Graphics Reference
In-Depth Information
Theorem
qns 6, 9
n
(
a
If
f
(
x
)
1/
x
, then
f
f
lim
1).
Theorem
qns 26, 27,
28
If
f
:[
a
,
b
]
R has integral
F
, then
k
·
f
has
integral
k
·
F
for any real number
k
.
Theorem
qns 33, 35
If
f
:[
a
,
b
]
R is integrable, then
f
is also
integrable, and
f
f
.
R
has integral
G
, then
f
g
has integral
F
G
.
R
Theorem
qn 29
If
f
:[
a
,
b
]
has integral
F
and
g
:[
a
,
b
]
Theorem
qn 39
If
f
:[
a
,
b
]
R
is continuous, then
f
is integrable.
Mean
V
alue
heorem for integrals
qn 45
T
If
f
:[
a
,
b
]
R is continuous,
then
f
f
(
c
) · (
b
a
) for some
c
[
a
,
b
].
Theorem
qn 46
If
f
:[
a
,
b
]
R is integrable and
a
c
d
b
,
then
f
:[
c
,
d
]
R is integrable.
Theorem
qn 47
If
f
:[
a
,
b
]
R is integrable and
a
c
b
, then
f
f
f
.
Definition
qn 48
f
0 and
f
f
.
Indefinite integrals
49 Give a reason why the integer function defined by
f
(
x
)
[
x
]is
integrable on any closed interval.
Givea formula for
F
(
x
)
f
(i) when 0
x
1,
(ii) when 1
2,
(iii) when 2
x
3.
x
Is thefunction
F
continuous at
x
1 and
x
2?
50 (
Darboux
, 1875) Thefunction
f
is given to be integrable on the
interval [
a
,
b
] and a function
F
is defined on the interval [
a
,
b
]by
F
(
x
)
f
.
Provethat
F
is continuous at every point of [
a
,
b
].
