Finding Error on Taylor Polynomials using Taylor's Theorem

It can be used to estimate the error in the approximation given above, and it should give an answer of 1/2 * 8^{-7/2}. For the second part, use the Taylor series for sin(x) and the remainder term to find a bound on the difference between sin(x) and x- x^{3}/6 + x^{5}/120 for x in [0,1]. This will help to find a solution for the problem on the sheet. In summary, to estimate the error in using the Taylor Polynomial of f(x)=sqrt{x} of degree 2 to approximate sqrt{8}, use Taylor's Theorem and the formula for the error after the nth term. For the second part, use the Taylor
  • #1
meichberg92
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(a) Use Taylor's Theorem to estimate the error in using the Taylor Polynomial of f(x)=sqrt{x} of degree 2 to approximate sqrt{8}. (The answer should be something like 1/2 * 8^{-7/2}.

(b) Find a bound on the difference of sin(x) and x- x^{3}/6 + x^{5}/120 for x in [0,1]This is a problem on a problem sheet that isn't for homework but there are NO solutions. Any help toward a solution would be appreciated.
 
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  • #2
meichberg92 said:
(a) Use Taylor's Theorem to estimate the error in using the Taylor Polynomial of f(x)=sqrt{x} of degree 2 to approximate sqrt{8}. (The answer should be something like 1/2 * 8^{-7/2}.

(b) Find a bound on the difference of sin(x) and x- x^{3}/6 + x^{5}/120 for x in [0,1]


This is a problem on a problem sheet that isn't for homework but there are NO solutions. Any help toward a solution would be appreciated.

Use the formula for the error in a Taylor series after the nth term. It is widely available; just Google 'Taylors Theorem'.
 

FAQ: Finding Error on Taylor Polynomials using Taylor's Theorem

1. How do I determine the error on a Taylor polynomial using Taylor's Theorem?

To determine the error on a Taylor polynomial using Taylor's Theorem, you will need to first find the value of the remainder term, also known as the error term. This can be done by using the formula for the remainder term in Taylor's Theorem, which is given by R_n(x) = \frac{f^{(n+1)}(z)(x-c)^{n+1}}{(n+1)!}, where c is a value between the center of the polynomial and the point at which you want to find the error, and z is a value between c and x. Once you have found the remainder term, you can then use it to calculate the error by substituting in the appropriate values.

2. Can Taylor's Theorem be used to find the error on any Taylor polynomial?

Yes, Taylor's Theorem can be used to find the error on any Taylor polynomial. However, it is important to note that the theorem only applies when the function is infinitely differentiable at the center of the polynomial and all of its derivatives exist and are continuous in the interval between the center and the point at which you want to find the error.

3. Is it possible for the error on a Taylor polynomial to be zero?

Yes, it is possible for the error on a Taylor polynomial to be zero. This means that the Taylor polynomial is an exact representation of the function and there is no error in using it to approximate the function.

4. How do I know if the error on a Taylor polynomial is significant?

The significance of the error on a Taylor polynomial depends on the context in which it is being used. If the function being approximated is highly sensitive to small changes, then even a small error may be significant. However, if the function is relatively smooth and not highly sensitive to small changes, then a larger error may be acceptable. It is important to consider the context and purpose of the approximation when determining the significance of the error.

5. Can the error on a Taylor polynomial be negative?

No, the error on a Taylor polynomial cannot be negative. This is because the remainder term in Taylor's Theorem is always positive, as it involves taking the absolute value of the n+1st derivative of the function. Therefore, the error on a Taylor polynomial can only be equal to or greater than zero.

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