Enter the maximum value of the (n+1)th derivative on the interval and the radius to determine the Taylor inequality error. This calculator helps to estimate the error bound of a Taylor polynomial approximation.
Taylor Inequality Error Formula
The following formula is used to calculate the Taylor inequality error.
E = frac{|f^{(n+1)}(x)| cdot r^{n+1}}{(n+1)!}
Variables:
- E is the error of the Taylor polynomial approximation
- |f^(n+1)(x)| is the maximum value of the (n+1)th derivative on the interval
- r is the radius of the interval around the point of approximation
- (n+1)! is the factorial of n+1
To calculate the Taylor inequality error, multiply the maximum value of the (n+1)th derivative by the radius raised to the power of n+1, and then divide by the factorial of n+1.
What is Taylor Inequality Error?
Taylor inequality error is an estimation of the error made when using a Taylor polynomial to approximate a function. It is based on the remainder term of Taylor’s theorem, which provides a bound on the error. This error bound is useful for determining the accuracy of the approximation and for deciding how many terms of the Taylor series are needed to achieve a desired level of precision.
How to Calculate Taylor Inequality Error?
The following steps outline how to calculate the Taylor Inequality Error.
- First, determine the maximum value of the (n+1)th derivative on the interval |f^(n+1)(x)|.
- Next, determine the radius (r) of the interval around the point of approximation.
- Next, gather the formula from above = E = |f^(n+1)(x)| * r^(n+1) / (n+1)!.
- Finally, calculate the Taylor Inequality Error (E).
- After inserting the variables and calculating the result, check your answer with the calculator above.
Example Problem :
Use the following variables as an example problem to test your knowledge.
Maximum value of the (n+1)th derivative on the interval |f^(n+1)(x)| = 24
Radius (r) = 0.5
Assuming n = 3 for this example, calculate the Taylor Inequality Error (E).
