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t
n
D
nt
for n
D
0;1;:::N:The trapezoidal approximation then gives
r.t/
r
0
C
t
1
2
f.t/
!
:
2
f.0/
C
N
X
nD1
1
f.t
n
/
C
(2.25)
Example 2.2.
Let us again consider the case of
f.t/
D
t
2
;
r.0/
D
0;
where we want an approximation of r.1/: We choose N
D
10 and get
r.1/
t
1
2
f.t/
!
9
X
1
2
f.0/
C
f.t
n
/
C
nD1
9
X
!
1
2
10
2
D
t
3
n
2
C
nD1
9
10
19
6
C
50
1
10
3
D
D
0:335;
which is a very good approximation to the correct value, 1=3: If we choose N
D
100;
we get
r.1/
t
1
2
f.t/
!
9
X
1
2
f.0/
C
f.t
n
/
C
nD1
9
X
!
1
2
100
2
D
t
3
n
2
C
nD1
99
100
199
6
C
5000
1
100
3
D
D
0:33335:
2.2.2
Numerical Approximation of Exponential Growth
Our next goal is to derive a numerical method for the initial value problem
r
0
.t /
D
ar
.t /;
t
2
.0; T /;
(2.26)
