Looking for insight in the special case of antiderivatives

Turning Corners (or Not!!!) Euler’s method is very bad at turning corners. Think about a solution curve like this one...

Turning Corners (or Not!!!) Euler’s method is very bad at turning corners. When the curve nears a maximum, Euler’s method “overshoots.” Likewise, when the curve nears a minimum, Euler’s method drops too far.

Point of View tt tt When our differential equation is of the form Euler’s method is a generalization of the left end-point Riemann sum!

Midpoint Approximations tt The midpoint Riemann sum is much more accurate.  tt tt We use this insight to improve on Euler’s method.

Improved Euler’s Method The idea obviously has merit. There’s only one problem... We don’t know the value of the function at the midpoint. We only know the value of the function at the left endpoint. But we can approximate the value of the function at the midpoint using the ordinary Euler approximation!  tt tt