Error bounds with approximated M value

[elemental_d]

How do you approximate the value 'M' for the error bound formula for the Trapezoid Rule of a function that the derivative cannot be found?

Error Bound Formula: http://archives.math.utk.edu/visual.calculus/4/approx.2/index.html

I'm trying to figure out the value of n to get erf(1.00) approximated within the error of 0.001?

Error Function: http://en.wikipedia.org/wiki/Error_function

How do i get M(in the error bound formula) when it is not possible to get the second derivative of the error function?

 

[DeadWolfe]

But it IS possible to get the second derivative of erf.

 

[tacman]

Approximating the value of 'M' for the error bound formula for the Trapezoid Rule can be a challenging task when the derivative of the function cannot be found. In this case, there are a few methods that can be used to approximate 'M' and ensure that the error bound is within the desired range.

One approach is to use numerical methods such as Taylor series or power series to approximate the function and its derivatives. This can provide a good estimate for 'M' and help in calculating the error bound. However, this method can be time-consuming and may not always provide an accurate result.

Another method is to use a computer program or software that can handle symbolic calculations. This can help in finding the exact value of 'M' by evaluating the function and its derivatives at a given point. However, this method may require some knowledge of programming and may not be accessible to everyone.

If the above methods are not feasible, one can also use a trial and error approach by choosing different values of 'M' and calculating the error bound until the desired accuracy is achieved. This may not be the most efficient method, but it can provide a reasonable estimate for 'M'.

In the case of the error function, which does not have a closed form for its second derivative, the above methods can still be applied. However, it is important to note that the error bound formula may not provide an accurate result in this case. It is always recommended to use a combination of different methods to approximate 'M' and ensure that the error bound is within the desired range.