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Suppose f is a function such that f(n+1)(t) is continuous on an interval containing a and x. Then f) (a) (x – a)i j! f (n+1)(c) (n + 1)! n+1 f(x) – -(z - a)까1 - j=0 where c is some number between a and x.

Suppose f is a function such that f(n+1)(t) is continuous on an interval containing a and x. Then f) (a) (x – a)i j! f (n+1)(c) (n + 1)! n+1 f(x) – -(z - a)까1 - j=0 where c is some number between a and x.

Advanced Engineering Mathematics
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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This gives us f (n+1) (c) I pla+)(t)(x – t)" dt (х — а)"+1. n +1 And the result follows. Problem 8.2.2. Prove Theorem 8.2.1 for the case where x < a. Theorem 8.2.1. Lagrange's Form of the Remainder Suppose f is a function such that f(n+1)(t) is continuous on an interval containing a and x. Then f(1) (a) f (n+1) (c) n f(x) – a)' (x – a)"+1 - j! (n + 1)! where c is some number between a and x. in-context Hint. Note that I fla+1) (t)(x – t)" dt = (-1)"+1 / f(a+)(t)(t – æ)" dt. f(n+1) (t)(t – x)" dt. t=a t=x Use the same argument on this integral. It will work out in the end. Really! You just need to keep track of all of the negatives.
Transcribed Image Text:This gives us f (n+1) (c) I pla+)(t)(x – t)" dt (х — а)"+1. n +1 And the result follows. Problem 8.2.2. Prove Theorem 8.2.1 for the case where x < a. Theorem 8.2.1. Lagrange's Form of the Remainder Suppose f is a function such that f(n+1)(t) is continuous on an interval containing a and x. Then f(1) (a) f (n+1) (c) n f(x) – a)' (x – a)"+1 - j! (n + 1)! where c is some number between a and x. in-context Hint. Note that I fla+1) (t)(x – t)" dt = (-1)"+1 / f(a+)(t)(t – æ)" dt. f(n+1) (t)(t – x)" dt. t=a t=x Use the same argument on this integral. It will work out in the end. Really! You just need to keep track of all of the negatives.
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