Thread: Lagrange Error Bound Question/Problem

My Calculus teacher gave me the following problem:

The function f has derivatives of all orders for all real numbers, and f⁽⁴⁾ (x) = e^sin(x). If the third degree Taylor polynomial for f about x=0 is used to approximate f on the interval [0,1], what is the Lagrange error bound for the maximum error on the interval [0,1]?

My peers and I had a disagreement on the correct answer and I'm curious which one of us is right.

Originally Posted by Xeritas

My Calculus teacher gave me the following problem:

The function f has derivatives of all orders for all real numbers, and f⁽⁴⁾ (x) = e^sin(x). If the third degree Taylor polynomial for f about x=0 is used to approximate f on the interval [0,1], what is the Lagrange error bound for the maximum error on the interval [0,1]?

My peers and I had a disagreement on the correct answer and I'm curious which one of us is right.

so ... what say the both of you w/ regard to the maximum error?

I'm saying the Lagrange error bound should be .113 while the other individual says it should be .097.<--period

We're using the following definition:

|R_n(x)|<= (M/((n+1)!))|x-a|ⁿ⁺¹

for any ...



note that the error depends on the x-value in [0,1] where the function is being approximated.

Originally Posted by skeeter

for any ...



note that the error depends on the x-value in [0,1] where the function is being approximated.

Yes, but the maximum error should occur furthest from the origin, which is at x=1, correct?
And M is referring to the maximum value of the function, which would then make it:

Correct?
Or is the x value within M restricted to [0,1]?

Originally Posted by Xeritas

Yes, but the maximum error should occur furthest from the origin, which is at x=1, correct?
And M is referring to the maximum value of the function, which would then make it:

Correct?
Or is the x value within M restricted to [0,1]?

M is the maximum value of the (n+1)st derivative on the interval [0,1], in this case ... x is the specific value in [0,1] where the function is being approximated.

Ok, I guess I heard wrong from our teacher, thanks again.