Graphics Reference
In-Depth Information
58 If thefunction
f
has a continuous second derivative on [
a
,
b
], prove
that
f
(
b
)
f
(
a
)
(
b
a
)
f
(
a
)
(
b
x
)
f
(
x
)
dx
.
59 (
Cauchy
, 1823,
Taylor
'
s Theorem with integral form of the remainder
)
If thefunction
f
is infinitely differentiable on
, proveby induction,
using integration by parts, that for any real numbers
a
and
b
:
R
(
b
a
)
f
(
a
)
2!
(
b
a
)
f
(
a
)
n
!
f
(
b
)
f
(
a
)
(
b
a
)
f
(
a
)
...
1
n
!
(
b
x
)
f
(
x
)
dx
.
If
f
(
x
)
sin
x
and
a
0, show that the last term of this expansion
tends to 0, as
n
.
Apply the Mean Value Theorem for integrals (qn 45) to deduce the
Cauchy form of the remainder for a Taylor Series (qn 9.45).
Integration bysubstitution
60 If
f
is a continuous function on [
g
(
a
),
g
(
b
)] with
F
f
, and the
function
g
has a continuous derivative on [
a
,
b
], identify an
anti-derivative for the function (
f
g
) ·
g
.
Justify each of the equations
f
[
F
]
and
f
(
g
(
x
))·
g
(
x
)
dx
[
F
(
g
(
x
))]
.
What further condition must thefunction
g
satisfy if weareto be
sure that all four expressions are equal?
Leibniz' notation is helpfully suggestive here as it takes the form
f
(
g
(
x
))
dg
f
(
g
)
dg
dx
dx
.
Improper integrals
61 If
a
and
b
arepositivenumbrs, find thevalueof
f
where
f
(
x
)
1/
x
.
Determine lim
f
.
