Taylor's Upper Bound: f(x) 2x Diff. Function (0,∞)

In summary, Taylor's Upper Bound is a mathematical concept used to approximate the value of a function at a certain point. It is calculated by finding the maximum value of the higher order derivative of a function on a given interval and multiplying it by the remaining terms of the Taylor polynomial at that point. The function f(x) 2x Diff. Function (0,∞) represents the second derivative of the original function and is important in the calculation of Taylor's Upper Bound. It is often used in mathematics to estimate the error in a Taylor polynomial approximation and to prove the convergence of a Taylor series. However, it is limited to infinitely differentiable functions with bounded higher order derivatives on the given interval and may not always provide an accurate estimation of
  • #1
burak100
33
0
upper bound of taylor!

[itex]f(x)[/itex] is two times diff. function on [itex](0, \infty)[/itex] . [itex]\lim\limits_{x\rightarrow \infty}f(x) = 0[/itex] satisfy.
[itex]M=\sup\limits_{x>0}\vert f^{\prime \prime} (x) \vert[/itex] satisfy
. for each integer [itex]L[/itex] ,
[itex]g(L) = \sup\limits_{x\geq L} \vert f(x) \vert[/itex], and [itex]h(L) = \sup\limits_{x\geq L} \vert f^{\prime}(x) \vert[/itex]. for any [itex] \delta > 0[/itex], SHOW

[itex]h(L) \leq \dfrac{2}{\delta} g(L) + \dfrac{\delta}{2}M[/itex].

please helppppp...
 
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  • #2


So, what did you try already??
 

FAQ: Taylor's Upper Bound: f(x) 2x Diff. Function (0,∞)

1. What is Taylor's Upper Bound?

Taylor's Upper Bound is a mathematical concept that is used to approximate the value of a function at a certain point. It is a method of estimating the maximum difference between a function and its Taylor polynomial at a given point.

2. How is Taylor's Upper Bound calculated?

Taylor's Upper Bound is calculated by finding the maximum value of the absolute value of the higher order derivative of a function on a given interval. This value is then multiplied by the remaining terms of the Taylor polynomial at that point.

3. What is the significance of f(x) 2x Diff. Function (0,∞) in Taylor's Upper Bound?

The function f(x) 2x Diff. Function (0,∞) represents the second derivative of the original function. This is important because Taylor's Upper Bound relies on the higher order derivatives of a function to calculate the approximation.

4. How is Taylor's Upper Bound used?

Taylor's Upper Bound is often used in mathematics to estimate the error in a Taylor polynomial approximation. It can also be used to prove the convergence of a Taylor series for a given function.

5. What are the limitations of Taylor's Upper Bound?

Taylor's Upper Bound is only applicable for functions that are infinitely differentiable. It also requires the higher order derivatives to be bounded on the given interval. Additionally, it is only an estimation and may not always provide an accurate upper bound for the error in a Taylor polynomial approximation.

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