The graph below shows sections of the relationships y = sin(0) and y = 0. The graph suggests that sin(0) 0 when is small. The shaded section between the two curves serves to highlight the failure of this approximation as increases. 3 π 4 I - 12/22 I IT 4 E DIN I 4 FIN 2 15- 3 TT 4 Ө N Σ n=0 (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 10-00] ≤ 1. f(n) (00) (0-00)". n! (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) 0 when is small. The shaded section between the two curves serves to highlight the failure of this approximation as increases. 3 π 4 I - 12/22 I IT 4 E DIN I 4 FIN 2 15- 3 TT 4 Ө N Σ n=0 (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 10-00] ≤ 1. f(n) (00) (0-00)". n! (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 is small. The shaded section between the two curves serves to highlight the failure
of this approximation as increases.
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KIN
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(i) Consider f(0) = sin(0) and the truncated Maclaurin series SN (0) =>
n=0
+
e
100%
f(n) (00)
n!
(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
10-00| ≤ 1/1.
(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.
when
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