Post on 08-Jul-2018
8/19/2019 01_19_06h
1/1
Problem Session 2, January 19, 2006
Suppose m ≤ f (x) ≤ M on [a, b ] and that α is increasing on [a, b ]. Provethat
m (α (b) − α (a )) ≤ b
a
f dα ≤ M (α (b) − α (a ))
If f is continuous, nonconstant, and α is strictly increasing, prove that theinequality is strict.
June 2003: Let f be a nonnegative, continuous function on [ a, b ] and let αbe strictly increasing on [ a, b ]. Show that if
b
a f dα = 0, then f = 0 on [a, b ].
June 2003: Let α be given by:
α (x) =0; 0 ≤ x < 12; 1 ≤ x < e5; e ≤ x ≤ π
Either directly or by aid of a theorem, calculate the value of the integral
π
0
x 100 dα (x), and show all the details in your calculation.
June 2003: Let f : [− 1, 1] → R be a bounded function. Let α be given by:α (x) = 0 if x ≤ 0, and α(x) = 1 if x > 0. Prove that f ∈ R (α )[− 1, 1] if andonly if f (0+) = f (0). In this case show that
1
− 1 f dα = f (0).
Jan 2001 Dene g(x) =− 1; x = 0x ; 0 < x < 12 x = 1
Compute 1
0
1x + 3
dg(x).
1