MATHEMATICS

1. (20 pts) Solve the
a) Let f : [a;1) ! R be a function such that the improper integral R 1
a
f(x)dx converges. Assume
the existence of the (Önite) limit
limx!1
f(x) = :
Show that  = 0.
b) Assume that f : [a;1) ! R has a continuous Örst derivative and that the improper integrals
R 1
a
f(x)dx and R 1
a
f
0
(x)dx converge. Show that
limx!1
f (x) = 0:
2. (20 pts) Let f : [0; 1] ! R be deÖned by
f(x) = 
x sin

2x


if x 2 (0; 1]
0 if x = 0
Determine whether f is of bounded variation of not.
3. (20 pts) For a given function g : [a; b] ! R, 0 < a < b , compute its total variation Var
[a;b]
(g)
a) g (x) = sin x, where x 2 [a; b].
b) g (x) = x
3
3x
2 + 6x
2, x 2 [a; b].
4. (20 pts) Let f; g : [a; b] ! R be bounded functions and g is increasing (non-decreasing). Show that
for every partitions P and Q of [a; b] with P  Q, we have
s(f; P; g)  s(f; Q; g)  S(f; Q; g)  S(f; P; g):
5. (20 pts) Let f : [0; +1) ! R be deÖned by f(x) = x
x2+1 : Find
lim a!+1
Var
[a;
a]
(f):

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