Graphics Reference
In-Depth Information
R
29 A function
f
:[
a
,
b
]
is continuous on the closed interval [
a
,
b
]
and differentiable on the open interval (
a
,
b
).
By applying del'Hoˆ pital's ruleto
f
(
c
h
)
f
(
c
h
)
2
f
(
c
)
,
h
provethat if
f
exists at
c
(
a
,
b
) then
f
(
c
h
)
f
(
c
h
)
2
f
(
c
)
h
f
(
c
)
lim
.
By considering the function
f
for which
f
(
x
)
x
when
x
0, and
f
(
x
)
0, show that this limit may exist when
f
is
not twice differentiable.
x
when
x
It may happen that lim
f
(
x
)/
g
(
x
) does not exist,
when lim
f
(
x
)/
g
(
x
) does.
30 Let
f
(
x
)
x
sin(1/
x
), when
x
0,
f
(0)
0 and
g
(
x
)
x
.
Show that lim
f
(
x
)/
g
(
x
)
0.
Show also that lim
f
(
x
)/
g
(
x
) does not exist.
Check that the conditions for Cauchy's Mean Value Theorem hold
here for any interval [0,
x
], with
x
0. Why may theargument in
theproof of del'Ho ˆ pital's rule
not
be used to establish the existence
of lim
f
(
x
)/
g
(
x
) when lim
f
(
x
)/
g
(
x
) is known?
Rolle's Theorem and Mean Value Theorem
Rolle
'
s Theorem
qn 7
Summary
-
If
f
:[
a
,
b
]
R
(i) is continuous on [
a
,
b
]
(ii) is differentiable on (
a
,
b
), and
(iii)
f
(
a
)
f
(
b
),
then
f
(
c
)
0 for some
c
, with
a
c
b
.
An intermediate value theorem for derivatives
qn 12
If
f
:[
a
,
b
]
R is differentiable on [
a
,
b
] and
f
(
a
)
k
f
(
b
),
then
f
(
c
)
k
for some
c
, with
a
c
b
.
