Advanced Calculus Assignment 4

1. Advanced Calculus, Assignment 4
1. Let f : [0, 1] → R be defined by f(x) = x3
.
(a) Let P = {0, 0.1, 0.4, 1}. Compute L(P, f) and U(P, f).
(b) Show that f ∈ R([0, 1]) and compute
1
0
f using the definition of the
integral.
2. Let f : [a, b] → R be a bounded function. Suppose there exists a sequence
of partitions {Pk} of [a, b] such that
lim
k→∞
(U(Pk, f) − L(Pk, f)) = 0.
Show that f is Riemann integrable and that
b
a
f = lim
k→∞
U(Pk, f) = lim
k→∞
L(Pk, f).
3. (a) Let f, g ∈ R([a, b]). Prove that f + g is in R([a, b]) and
b
a
(f + g)(x)dx =
b
a
f(x)dx +
b
a
g(x)dx.
Hint: Find a single partition P such that
U(P, f) − L(P, f) <
ε
2
and U(P, g) − L(P, g) <
ε
2
.
(b) Let f ∈ R([a, b]). Prove that −f is in R([a, b]) and
b
a
−f(x)dx = −
b
a
f(x)dx.
4. (a) Prove the Mean Value Theorem for integrals. That is, prove that if
f : [a, b] → R is continuous, then there exists a c ∈ [a, b] such that
b
a
f(x)dx = f(c)(b − a).
(b) If g : [a, b] → R and h : [a, b] → R are continuous functions such
that
b
a
g =
b
a
h. Then show that there exists a c ∈ [a, b] such that
h(c) = g(c).
5. If f : [a, b] → R is a continuous function such that f(x) ≥ 0 for all x ∈ [a, b]
and
b
a
f = 0. Prove that f(x) = 0 for all x.
6. Let f : [a, b] → R be increasing.
(a) Show that f is Riemann integrable.
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2. (b) Use part (a) to show that a decreasing function is Riemann integrable.
7. (a) Compute
d
dx
x
−x
es2
ds
(b) Compute
d
dx
x2
0
sin(s2
)ds
8. Let f : [a, b] → R be a continuous function. Let c ∈ [a, b] be arbitrary.
Define
F(x) :=
x
c
f
Prove that F is differentiable and that F (x) = f(x) for all x ∈ [a, b].
9. Prove integration by parts. That is, suppose F and G are continuously
differentiable functions on [a, b]. Then prove
b
a
F(x)G (x) dx = F(b)G(b) − F(a)G(a) −
b
a
F (x)G(x) dx
10. Suppose f : [a, b] → R is continuous and
x
a
f =
b
x
f for all x ∈ [a, b].
Show that f(x) = 0 for all x ∈ [a, b].
11. Let y be any real number and b > 0. Define f : (0, ∞) → R and g : R → R
as f(x) = xy
and g(x) = bx
. Show that f and g are differentiable and find
their derivatives.
12. (a) Show that
ex
= lim
n→∞
1 +
x
n
n
.
(b) Use the logarithm to find limn→∞ n1/n
.
13. Can you interpret
1
−1
1
|x|
dx
as an improper integral? If so, compute its value.
14. Take f : [0, ∞) → R, Riemann integrable on every interval [0, b], b > 0,
and such that there exist M, a, and T such that |f(t)| ≤ Meat
for all
t ≥ T. Show that the Laplace transform of f exists. That is, for every
s > a the following integral converges:
F(s) =
∞
0
f(t)e−st
dt.
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3. 15. (a) Suppose f : [0, ∞) → R is nonnegative and decreasing
i. Show that if
∞
0
f < ∞, then limx→∞ f(x) = 0.
ii. Show that the converse does not hold.
(b) Find an example of an unbounded continuous function f : [0, ∞) → R
that is nonnegative and such that
∞
0
f < ∞. (Note that this means
limx→∞ f(x) does not exist.)
(Hint: on each interval [k, k + 1], k ∈ N, define a function whose
integral over this interval is less than say 2−k
).
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