Graphics Reference
In-Depth Information
Mean Value Theorem
qn 7
R
If
f
:[
a
,
b
]
is
(i) continuous on [
a
,
b
], and
(ii) differentiable on (
a
,
b
), then
f
(
b
)
f
(
a
)
b
a
f
(
c
) for some
c
,
with
a
c
b
.
Theorem
qn 15
If
f
:[
a
,
b
]
R satisfies the conditions of the
Mean Value Theorem and
f
(
x
)
0 for all
x
,
with
a
x
b
, then
f
is strictly monotonic
increasing.
Theorem
qn 17
If
f
:[
a
,
b
]
R satisfies the conditions of the
Mean Value Theorem and
f
(
x
)
0 for all
x
,
with
a
x
b
then
f
is constant.
Theorem
qn 18
If
f
:[
a
,
b
]
satisfies the conditions of the
Mean Value Theorem and
R
f
(
x
)
1 for all
x
,
with
a
x
b
, then
f
(
x
)
x
has exactly one
solution, which is the limit of any sequence (
a
)
defined by
a
f
(
a
).
satisfies the conditions of the
Mean Value Theorem and if, for some
c
, with
a
c
b
,
f
(
c
)
0 and
f
(
c
)
0, then
f
has a
local minimum at
c
.
Cauchy
'
s Mean Value Theorem
qn 25
Theorem
qn 22
If
f
:[
a
,
b
]
R
If
f
:[
a
,
b
]
R and
g
:[
a
,
b
]
R arefunctions
such that
(i) both arecontinuous on [
a
,
b
],
(ii) both are differentiable on (
a
,
b
), and
(iii)
g
(
x
)
0 for any
x
, with
a
x
b
, then
f
(
b
)
f
(
a
)
b
a
f
(
c
)
g
(
c
)
for some
c
, with
a
c
b
.
de l
'
Hoˆ pital
'
s rule
qn 26
If
f
:[
a
,
b
]
R
and
g
:[
a
,
b
]
R
arefunctions
such that
(i) both arecontinuous on [
a
,
b
]
(ii) both are differentiable on (
a
,
b
),
(iii)
f
(
a
)
g
(
a
)
0, and
f
(
x
)
f
(
x
)
g
(
x
)
l
.
(
x
)
l
,
(iv) lim
then
lim
g
