BC 1-2
Problem Set #5
Spring 2012
Name
Due Date: Tuesday, 3/23. (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.
For each of the following conjectures, decide if it is true or false. For the true statements, give
some coherent justification for the statement using complete sentences. For the false statement,
draw or otherwise describe a function f which provides a counterexample, and explain why it does.
For intervals and unions of intervals, always assume that [a, b] implies that a  b and
(a, b] [c, d ) implies that a  b  c  d .
a. Let a function f on [a, b] be given which is differentiable on (a, b) and continuous on [a,b]. If
f ( s)  f (r )
c  (a, b) , then there are r , s  [a, b] such that r < c < s and f (c) 
.
sr
b. Let f be a continuous function on (a, b) with the property that for all r , s  (a, b) , f has a
maximum value on [ r , s ] , then f has a maximum value on [a,b].
c. Suppose that f is differentiable on (a, b) . Then for r , s  (a, b) and a y between f (r ) and f ( s)
there is some c  (r , s ) such that f (c)  y .
d. Suppose that f is defined on [a, b] and satisfies that for all r , s  [a, b] , if y is between
f (r ) and f ( s) there is some c  (a, b) such that f (c)  y . Then f is continuous on [ a, b] .
2.
BC 1-2
Problem Set #5
Spring 2012
Name
Due Date: Tuesday, 3/23. (at beginning of class)
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 the rumor). Find an
dP
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
Problem Set #5
Spring 2012
Name
Due Date: Tuesday, 3/23. (at beginning of class)
 x  cosh t
3. Look at the parametric equations: 
, for t  0 .
 y  sinh t
a. Eliminate the parameter from the equations (to get an equivalent equation in terms of x and y).
b. Sketch the graph (Neatly and accurately).
c. Find all value(s) of t, such that the slope of the tangent line to the graph is m 
5
, then find the
3
corresponding points ( x, y ) . [Sketch the tangent line(s) on the graph in part b.]
4. Prove that if f  is increasing then each tangent line to f intersects the graph of f exactly once.
BC 1-2
Problem Set #5
Spring 2012
Name
Due Date: Tuesday, 3/23. (at beginning of class)
5. Find the derivative of:
f ( x)  22  2x  x x
x
6.
2
2
 2
1
 x sin   x  0
Consider the function given by f ( x)  
x

0
x0

a.
Show that f(x) is continuous at x  0 . (I believe you’ll need a limit here).
b.
Find a formula for the derivative f(x) for all x. (The point at x  0 is of particular interest
here).
c.
Determine whether f(x) is continuous at x = 0. (I will refrain from making parenthetical
remarks here ….oops).