What is the correct range for c in Taylor's Theorem error bound?

In summary, the error bound for Taylor's theorem states that for the n+1 derivative evaluated at some c, which maximizes the derivative, c must be between a and x. However, a question arises when calculating the error of using a second degree Taylor polynomial, where the n+1 derivative would be e^x. In this case, the question is whether to use 0 or 1 as the value for c, as absolute value x is 1 and maximizes e^x on the interval [-1,1]. While the textbook and previously learned information suggest using a and x, the teacher insists on using absolute value x and negative absolute value x. This can lead to a weaker estimate, but may be necessary in certain
  • #1
member 508213
For the error bound for taylor's theorem, for the n+1 derivative evaluated at some c which maximizes the derivative my textbook says c must be between a and x..but today my teacher said that c must be between absolute value x and negative absolute value x, which is different than I thought.

An example would be calculating the error of using a second degree taylor polynomial to estimate e^x at x=-1...the n+1 derivative would be e^x, so the question would be do I use 0 because 0 maximizes e^x on [-1,0] or do I use 1 because of absolute value x being 1 and 1 maximizes e^x on [-1,1].

Hopefully my question makes sense, just to reiterate I am wondering if c is between a and x (which is what textbook says and is what I thought in the past) or between absolute value x and negative absolute value x.

Additionally I already tried to talk to my teacher to clarify and he insisted it must be between absolute value x and negative absolute value x...but in the past I learned it was x and a which is confirmed by my book.

Any help on this is appreciated
 
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  • #2
How did you define a and the variable you call x (despite using it as free variable at the same time)?
 
  • #3
mfb said:
How did you define a and the variable you call x (despite using it as free variable at the same time)?
Sorry for not being more clear, a is where the polynomial is centered at and x is where it is being evaluated at... In my example above the polynomial is centered at 0 (forgot to say that sorry) and we are evaluating it at x=-1
 
  • #4
Between a and x then.

If a=0, then "between -|x| and |x|" is a weaker statement. It is not wrong, but sometimes (like here) it leads to a weaker estimate.
 
  • #5
mfb said:
Between a and x then.

If a=0, then "between -|x| and |x|" is a weaker statement. It is not wrong, but sometimes (like here) it leads to a weaker estimate.
Yes thank you that's what I thought, is there ever a situation where you would need to use abs value x to negative abs value x? Because I am trying to figure out why my teacher said that when it seems like a to x works and is more accurate
 
  • #6
Maybe if you want an upper limit for the whole range of the expansion.
 

Related to What is the correct range for c in Taylor's Theorem error bound?

1. What is Taylor's Theorem Error Bound?

Taylor's Theorem Error Bound is a mathematical formula that approximates the error between a given function and its Taylor polynomial approximation. It gives an upper bound on the absolute value of the difference between the actual value of the function and its approximation.

2. How is Taylor's Theorem Error Bound calculated?

Taylor's Theorem Error Bound involves taking the nth derivative of the function and evaluating it at a specific point. This value is then multiplied by the x to the power of n+1, divided by (n+1)! and multiplied by the distance between the given point and the center of the approximation. The resulting value is the error bound.

3. What is the significance of Taylor's Theorem Error Bound?

Taylor's Theorem Error Bound is significant because it allows us to estimate the accuracy of a Taylor polynomial approximation. It helps us determine how close the approximation is to the actual function value and can be used to determine the number of terms needed for a desired level of accuracy.

4. What are the assumptions for using Taylor's Theorem Error Bound?

The main assumption for using Taylor's Theorem Error Bound is that the function must be infinitely differentiable at the given point. This means that all of its derivatives must exist and be continuous at that point. Additionally, the interval of interest must contain the center of the approximation.

5. How can Taylor's Theorem Error Bound be used in real-life applications?

Taylor's Theorem Error Bound has many real-life applications, especially in fields such as engineering, physics, and finance. It can be used to approximate the behavior of a system, determine the accuracy of numerical methods, and evaluate the performance of financial models. It is also used in error analysis and in the development of more efficient algorithms.

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