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Let f(x) = ln(2x+1), x = (2,4). 1. The Taylor polynomial T₂ (2) at the point a = 3 is In(7)+(2/7)*(x-3)-(2/49)*(x-3)^ 2. The smallest value of M that occurs in Taylor's inequality is 16/125 3. With M having the above value, Taylor's inequality assures that the error in the approximation f(x) ≈ T2(x) is less than 8/375 for all x € (2,4). 4. If Є (3, 4) the Alternate Series Estimation Theorem assures that the error in the approximation f(x) T₂(2) is less than 16/343 Notice: Your input in 3. and 4. should contain the smallest possible value, as indicated by Taylor Inequality and ASET.

Let f(x) = ln(2x+1), x = (2,4). 1. The Taylor polynomial T₂ (2) at the point a = 3 is In(7)+(2/7)*(x-3)-(2/49)*(x-3)^ 2. The smallest value of M that occurs in Taylor's inequality is 16/125 3. With M having the above value, Taylor's inequality assures that the error in the approximation f(x) ≈ T2(x) is less than 8/375 for all x € (2,4). 4. If Є (3, 4) the Alternate Series Estimation Theorem assures that the error in the approximation f(x) T₂(2) is less than 16/343 Notice: Your input in 3. and 4. should contain the smallest possible value, as indicated by Taylor Inequality and ASET.

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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Let f(x) = ln(2x+1), x = (2,4). 1. The Taylor polynomial T₂ (2) at the point a = 3 is In(7)+(2/7)*(x-3)-(2/49)*(x-3)^ 2. The smallest value of M that occurs in Taylor's inequality is 16/125 3. With M having the above value, Taylor's inequality assures that the error in the approximation f(x) ≈ T2(x) is less than 8/375 for all x € (2,4). 4. If Є (3, 4) the Alternate Series Estimation Theorem assures that the error in the approximation f(x) T₂(2) is less than 16/343 Notice: Your input in 3. and 4. should contain the smallest possible value, as indicated by Taylor Inequality and ASET.
Transcribed Image Text:Let f(x) = ln(2x+1), x = (2,4). 1. The Taylor polynomial T₂ (2) at the point a = 3 is In(7)+(2/7)*(x-3)-(2/49)*(x-3)^ 2. The smallest value of M that occurs in Taylor's inequality is 16/125 3. With M having the above value, Taylor's inequality assures that the error in the approximation f(x) ≈ T2(x) is less than 8/375 for all x € (2,4). 4. If Є (3, 4) the Alternate Series Estimation Theorem assures that the error in the approximation f(x) T₂(2) is less than 16/343 Notice: Your input in 3. and 4. should contain the smallest possible value, as indicated by Taylor Inequality and ASET.
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