Solve 3rd Taylor Polynomial for (x-1)In(x) About x0=1 - Find Error Bound

In summary, the conversation is about finding the third Taylor polynomial for the function f(x)=(x-1)ln(x) about x0=1 and using p3(0.5) to approximate f(0.5). The conversation also mentions finding the upper bound for error and comparing it to the actual error. One person is struggling with the problem and asks for help, while another person clarifies a mistake in the function notation and provides guidance on how to solve the problem. The conversation ends with a question about finding the actual error.
  • #1
Newbatmath
12
0
Hey guys! I am attempting to do this problem and have been working with it for awhile now. Once again, it is an issue of the textbook not being very clear and making me more confused than ever. Sadly, our teacher is still MIA.

Find the third Taylor polynomial P3(x) for the function f(x)= (x-1)In(x) , about x0 = 1; and use p3(0.5) to approx. f(0.5). Find the upper bound for error /f(0.5) – p3(0.5)/, then compare it to the actual error.

I know this is a tall order but any help you could give me that I could compare to my work here would be such a help! Thanks a bunch!
 
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  • #2
What is the 'In(x)' in your function. Do you mean 'ln(x)'?

Otherwise this question is very straightforward. Just calculate the first 3rd Taylor polynomial and plug in numbers.

I don't see the problem?
 
  • #3
Yep, you are right. ln(x). Sorry about that. :) Thanks so much!
 
  • #4
Okay so I somehow got -4.25. Does anyone else think that is correct?

Anyone know how to find the actual error?
 

1. What is a Taylor polynomial?

A Taylor polynomial is a function that approximates another function near a given point. It is a finite sum of terms that are derived from the derivatives of the function at that point.

2. What is the 3rd Taylor polynomial for (x-1)In(x) about x0=1?

The 3rd Taylor polynomial for (x-1)In(x) about x0=1 is: (x-1) - (x-1)^2 + (x-1)^3. This can be simplified to x - 2x^2 + 3x^3.

3. How do you find the error bound for a Taylor polynomial?

The error bound for a Taylor polynomial can be found using the remainder term, which is given by: Rn(x) = f(n+1)(c)(x-a)^(n+1)/ (n+1)! where n is the degree of the polynomial, c is a number between x and a, and a is the point about which the polynomial is centered. The error bound is then given by the absolute value of the remainder term.

4. What is the error bound for the 3rd Taylor polynomial for (x-1)In(x) about x0=1?

The error bound for the 3rd Taylor polynomial for (x-1)In(x) about x0=1 is given by: R3(x) = f(4)(c)(x-1)^4/ 4! where c is a number between x and 1. The exact value of the error bound will depend on the value of c chosen.

5. How can the error bound be used to determine the accuracy of a Taylor polynomial approximation?

The error bound can be used to determine the maximum possible difference between the actual value of the function and the value obtained using the Taylor polynomial approximation. This allows us to gauge the accuracy of the approximation and make adjustments if needed. The smaller the error bound, the more accurate the approximation will be.

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