Graphics Reference
In-Depth Information
F
(
x
)
F
(
y
)
L
x
y
Identify a real number
L
such that
ยท
for all
x
and
y
in [
a
,
b
].
Thefunction
F
is called an
indefinite integral
for
f
.
51 (
Darboux
, 1875) Let
f
bea function which is intgrableon [
a
,
b
],
and suppose there exists a function
F
such that
F
(
x
)
f
(
x
) for all
x
[
a
,
b
].
Supposealso that
a
x
x
x
...
x
b
is a subdivision
of [
a
,
b
].
(i) How do you know that there exists a
c
(
x
,
x
) such that
F
(
x
)
F
(
x
)
F
(
c
)?
x
x
(ii) By adding equations of the type
F
(
x
)
F
(
x
)
f
(
c
)(
x
x
),
show that
F
(
b
)
F
(
a
)
f
(
c
)(
x
x
).
(iii)
Deduce that
F
(
b
)
F
(
a
)
f
.
It is customary to write
F
(
b
)
F
(
a
)
[
F
(
x
)]
.
There was no requirement in qn 51 that
f
becontinuous.
52 Apply qn 51 on [0,
x
]to
F
(
x
)
x
sin(1/
x
) for
x
0,
F
(0)
0.
In qn 51, two conditions were postulated, namely
(i)
f
is integrable on [
a
,
b
], and
(ii)
F
(
x
)
f
(
x
).
This second condition appears to imply that
f
has an anti-derivative as
will be defined after qn 54, and therefore might be thought to imply
that
f
is integrable, but this is not so. Conditions (i) and (ii) are
independent, as qn 53 will show. There are even bounded derivatives
which are not integrable as Volterra found in 1881.
53 Suppose
F
(
x
)
x
sin(1/
x
) when
x
0 and
F
(0)
0. Check that
F
is differentiable on [0, 1]. Use computer graphics to examine the
graph of
F
. Show that
F
is unbounded on [0, 1] and so not
integrable.