1 / 5

Euler’s Method

Euler's method is a numerical technique used to approximate the antiderivative of a function. By computing the derivative and knowing the value of the function at a specific point, Euler's method constructs an approximation for the function. The sum of these approximations is equivalent to a left Riemann sum approximation for the area under the graph of the derivative. Additionally, adding the approximations to the initial value yields the final value of the function. This method is particularly useful when the derivative formula is given in terms of a single variable.

cgoulet
Download Presentation

Euler’s Method

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 If we have a formula for the derivative of a function and we know the value of the function at one point, Euler’s method lets us build an approximation to the function f. Euler’s method is numerical antidifferentiation. Dt

  2. Point of View Dt Dt Dy = f’(point)*Dt Area = f’(point)*Dt

  3. Total Change The sum of the Dy’s is a left Riemann sum approximation to the (signed) area under the graph of f ’. Furthermore, adding the Dy’s to the original y0 in Euler’s method, yields the final y-value. (Why?) That is, to say, the sum of the Dy’s in Euler’s method is an approximation of the total change in the function f over the entire interval.

  4. The sum of the Dy’s is a left Riemann sum approximation to the (signed) area under the graph of f ’. The sum of the Dy’s in Euler’s method is and approximation of the total change in the function f over the entire interval. The integral of f’ over the interval [a,b] represents both the (signed) area under the graph of f’ and the total change in the function f over [a,b].

  5. Suppose the formula for the derivative of y=f(t) is given in terms of t only. (E.g. y’ = sin(t2).) At each stage of Euler’s method, we compute the change in y by multiplying the slope of function at the (left) point by Dt. This same quantity represents the area of the left Riemann rectangle at the corresponding point on the graph of f ’! Euler’s Method and Riemann Sums Euler’s method computes the total change in f over the interval. The left Riemann Sums of f’ compute the same thing.

More Related