Assignment 3 of Math 5302
1. Let
f(t) =
0 if t < 0;
t if 0 ≤ t ≤ 1;
4 if t > 1.
Let F(x) = R x
0
f(t)dt.
(a) Find F(x).
(b) Where is F continuous?
(c) Where is F differentiable? Calculate F
0
at the points of differentiability.
2. Let f be a continuous function on R and define
F(x) = Z sin x
0
f(t)dt for x ∈ R.
Show that F is differentiable on R and compute F
0
.
3. Let
F(x) =
x
2
sin( 1
x2 ) if 0 < x ≤ 1;
0 if x = 0.
(a) Show that F has a derivative at every x ∈ [0, 1].
(b) Show that F
0
is not Riemann integrable on [0, 1]. (So F is not the integral of its derivative.)
4. Show that for each p > 0,
R ∞
1
sin x
xp dx converges. Hint: For 0 < p < 1, you may find it helpful to use
integration by parts.
5. Consider R ∞
1
x
p
1+xq dx.
(a) For what values of p and q are the integral convergent?
(b) For what values of p and q are the integral absolutely convergent?
1
