3. Let I = 0 arctan(r)dr. X (a) Find the Taylor polynomial of order 2, P2(x), about x = 0 for the function arctan(z). (b) Use Lagrange's formula for the remainder R₂(x) = arctan(x) - P₂(x) to show that - S² P²2(2) dx = = x arctan(r) S X -dx- (c) Hence calculate I with an error up to 1.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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3. Let
I =
[arctan (r)
-dx.
(a) Find the Taylor polynomial of order 2, P2(x), about x = 0 for the function
arctan(r).
(b) Use Lagrange's formula for the remainder R₂(x) = arctan(x) - P₂(x) to show that
S
¹ arctan(x) dx - P(x) dx ≤ 1
<
X
x
9
(c) Hence calculate I with an error up to 3.
Transcribed Image Text:3. Let I = [arctan (r) -dx. (a) Find the Taylor polynomial of order 2, P2(x), about x = 0 for the function arctan(r). (b) Use Lagrange's formula for the remainder R₂(x) = arctan(x) - P₂(x) to show that S ¹ arctan(x) dx - P(x) dx ≤ 1 < X x 9 (c) Hence calculate I with an error up to 3.
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