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

  • Context: Graduate 
  • Thread starter Thread starter member 508213
  • Start date Start date
  • Tags Tags
    Bound Error Theorem
Click For Summary

Discussion Overview

The discussion revolves around the correct range for the variable c in the error bound of Taylor's Theorem. Participants explore the implications of different definitions of c, particularly in the context of evaluating Taylor polynomials and their derivatives.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant states that their textbook indicates c must be between a and x, while their teacher suggests c should be between negative absolute value x and absolute value x.
  • Another participant questions the definitions of a and x, seeking clarification on their roles in the context of Taylor's Theorem.
  • A participant clarifies that a is the center of the polynomial and x is the point of evaluation, providing an example where a=0 and x=-1.
  • Some participants argue that if a=0, the statement about c being between negative absolute value x and absolute value x is a weaker estimate compared to being between a and x.
  • One participant inquires about scenarios where using absolute values might be necessary, questioning the rationale behind the teacher's assertion.
  • Another participant suggests that using absolute values could be relevant for determining an upper limit for the entire range of the expansion.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate range for c, with no consensus reached on whether it should be between a and x or between negative absolute value x and absolute value x.

Contextual Notes

There are unresolved definitions and assumptions regarding the variables a and x, as well as the implications of using absolute values in the context of Taylor's Theorem.

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
 
Physics news on Phys.org
How did you define a and the variable you call x (despite using it as free variable at the same time)?
 
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
 
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.
 
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
 
Maybe if you want an upper limit for the whole range of the expansion.
 

Similar threads

High School Mean-Value Theorem, Taylor's formula, and error estimation
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Error bounds for Newton's Method
  • · Replies 13 ·
Replies
13
Views
3K
Undergrad What is the difference between these two power series theorems?
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Understanding Theorem 13 from Calculus 7th ed, R. Adams, C. Essex, 4.10
  • · Replies 10 ·
Replies
10
Views
3K
High School Why do I not understand what seems to be basic Calculus?
  • · Replies 11 ·
Replies
11
Views
3K
High School Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?
  • · Replies 19 ·
Replies
19
Views
4K
Undergrad Regarding Error Bound of Taylor Series
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Can someone explain the Taylor's Theorem error bound?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Finding the Largest (or smallest) Value of a Function - given some constant symmetric errors
  • · Replies 3 ·
Replies
3
Views
2K