Calculus Methods
18 Euler's Method
This is a method for finding the value of a function, given
another value of the function. Is it useful? Not so much.
There are much better methods out there at this time in
history, and any graphing calculator will do it as well. Why
include it, then? Well, it's on the AP Calculus BC test, so
here it is. The premise on which it is based is that all
functions with continuous derivatives look like lines when
one zooms in close enough. Thus, to follow a curve, from one
point to another, one simply moves a certain distance in the
same direction as the starting slope, and then repeats.
Obviously, this method works best with functions with only
gentle slopes and when many steps are taken.
1) Find the derivative of the function.
2) Pick the starting value. Choose an x value for which you can
calculate the function value.
3)
Calculate the width of each step, 
,
where 
and 
are
the starting and ending values of the function and 
is the number
of steps to be taken.
4)
Successive values of y are found by 
. If your
starting value is 
,
your ending value will be 
.
Example:
For the function defined by 
, solve for 
using six steps.
1) For this particular problem, the derivative is given.
2)
3)
4)
This differential equation is solvable, and the solution is
(try plugging it into the differentiable equation to
check it). Using this definition, we know
that 
.
So you can see the approximation in this case was not spectacular.
Part of the reason for this is that the function is an exponential,
and therefore varies rapidly.