Transcript Euler’s Method - Kenyon College

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.
f
Euler’s method is numerical
antidifferentiation.
Dt
Point of View
f
f
Dt
Dt
Dy = f’(point)*Dt
Area = f’(point)*Dt
Total Change
f
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 yvalue. (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.
b
 f   f(b) - f(a)
a
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].
Euler’s
Method
and
Riemann
Sums
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 computes the total change in f
over the interval. The left Riemann Sums of f’
compute the same thing.