The graph below shows sections of the relationships y = sin(0) and y = 0. The graph suggests that sin(0) when is small. The shaded section between the two curves serves to highlight the failure of this approximation as increases. -37-4 플 л I π 4 I 4 Д 4 KIN Д 2 3 π 4 N f(n) (00) n! (i) Consider f(0) = sin(0) and the truncated Maclaurin series SN (0) = Σ n=0 -(0-00)". Show that S₁ (0) and S₂(0) give the approximation sin(0) ≈ 0. (ii) For the approximation S₂(0), use Taylor's Inequality to calculate the error bound on the interval |0-00 ≤ (iii) If we want to increase the number of terms in our Maclaurin series approximation to include 05 whilst ensuring a maximum absolute error bound of 0.02, determine the corresponding interval of convergence.
The graph below shows sections of the relationships y = sin(0) and y = 0. The graph suggests that sin(0) when is small. The shaded section between the two curves serves to highlight the failure of this approximation as increases. -37-4 플 л I π 4 I 4 Д 4 KIN Д 2 3 π 4 N f(n) (00) n! (i) Consider f(0) = sin(0) and the truncated Maclaurin series SN (0) = Σ n=0 -(0-00)". Show that S₁ (0) and S₂(0) give the approximation sin(0) ≈ 0. (ii) For the approximation S₂(0), use Taylor's Inequality to calculate the error bound on the interval |0-00 ≤ (iii) If we want to increase the number of terms in our Maclaurin series approximation to include 05 whilst ensuring a maximum absolute error bound of 0.02, determine the corresponding interval of convergence.
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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The graph below shows sections of the relationships y sin(0) and y 0. The graph suggests that
sin(0) 0 when 0 is small. The shaded section between the two curves serves to highlight the failure
of this approximation as increases.
3π
4
π
4
A
I
2
4
4
I
KI+
KIN
2
3 π
4
N
f(n) (00) (0 - 00)".
n!
(i) Consider f(0) = sin(0) and the truncated Maclaurin series SN (0):
Show that S₁ (0) and S₂(0) give the approximation sin(0) ≈ 0.
(ii) For the approximation S₂(0), use Taylor's Inequality to calculate the error bound on the interval
|0-00| ≤7.
n=0
(iii) If we want to increase the number of terms in our Maclaurin series approximation to include 05
whilst ensuring a maximum absolute error bound of 0.02, determine the corresponding interval
of convergence.
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