Math 165
Fall 2002
Exam 3
- (20 pt) Find the
following limits:
a) 
where: 
b) 
c) 
d) 
- (20 pt) Sketch the
graph of the function
. The first two derivatives of this are 
and 
- (20 pt) Sketch the
graph of the function
The first two
derivatives of this are 
and 
- (20 pt) A silo is to be
built by putting a hemisphere on top of a cylinder (no floor is to be
constructed). If the top (hemispherical dome) costs twice as much to
produce per square foot as the side, and the silo is to have a volume of
72p ft3, how
should the silo be constructed to minimize the cost?
- (20 pt) For this
problem, we consider the right half of the parabola
For each value
of 
we form a triangle bounded by the line 
, the 
axis, and the tangent line to 
at the point 
What is the minimal area that can be formed this way?
- (10 pt) Suppose that
is always decreasing (you can assume that 
).
a)
Explain
why if you want to maximize 
on some interval, you should minimize 
on this interval (you don’t need, but may assume, that 
is differentiable).
b)
Use
the result from part a) to find the 
value where 
attains its maximum
value (hint: what kind of function is 
)?