Graphics Reference
In-Depth Information
1
C
a
.
1
r
r
...
r
Find the limit of each of these bounds as
r
1. Sincethsetwo
inequalities hold for every
r
with 0
r
1, theshared limit,
a
/(
k
1), is theonly possiblevaluefor
C
.
5 Use the method of qns 3 and 4 to find the area bounded by the
curve
y
1/
x
, the
x
-axis and theline
x
1, where
k
is a positive
integer greater than 1.
6 (
Gregory of St Vincent
, 1647) Let
D
be the area bounded by the
curve
y
1, and
x
a
and the
x
-axis, where
a
1. Usethemthod of qn 3 to show that
1/
x
, thelins
x
n
(
a
1)
a
D
n
(
a
1).
From qn 4.41 weknow that (
n
(
a
1)) has a limit as
n
.Use
qns 3.57(d), 3.67, the quotient rule, and 3.78, the closed interval
property, to deduce that that limit is
D
.
Monotonic functions
7 (
Newton
, 1687) Let
f
:[
a
,
b
]
R
bea function which is positiv,
monotonic increasing and continuous.
Let
c
(
a
b
).
(i) Explain why
(
b
a
)
f
(
a
)
(
b
c
)
f
(
c
)
(
b
a
)
f
(
a
)
(
c
a
)
f
(
c
)
(
b
c
)
f
(
b
)
(
b
(
c
a
)
f
(
a
)
a
)
f
(
a
)
(
b
a
)
f
(
b
).
(ii) Dividetheintrval [
a
,
b
] into
n
equal parts each of length
(
b
a
)/
n
.Let
I
thesum of theareas of the
n
rectangles
inscribed under the graph of
y
f
(
x
), with bases
(
a
, 0)(
a
l
/
n
, 0), (
a
l
/
n
, 0)(
a
2
l
/
n
, 0), . . .,
(
a
(
n
l)
l
/
n
, 0)(
b
, 0), where
l
b
a
; and let
C
thesum
of theareas of
n
rectangles just covering the graph of
y
f
(
x
),
with bases (
a
, 0)(
a
l
/
n
, 0), (
a
l
/
n
, 0)(
a
2
l
/
n
, 0), . . .,
(
a
(
n
1)
l
/
n
, 0)(
b
, 0). Use the argument at the beginning of
this question to show that
I
I
C
C
.
(iii) Provethat (
I
) is a monotonic increasing sequence.
Deduce that this sequence is convergent to a limit
I
, say.
