1 / 6

Euler’s Method and Riemann Sums

Euler’s Method and Riemann Sums. 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!!!).

liang
Download Presentation

Euler’s Method and Riemann Sums

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Euler’s Method and Riemann Sums Looking for insight in the special case of antiderivatives

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

  3. 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.

  4. Point of View Dt Dt When our differential equation is of the form Euler’s method is a generalization of the left end-point Riemann sum!

  5. Dt Dt Dt 2 Midpoint Approximations We use this insight to improve on Euler’s method. The midpoint Riemann sum is much more accurate.

  6. Dt Dt 2 Improved Euler’s Method We don’t know the value of the function at the midpoint. We only know the value of the function at the left endpoint. The idea obviously has merit. There’s only one problem . . . But we can approximate the value of the function at the midpoint using the ordinary Euler approximation!

More Related