Information Technology Reference
In-Depth Information
1.2
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
D
ln
10
sin.t /
e
t
for t
Fig. 5.15
The graph of the function y.t/
C
2
Œ0; 1
p.t/
D
t;
(5.103)
we see from Fig.
5.16
that p is a good approximation of y on the interval Œ0; 1.But
is it the best possible linear approximation? In Fig.
5.17
we have plotted y and p for
t
2
Œ0; 10 and now we see that the approximation is really good. If we consider y a
bit closer, we see that as t increases,
1
10
sin.t /
C
e
t
e
t
:
For instance, at t
D
10,wehave
1
10
sin.10/
C
e
10
22026:41
and
e
10
22026:46:
So for large t we have
y.t/
D
ln
1
10
sin.t /
C
e
t
ln.e
t
/
D
t:
This explains the good approximation.
