BC 1-2
Spring 2016
Problem Set #4
Name
Due Date: Friday, 3/18 (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. Let f ( x)  x  g ( x) where g is a continuous function defined on [1,1] . Prove that f is
differentiable at x  0 and find f (0) in terms of g. (The hardest part of this problem will
be writing all of the details very carefully. Justify all your equalities.)
2.
.
Show that the graphs of the curves x 2  4 xy  y 2  1 and x 2  xy  y 2  1 intersect at
right angles. (Hint: For graphs to intersect at right angles, think about what must occur
at the point(s) of intersection of the graphs?)
BC 1-2
Spring 2016
Problem Set #4
Name
Due Date: Friday, 3/18 (at beginning of class)
3. For each function f(x) below, find a function F(x) so that dF/dx = f(x). Explain briefly.
a.
f(x) = 2 cos(2x)
b.
f(x) = cos(2x) – 2x sin(2x) (Think product rule)
c.
f(x) = x sin(2x) (Combine the previous parts)
d.
f ( x) 
3x 2
1  x 
3
3
 2
1
 x sin  
4. In this problem we investigate the behavior of the function f ( x)  
 x

0

near x  0 .
a. Find the derivative of f for x  0 .
if x  0
if x  0
b. Find f (0) using the limit definition.


c. We now have a piece-wise definition of f ( x)  


d. Is f  continuous at x  0 ? Explain.
if x  0
if x  0
.
BC 1-2
Spring 2016
Problem Set #4
Name
Due Date: Friday, 3/18 (at beginning of class)
5.
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
f ( s)  f (r )
[a,b]. If 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] .