Lecture 4 - Computer Science Department - University of Kentucky

We have
Example 2
+ 2
3
a h
h h
∫ f
( x
) dx = hf
( a
) +
f '(
a
) +
f "(
a
) +L
3!
(1)
a
2
We can also apply the Taylor series directly on f(t) as
2
3
(2)
h h
f
( a
+ h
) =
f
( a
)
+
hf
'( a
)
+
f
"( a
)
+
f
"'( a
)
+L
2 3!
Adding f(a) on both sides of (2) and multiplying it by h/2, we obtain
(3)
h
[
2
2
3
h
h
f ( a)
+ f ( a+
h)]
= hf(
a)
+ f '( a)
+ f "( a)
+L
2 4
Subtracting (3) from (1), we finally get
a
∫ + h h
1 3
f ( x)
dx−
[ f ( a)
+ f ( a+
h)]
=− h f"(
a)
+L
a
2
12
20