Let F(x) E C'[0, 1] and F(x) be twice differentiable in (0, 1). Suppose that f(x) = F'(x) E R[0,1] and that |F"(x)| < M for all æ E (0, 1). Show that for some constant K > 0, i – 1 < Kn-1. i=1
Let F(x) E C'[0, 1] and F(x) be twice differentiable in (0, 1). Suppose that f(x) = F'(x) E R[0,1] and that |F"(x)| < M for all æ E (0, 1). Show that for some constant K > 0, i – 1 < Kn-1. i=1
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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![Let F(x) E C'[0, 1] and F(x) be twice differentiable in (0, 1). Suppose that f(x) = F'(x) E R[0, 1] and that
|F"(x)| < M for all x E (0,1). Show that for some constant K > 0,
n
< Kn-1.
Hint: apply Taylor expansion with the Lagrange Remainder on each interval [(i – 1)/n, i/n].
Remark 1: This gives the speed of convergence of the left Riemann sum.
Remark 2: We will learn that continuous functions are always Riemann-integrable so the assumption f E R[0, 1] is
unnecessary.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1de81170-e7cf-4e92-bed5-a7f0f5dfddb4%2F65a07068-dd60-495a-b12e-dc8488dba3c7%2Fgr5wpr6_processed.png&w=3840&q=75)
Transcribed Image Text:Let F(x) E C'[0, 1] and F(x) be twice differentiable in (0, 1). Suppose that f(x) = F'(x) E R[0, 1] and that
|F"(x)| < M for all x E (0,1). Show that for some constant K > 0,
n
< Kn-1.
Hint: apply Taylor expansion with the Lagrange Remainder on each interval [(i – 1)/n, i/n].
Remark 1: This gives the speed of convergence of the left Riemann sum.
Remark 2: We will learn that continuous functions are always Riemann-integrable so the assumption f E R[0, 1] is
unnecessary.
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