Graphics Reference
In-Depth Information
Use computer graphics to examine the graph of
A
for
a
1.5, 2 and 3.
For qns 10
—
28, wewill always assumethat
a
1.
10 Verify from qn 3.57 that (
A
(1/
n
))
1
A
(0) as
n
. Deduce that
(
A
(
A
(0) as
n
. Usethefact that
A
is strictly
increasing, established in qn 9, to show that
A
is continuous at 0.
1/
n
))
1
11 Provethat
A
is continuous at
q
Q by considering that
A
(
x
)
A
(
q
)
A
(
q
)(
a
1)
0as
x
q
.
So far we have only given a meaning to rational indices, and found
that for such indices the function
x
a
is continuous. Because the
rational numbers are dense on the real line we can complete the
definition of this function on
simply by insisting that it shall be
continuous. Questions 12
—
18 providethetools for showing that this
extension may only be done in one way and for finding the derivative
of thersulting function.
R
Keeping
a
1, define
D
: Q
0
R
by
a
1
D
(
x
)
.
x
Use computer graphics to examine the graph of
D
for
a
1.5, 2 and 3.
12 Why is
D
continuous where it has been defined?
13 By considering qn 2.50(iii), show that, when
m
,
n
Z
,
m
n
D
(
m
)
D
(
n
). Use the idea of qn 2.50(iv) to deduce that,
when
r
,
s
Q
,
r
s
D
(
r
)
D
(
s
).
Weproved in qn 4.41(i) that (
D
(1/
n
)) is convergent as
n
.
Let (
D
(1/
n
))
L
(
a
)as
n
.
14 Usethefact that
D
is strictly increasing for
x
0 to show that
lim
D
(
x
)
L
(
a
).
15 Usethefact that, for non zro
x
,
D
(
x
)
D
(
x
)/
A
(
x
), to show that
lim
D
(
x
)
L
(
a
).
16 How should
D
(0) be defined so that
D
is continuous on
Q
?
