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 xi-1 ≤ c ≤ 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) S a i=1 Suppose ƒ : [a, b] → R is Riemann integrable. Show that for all e > 0, there exists a partition P such that for any tagging 7, 71 f-f(c)A i=1 f(0,)Az,| < E : Given a bounded function f [a, b] → R and n € N, define the right unifo Riemann sum R₂(f, [a, b]) as Rn(f, [a, b]) := f(xi)▲x f(x):= 71 where Ar= (b-a)/n and x₁ = a + (b − a)(i/n). Let f: R→ R be the Dirichlet function, which satisfies i=1 20 (1 TEQ x & Q 0x Show that the sequences {R₂(f, [0, 1])}_₁, {R₂(ƒ, [1, 1+√√2])}_₁, and {R₂(ƒ, [0, 1+ √2])} converge, but lim (R₂(ƒ, [0, 1]) + R„(ƒ, [1,1 + √2])) ‡ lim R₂(ƒ, [0,1 + √2]) 888 848 (Remark: You may use the fact that if r Q and x Q, then r + x Q, and rx & Q if r ‡0.)
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 xi-1 ≤ c ≤ 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) S a i=1 Suppose ƒ : [a, b] → R is Riemann integrable. Show that for all e > 0, there exists a partition P such that for any tagging 7, 71 f-f(c)A i=1 f(0,)Az,| < E : Given a bounded function f [a, b] → R and n € N, define the right unifo Riemann sum R₂(f, [a, b]) as Rn(f, [a, b]) := f(xi)▲x f(x):= 71 where Ar= (b-a)/n and x₁ = a + (b − a)(i/n). Let f: R→ R be the Dirichlet function, which satisfies i=1 20 (1 TEQ x & Q 0x Show that the sequences {R₂(f, [0, 1])}_₁, {R₂(ƒ, [1, 1+√√2])}_₁, and {R₂(ƒ, [0, 1+ √2])} converge, but lim (R₂(ƒ, [0, 1]) + R„(ƒ, [1,1 + √2])) ‡ lim R₂(ƒ, [0,1 + √2]) 888 848 (Remark: You may use the fact that if r Q and x Q, then r + x Q, and rx & Q if r ‡0.)
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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