BC 1-2
Spring 2013
Problem Set #6
Name
Due Date: Tuesday, 4/23. (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. 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. 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

BC 1-2
Spring 2013
3.
Problem Set #6
Name
Due Date: Tuesday, 4/23. (at beginning of class)
Differentiate each function. Don’t even think of simplifying or using the TI-89!
 sec x 4 
tan 1  e





f ( x) 
 
x  x 2  x3  1
4.
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 2013
5.
Problem Set #6
Name
Due Date: Tuesday, 4/23. (at beginning of class)
 ax 4  bx 3  1 
Determine values for a, b, and c so that lim 
c.
x 1  x  1  sin   x 


6. 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
BC 1-2
Spring 2013
7.
Problem Set #6
Name
Due Date: Tuesday, 4/23. (at beginning of class)
(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.