Let [a, b] and [c, d] be intervals satisfying [c, d] C [a, b]. Show that if f E R(a) over [a, b] then ƒ € R(a)over [c, d].

Answered: Let [a, b] and [c, d] be intervals…

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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Question

Let [a, b] and [c, d] be intervals satisfying [c, d] C [a, b]. Show that if f E R(a) over [a, b] then ƒ € R(a)
over [c, d].

Expert Solution

Step 1

Given that $[a, b]$ and $[c, d]$ are intervals satisfying

$[c, d] \subset [a, b]$

we have to show that if $f \in R (α)$ over $[a, b]$ then $f \in R (α)$ over $[c, d]$

now,

$\begin{array}{rcl} α & \in & [c, d] \\ α & \in & [a, b]; [c, d] \subset [a, b] \\ f & \in & R (α) o v e r [a, b] t h e n \\ f & \in & R (α) o v e r [c, d] \end{array}$

Again,

$\begin{array}{rcl} α & \in & [a, b] \\ α & \in & [c, d]; [c, d] \supset [a, b] \\ f & \in & R (α) o v e r [c, d] t h e n \\ f & \in & R (α) o v e r [a, b] \end{array}$

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