt
b. If the number of people who have heard the rumor is initially growing at 4 people/day,
find the proportionality constant k.
c. Using the k from part b, find an expression for
dP
d 2P
in terms of
and P .
2
dt
dt
d. Sketch as accurate a graph as possible for P  P (t ) . Label scales if possible.
e. Does this seem to be a reasonable model for how a rumor spreads? What might affect
the value of k? Explain answer convincingly.
BC 1-2
Spring 2016
Problem Set #6
Name
Due Date: Tuesday, 4/5. (at beginning of class)
5. Let f ( x) be continuous on [0, 1] such that f(0) = f(1). Show that there exists at least one
1

c  [0,1] such that f (c)  f  c   .
2
