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(a) Use the first three terms of Taylor polynomial to predict f(3) for f(x) = vx² +1+ x using a = 1. (b) Determine the error bound of using the first three terms of Taylor Polynomial determined in part (a), the range for x is 0

(a) Use the first three terms of Taylor polynomial to predict f(3) for f(x) = vx² +1+ x using a = 1. (b) Determine the error bound of using the first three terms of Taylor Polynomial determined in part (a), the range for x is 0

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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The square root is over the (x^2+1)
(a) Use the first three terms of Taylor polynomial to predict f(3) for f(x) = Vx² + 1 + x using a = 1. (b) Determine the error bound of using the first three terms of Taylor Polynomial determined in part (a), the range for x is 0 < x< 4.
Transcribed Image Text:(a) Use the first three terms of Taylor polynomial to predict f(3) for f(x) = Vx² + 1 + x using a = 1. (b) Determine the error bound of using the first three terms of Taylor Polynomial determined in part (a), the range for x is 0 < x< 4.
Expert Solution
Step 1

Let us consider a function f(x)  then the Taylor series expansion of the function around x=a is given by 

f(x)≈P(x)=f(a)0!(x-a)0+f'(a)1!(x-a)1+f''(a)2!(x-a)2+------+f'''(a)n!(x-a)n+Rn+1

The error  value of the function is obtained   as 

If En(x)=f(x)-Pn(x) be the error using the approximate polynomial expansion P(x) then  

|En(x)|≤M(n+1)!(x-a)n+1 ≤M(n+1)!(x-a)n+1

where M is determined as f(n+1)(t)≤M for all t in the interval of x 

Here, n is number of terms required 

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