Euler

1
Eulers Method
If we have a formula for the derivative of a
function and we know the value of the function at
one point, Eulers method lets us build an
approximation to the function f.
Eulers method is numerical antidifferentiation.
Dt
2
Point of View
Dt
Dt
Area f(point)Dt
Dy f(point)Dt
3
Total Change
The sum of the Dys is a left Riemann sum
approximation to the (signed) area under the
graph of f .
Furthermore, adding the Dys to the original y0
in Eulers method, yields the final y-value.
(Why?)
That is, to say, the sum of the Dys in Eulers
method is an approximation of the total change in
the function f over the entire interval.
4
The sum of the Dys is a left Riemann sum
approximation to the (signed) area under the
graph of f .
The sum of the Dys in Eulers 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 yf(t)
is given in terms of t only. (E.g. y
sin(t2).) At each stage of Eulers 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 !
Eulers Method and Riemann Sums
Eulers method computes the total change in f
over the interval. The left Riemann Sums of f
compute the same thing.