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

3. Total Change
The sum of the areas of these rectangles is a left Riemann sum for the (signed) area under the graph of f’.
It is also the sum of the Dy’s when Euler’s method finds f !
Adding the Dy’s to the original y0 in Euler’s method, yields the final y-value. (Why?) Thus adding up the little changes in y gives us the total change in f over the interval.
Moreover, the sum of the Dy’s in Euler’s method is an approximation to the area under the graph of 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 also an approximation of the total change in the function f over the interval.
This is the Fundamental Theorem of Calculus

5. Special case of Euler’s method.
Left Riemann Sums are a
Special case of Euler’s method.
Suppose the formula for the derivative of y = f (t) is given in terms of t only. (E.g )
At each stage of Euler’s method, we compute the change in y by multiplying the slope of the function at the (left) point by Dt. Adding these up gives us a left endpoint Riemann sum for f ’ over the interval.