BC 1-2
Problem Set #4
Name
Spring 2013
Due Date: Thursday, 14 March
(Though I would appreciate having these at the beginning of class, I will accept them any time before 3:14)
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  ( r , s ) such that f (c)  y . Then f is continuous on [ a, b] .
 x  cosh t
2. 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 
corresponding points ( x, y ) . [Sketch the tangent line(s) on the graph in part b.]
5
, then find the
3
3. Prove that if f  is increasing then each tangent line to f intersects the graph of f exactly once.
4. Find the derivative of:
f ( x)  22  2x  x x
x
2
2