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(d) Note that the problem of bounding the error of the trapezoidal method is now
reduced to bounding the difference between the function f
D
f.x/ and the
linear function y
D
y.x/, which coincides with f in the endpoints x
D
a and
x
D
b. We will bound this difference by subsequent use of the Taylor series
given in (
1.41
)below.
Use the Taylor series with n
D
1 to show that there is a value between a and
x such that
f.x/
D
f.a/
C
.x
a/f
0
.a/
C
1
2
.x
a/
2
f
00
./;
(1.30)
and use this expansion to argue that
f.x/
y.x/
D
f
0
.a/
.x
a/
C
f.b/
f.a/
b
a
1
2
.x
a/
2
f
00
./:
(e) Use the Taylor expansion with n
D
1 once more to show that
f.b/
f.a/
b
a
1
2
.b
a/f
00
./
D
f
0
.a/
C
for some in the interval from a to b. Apply this relation to show that
f.x/
y.x/
D
1
1
2
.x
a/
2
f
00
./:
2
.x
a/.b
a/f
00
./
C
(f) Let
ˇ
f
00
.x/
ˇ
;
M
D
max
axb
(1.31)
and use the triangle inequality (1.37) to show that
j
f.x/
y.x/
j
M.b
a/
2
:
(g) Show that
E
M.b
a/
3
:
(h) Next we consider the composite trapezoidal rule. Recall that the composite
trapezoidal rule is defined as follows:
T
n
D
h
"
1
2
f.x
n
/
#
;
2
f.x
0
/
C
n
X
i D1
1
f.x
i
/
C
as in (1.16). Show that
n
X
T
n
D
T.f;x
i 1
;x
i
/
i D1
