Graphics Reference
In-Depth Information
59 Apply qn 57 to obtain
(
b
x
)
f
(
x
)
dx
[(
b
x
)
f
(
x
)]
n
(
b
x
)
f
(
x
)
dx
(
b
a
)
f
(
a
)
n
(
b
x
)
f
(
x
)
dx
.
Question 57 must be applied
n
times in all.
If
f
(
x
)
sin
x
, then
f
cos
x
sin(
x
(
x
)
),
so
f
(
x
)
sin(
x
n
).
1
n
!
b
(
n
1)
)
b
n
!
So
(
b
x
)
f
(
x
)
dx
n
!
sin(
c
.
Thelast trm
0as
n
from qn 3.74(ii).
So (sin
x
partial sum of thefirst
n
terms)
0. So the series tends to sin
x
.
60 (
F
. The first equation comes from the
application of the Fundamental Theorem to
F
f
on [
g
(
a
),
g
(
b
)]. The
second equation comes from the application of the Fundamental theorem
to (
F
g
)
(
f
g
) ·
g
on [
a
,
b
].
All four expressions are equal provided
g
is an injection which follows, for
example, if
g
is positiveon [
a
,
b
].
g
)
(
F
g
) ·
g
(
f
g
) ·
g
61 Using the Fundamental Theorem, the integral
1/
b
1/
a
.As
b
,
integral
1/
a
. Comparewith qn 3.
62 If
I
(
a
)
f
exists for
a
negative and unbounded below, and
lim
I
(
a
)
L
then we write
f
L
. Sameexampleas qn 61.
63 Integral
2
b
2
a
. lim
(2
b
2
a
)
2
b
.
64
lim
[2
(1
x
)]
2.
1
x
0
1/
2
1/
(1
x
)
1
0
1/
(1
x
)
1/
(1
x
).
dx
Now
I
(
a
)
)
increases as
a
1
, but is bounded
(1
x
aboveby 2.
Let sup
I
(
a
)
1
a
0
L
, then as
a
1
,
I
(
a
)
L
and
dx
)
L
.
(1
x
