Let f: [a, b] → R be a bounded function, and let P = {xo,...,n} be a partition of [a, b]. We say a set of points 7:= {C₁,..., Cn} is a tagging of P if x₁-1 ≤ i ≤ xi for all i = 1,..., n. Given any partition P and tagging r of P, show that 71 L(P, f) ≤ f(c₂) Ax¡ ≤U(P, f) i=1 Suppose f: [a, b] →→ R is Riemann integrable. Show that for all e > 0, there exists a partition P such that for any tagging 7, n f-f(c)Ax;
Let f: [a, b] → R be a bounded function, and let P = {xo,...,n} be a partition of [a, b]. We say a set of points 7:= {C₁,..., Cn} is a tagging of P if x₁-1 ≤ i ≤ xi for all i = 1,..., n. Given any partition P and tagging r of P, show that 71 L(P, f) ≤ f(c₂) Ax¡ ≤U(P, f) i=1 Suppose f: [a, b] →→ R is Riemann integrable. Show that for all e > 0, there exists a partition P such that for any tagging 7, n f-f(c)Ax;
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![4. In lecture, we've mentioned how the lower and upper Darboux sums 'bound' possible
Riemann sums. In this problem, let's make that statement more rigorous.
(a) Let f: [a, b] → R be a bounded function, and let P = {xo,...,n} be a partition
of [a, b]. We say a set of points 7 := {₁,..., Cn} is a tagging of P if x₁-1 ≤ Cj ≤ xi
for all i= 1,..., n.
Given any partition P and tagging 7 of P, show that
71
L(P, f) ≤ f(c) Ax¡ ≤ U(P, ƒ)
i=1
(b) Suppose f: [a, b] → R is Riemann integrable. Show that for all e > 0, there exists
a partition P such that for any tagging 7,
T
[1-Monce
-Σ
Δε; <ε
i=1
(c) Given a bounded function f [a, b] → R and n € N, define the right uniform
Riemann sum Rn(f, [a, b]) as
Rn(f, [a, b])
E
f(x):=
72
Σf(xi) Ax
i=1
where Ar= (b-a)/n and x₁ = a + (b-a)(i/n).
Let f: RR be the Dirichlet function, which satisfies
xEQ
{
10x & Q
Show that the sequences {R₂(f, [0, 1])}_₁, {R₂(ƒ, [1, 1+√2])}_₁, and {R₂(f, [0, 1+
√2])} converge, but
lim (R₂ (ƒ, [0, 1]) + R₂(ƒ, [1,1 + √2])) ‡ lim R₁(f, [0,1 + √2])
84x
84x
(Remark: You may use the fact that if r € Q and x & Q, then r + x & Q, and
rx Q if r #0.)
:
(d) Prove or disprove (i.e. by counterexample): A bounded function f [a, b] → R
is Riemann integrable if and only if the sequence of right uniform Riemann sums
{R₂(f, [a, b])} converges as n → ∞o.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97e612ef-1556-436b-b62c-352b280e9e69%2Fd1ea34d2-7e57-4928-9d39-2da29f8021ba%2Fk97oieo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. In lecture, we've mentioned how the lower and upper Darboux sums 'bound' possible
Riemann sums. In this problem, let's make that statement more rigorous.
(a) Let f: [a, b] → R be a bounded function, and let P = {xo,...,n} be a partition
of [a, b]. We say a set of points 7 := {₁,..., Cn} is a tagging of P if x₁-1 ≤ Cj ≤ xi
for all i= 1,..., n.
Given any partition P and tagging 7 of P, show that
71
L(P, f) ≤ f(c) Ax¡ ≤ U(P, ƒ)
i=1
(b) Suppose f: [a, b] → R is Riemann integrable. Show that for all e > 0, there exists
a partition P such that for any tagging 7,
T
[1-Monce
-Σ
Δε; <ε
i=1
(c) Given a bounded function f [a, b] → R and n € N, define the right uniform
Riemann sum Rn(f, [a, b]) as
Rn(f, [a, b])
E
f(x):=
72
Σf(xi) Ax
i=1
where Ar= (b-a)/n and x₁ = a + (b-a)(i/n).
Let f: RR be the Dirichlet function, which satisfies
xEQ
{
10x & Q
Show that the sequences {R₂(f, [0, 1])}_₁, {R₂(ƒ, [1, 1+√2])}_₁, and {R₂(f, [0, 1+
√2])} converge, but
lim (R₂ (ƒ, [0, 1]) + R₂(ƒ, [1,1 + √2])) ‡ lim R₁(f, [0,1 + √2])
84x
84x
(Remark: You may use the fact that if r € Q and x & Q, then r + x & Q, and
rx Q if r #0.)
:
(d) Prove or disprove (i.e. by counterexample): A bounded function f [a, b] → R
is Riemann integrable if and only if the sequence of right uniform Riemann sums
{R₂(f, [a, b])} converges as n → ∞o.
![Darboux Integral
Def A partition P of an interval [9,6] is a finite set of real numbers
2x0, x₁,...,x₁² such that
a=Xo < x₁ < x₂ < < x₁ = 6
we write
Axi = xi-xi-1
Def let fila, b1-R be a bounded function, and let P be a partition
of [a,b]. Define
m₁ := inf {f(x)= xe [xi-1, ^;]}
Misụp {f(x): XE LX, x3
L(P, f):=
U (P₁ f) :=
MAX: "lower Darboux sum"
M₁0x₁ & "upper Parboux sum
6=1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97e612ef-1556-436b-b62c-352b280e9e69%2Fd1ea34d2-7e57-4928-9d39-2da29f8021ba%2F4fd60by_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Darboux Integral
Def A partition P of an interval [9,6] is a finite set of real numbers
2x0, x₁,...,x₁² such that
a=Xo < x₁ < x₂ < < x₁ = 6
we write
Axi = xi-xi-1
Def let fila, b1-R be a bounded function, and let P be a partition
of [a,b]. Define
m₁ := inf {f(x)= xe [xi-1, ^;]}
Misụp {f(x): XE LX, x3
L(P, f):=
U (P₁ f) :=
MAX: "lower Darboux sum"
M₁0x₁ & "upper Parboux sum
6=1
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