Title: Euler
1Eulers 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
2Point of View
Dt
Dt
Area f(point)Dt
Dy f(point)Dt
3Total 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.
4The 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.
5Suppose 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.