
194 Ñ Numerical Methods, Calculus, and Statistics
If you want to find the value of x with respect to time t, then consider the small interval del (t) » D t,
which has a time range of (t
0
, t
0
+ D t ). Now, suppose the curve x is a continuous function of
time t, as shown in Figure 6.8 with the initial point in the interval shown as (t
0
, x
0
). Then the
good approximation for the value of x at the end of this interval is:
xx
dx
dt
t
x
10
0
=+ .∆
where
dx
dt
fx t
x
0
= (,)
is the actual value of the slope of the curve at the initial point (t
0
, x
0
).
The accuracy of the approximation, however, depends upon the nature of the actual curve and
the duration of the interval ...