BC 1-2
Spring 2012
Problem Set #6
Name
Due Date: Monday, 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 for any n  Z  there exists
1

at least one c  [0,1] such that f (c)  f  c   .
n

BC 1-2
Spring 2012
3.
Name
Due Date: Monday, 4/23. (at beginning of class)
Differentiate each function. Don’t even think of simplifying or using the TI-89!
 sec x 4 
a.
f ( x)  tan 1  e





 
b.
4.
Problem Set #6
g ( x)  x  x 2  x3  1
Show that the graphs of the curves x2 + 4xy + y2 = 1 and x2 – xy + y2 = 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 2012
5.
Problem Set #6
Name
Due Date: Monday, 4/23. (at beginning of class)
A tightrope is stretched 30 feet above the ground between
the Em and the Saw buildings, which are 50 feet apart. A
tightrope walker, walking at a constant rate of 2 feet per
second from point A to point B, is illuminated by a
spotlight 70 feet above point A, as shown in the diagram at
the right.
a.
How fast is the shadow of the tightrope walker’s
feet moving along the ground when she is halfway
between the buildings?
Em
Saw
50 ft
70 ft
A
B
30 ft
ground
b.
6.
How fast is the shadow of the tightrope walker’s feet moving up the wall of the Saw
building when she is 10 feet from point B?
ax 4  bx 3  1
 c.
Determine values for a, b, and c so that lim
x 1
x 1 sin x

BC 1-2
Spring 2012
Problem Set #6
Name
Due Date: Monday, 4/23. (at beginning of class)
7. 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
8.
3
(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.