Thread: Riemann Sum

Is it possible to have a function on the interval [-1,2] so that the Riemann sum using left endpoints is a better approximation than the one using midpoints? What would this be?

I am assuming that it is not possible, but i need an explanation and i am unsure of how to put those ideas on paper

Originally Posted by mistykz

Is it possible to have a function on the interval [-1,2] so that the Riemann sum using left endpoints is a better approximation than the one using midpoints? What would this be?

I am assuming that it is not possible, but i need an explanation and i am unsure of how to put those ideas on paper

I think this is not a good question.

1) Does the interval have anything to do with it?
2) "THE" Riemann Sum?

Given an interval size, it would not be difficult to construct such an animal. It would not be very smooth, I would think. Change the interval size and you must start the construction from the beginning.

Of course, it is trivial to construct an example where the left, mid, right, min, and max are exactly the same.

yes, the interval is part of the question. what would be an example of a function that you could graph like that?

In order to create a Riemann Sum in which the left end point may be more accurate than the midpoint you may want to create an oscillation which under close reveiw would show the mid point underestemating and both the left and right points slightly overestemating. Or you scould try something simple like ex: -(x^3)cos(x)