Math 160
Fall 1999
Exam 3
1.
(20
pt) Evaluate the following limits
a) 
b) 
c) 
d) 
e) 
2.
(20
pt) Show that the area of the largest rectangle that can be inscribed in an
isosceles triangle has exactly half the area of the triangle. Hint: consider
the following picture to get started.
(0,h)
(0,0) (a,0)
3.
(20
pt) A square piece of cardboard (say each side has length a) has equal squares
cut out of the corners and is folded up to make a box. How do you choose x so
that the resulting volume is a maximum?
x
x
4.
(20
pt) A large can is to hold a volume of 
cubic meters of material. Find the dimensions of the can that
minimize the cost if making the top and bottom costs 4 cents per square meter
and the cost of the side is 2 cents per square meter.
5.
(20
pt) Sketch the graph of the function 
. For your convenience, the derivatives are: 
and 
6. (10 pt) a) Use Newton’s method with initial
approximation 
to estimate 
(find
).
b) Assume that 
and that you want to use Newton’s method to find 
. Show that if you begin with a “reasonable approximation” 
, then all of the other approximations are greater than or
equal to 
. (Hint: Draw a picture).