Erroneously Finding the Lagrange Error Bound Consider f(x)=sin(5x+pi/4) and let P(x) be the third-degree Taylor polynomial for f about 0. I am asked to find the Lagrange error bound to show that |(f(1/10)−P(1/10))|<1/100. Because P(x) is a third-degree polynomial, I know the difference is in the fourth degree term.

Dalfelli8oy

Dalfelli8oy

Answered question

2022-12-14

Erroneously Finding the Lagrange Error Bound
Consider f ( x ) = sin ( 5 x + π 4 ) and let P(x) be the third-degree Taylor polynomial for f about 0. I
am asked to find the Lagrange error bound to show that |(f(1/10)−P(1/10))|<1/100. Because P(x) is a third-degree polynomial, I know the difference is in the fourth degree term.

Answer & Explanation

Joanna Vang

Joanna Vang

Beginner2022-12-15Added 7 answers

f ( x ) = sin ( 5 x + π 4 )
The third degree Taylor Polynomial is given by:
T 3 ( x ) = 125 x 3 6 2 25 x 2 2 2 + 5 x 2 + 1 2
The error term is given by:
R n + 1 = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x a ) n + 1 M ( n + 1 ) ! ( x a ) n + 1
We have n=3, thus:
d 4 d x 4 ( sin ( 5 x + π 4 ) ) = 625 sin ( 5 x + π 4 )
We have a = 0 , n = 3 , x = 1 10 and the max of sine is 1, so this yields:
R n + 1 M ( n + 1 ) ! ( x a ) n + 1 = 625 4 ! ( 1 10 ) 4 = 1 384

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