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Euler’s Method allows approximation of functions by numerical antidifferentiation using derivatives. Learn how to compute total change and areas under graphs efficiently.
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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
Point of View Dt Dt Dy = f’(point)*Dt Area = f’(point)*Dt
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.
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].
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.
