mearing
How do I determine if an approximated midpoint or trapezoidal sum is an upper or lower bound?
Example: Exam 1, Problem 4-C.
An archived instance of a Calc II forum
Upper or lower error of an approximating sum
mark
These are tied to the second derivative. In particular, a trapezoidal sum over an interval I is an upper bound whenever f''(x)>0 for all x\in I but it’s a lower bound if f''(x)<0 for all x\in I. This is pretty easy to remember, if you just remember the following picture:
Midpoints sums are the reverse.