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In-Depth Information
The Fundamental Theorem of Calculus
54 (
Cauchy
, 1823) Let
f
bea continuous function on [
a
,
b
] and let
F
(
x
)
f
.
(i) Why, for some
,0
1, must
F
(
x
h
)
h
·
f
(
x
h
)?
(ii)
Deduce that
F
(
x
h
)
F
(
x
)
F
(
x
)
f
(
x
)
f
(
x
h
)
f
(
x
).
h
(iii) Usethecontinuity of
f
to show that
F
f
(
x
).
This result is the
Fundamental theorem of Calculus
.
(
x
)
When
f
is continuous and
F
(
x
)
f
(
x
) it is customary, following
Leibniz, to write
F
(
x
)
f
(
x
)
dx
(echoing
f
(
x
)) and
F
is
called an
anti
-
derivative
for
f
.
F
may be differentiable even when
f
is
discontinuous, as in qn 36.
)(
x
x
55 If both
F
and
G
are anti-derivatives for
f
on a given interval, use
the Mean Value Theorem to deduce that
F
(
x
)
G
(
x
)is
independent of
x
. Usethis to vrify that
F
(
b
)
F
(
a
)
G
(
b
)
G
(
a
).
As a consequence of qn 55, it follows that, if
F
is an anti-derivative
of
f
, then every other anti-derivative of
f
has theform
x
F
(
x
)
c
, for
a constant real number
c
.
(56) A function
f
is integrable on a neighbourhood of
a
and
lim
f
(
x
)
L
. Show that
F
(
a
h
)
F
(
a
)
h
lim
L
,
where
F
is defined as in qn 54. Claim the corresponding result for
left-hand limits. Deduce that if
f
is continuous at
a
, then
F
is
differentiable at
a
, and
F
(
a
)
f
(
a
), but that if
f
has a jump
discontinuity at
a
(see qn 6.79), then
F
is not differentiable at
a
.
Integration byparts
57 If both thefunctions
f
and
g
have continuous derivatives, use the
fact that
f
·
g
is an anti-derivative of
f
·
g
f
·
g
to provethat
f
·
g
f
·
g
[
f
·
g
]
.
