Math 96:
Homework 3
Fall 2016
This homework is due in class on Friday, October 14th. Please complete one
of the following problems.
1979 B1 Prove or disprove: there is at least one straight line normal to the
graph of y = cosh(x) at a point (a, cosh(a)) and normal to the graph of y =
sinh(x) at a point (c, sinh(c)). [cosh(x) = (ex + e−x )/2, and sinh(x) = (ex −
e−x )/2.]
1981 A3 Find
Z tZ
lim e−t
t→∞
0
t
0
ex − ey
dxdy
x−y
or show that the limit doesn’t exist.
1982 B3 Let pn be the probability that c+d is a perfect square when c and d √
are
selected independently at random from {1,
2,
.
.
.
,
n}.
Show
that
lim
p
n→∞ n n
√
exists and express the limit in the form r( s − t) where s and t are integers and
r is a rational number.
1