MAT 472 โ Spielberg HOMEWORK 10 Due: 11/9/06
Write neatly, not too small, and not too lightly. You may discuss the problems with other
students from class, but you must write your own solutions. Reread your proofs before
copying them out to turn in; I really do mean that you should write (at least) one draft of
each solution.
37. Let f: [0,โ)โRbe continuous, and suppose that LโRsuch that
lim
tโโ
f(t) = L. Prove that lim
tโโ
1
tZt
0
f(x)dx =L.
38. Let f: [a, b]โRbe continuous. Prove that there exists cโ(a, b) such that
Zb
a
f(x)dx =f(c)(bโa).
39. Let f: [a, b]โRbe continuous and non-negative. Prove that
lim
nโโ Zb
a
f(x)ndx!
1
n
=kfksup.
(Hint: First prove it for functions of the form ฮปฯ[c,d].)
40. (i) Prove that if g: [a, b]โRis a step function, then
lim
nโโ Zb
a
g(x) sin(nx)dx = 0.
(ii) Let f: [a, b]โRbe a continuous function, and let ๎ > 0. Prove that there is a step
function g: [a, b]โRsuch that ๎๎f(x)โg(x)๎๎< ๎ for all xโ[a, b]. (Hint: use uniform
continuity.)
(iii) Prove that if f: [a, b]โRis a continuous function, then
lim
nโโ Zb
a
f(x) sin(nx)dx = 0.