Confusion in understanding Taylors theorem.

In summary, Taylors theorem can be used to expand f(a-h) in terms of f(a) and its derivatives over the interval (a-h,a) by defining g(x) = f(2a-x) and applying Taylor's theorem to g. This can be achieved by replacing the coefficients of odd powers of h in the expansion of x+h with their negatives.
  • #1
AAQIB IQBAL
11
0
I know that Taylors theorem is used to expand f(a+h) in terms of f(a) and its derivatives in the interval (a,a+h), but my question is that can we use it to expand f(a-h) in terms of f(a) and its derivatives over the interval (a-h,a).
If yes please give reason.
 
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  • #2
Hi AAQIB IQBAL! :smile:

Define g(x) = f(2a - x), so g(a + h) = f(a - h),

and then apply Taylor's theorem to g :wink:​
 
  • #3
Or- replace the coefficients of odd powers of h, in the expansion of x+h with their negatives.
 

1. What is Taylors theorem?

Taylor's theorem is a mathematical concept that allows us to approximate a function with a polynomial in order to simplify its evaluation.

2. Why is it important to understand Taylors theorem?

Taylor's theorem is important because it allows us to approximate complex functions with simpler polynomials, making it easier to understand and analyze them.

3. What is the difference between Taylor series and Taylor polynomial?

Taylor series is an infinite sum of terms that approximate a function, while Taylor polynomial is a finite sum of terms that approximate a function.

4. How is Taylor's theorem used in real-world applications?

Taylor's theorem is used in many scientific and engineering fields to approximate functions and make predictions. It is commonly used in physics, engineering, and economics.

5. What are the limitations of Taylors theorem?

Taylor's theorem is limited in its accuracy as it can only approximate a function within a certain range of values. It also assumes that the function is infinitely differentiable, which may not always be the case in real-world scenarios.

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