Graphics Reference
In-Depth Information
Deduce that
x
6
,
x
sin
x
x
and
x
2
x
24
cos
x
1
x
2
, on (0, 2
).
1
Use graph-drawing facilities on a computer to compare the graphs
of sine and cosine with those of the polynomials given here.
17 A function
f
:[
a
,
b
]
is continuous on the closed interval [
a
,
b
]
and differentiable on the open interval (
a
,
b
).
Provethat if
f
R
(
x
)
0 for all values of
x
, then the function
f
is
constant.
Construct a real function
f
which is not constant, but for which
f
(
x
)
0 for every point
x
of its domain.
18 A function
f
:[
a
,
b
]
[
a
,
b
] is continuous on the closed interval
[
a
,
b
] and differentiable on the open interval (
a
,
b
).
If
f
(
x
)
L
for all values of
x
, provethat
f
(
x
)
f
(
y
)
L
ยท
x
y
for all
x
and
y
in thedomain,
so that
f
satisfies a Lipschitz condition, and is uniformly continuous
as in qn 7.42.
If a sequence (
a
) is defined in [
a
,
b
]by
a
f
(
a
), provethat
a
a
L
a
a
L
b
a
.
If, further,
L
1, deduce that (
a
) is a Cauchy sequence, and hence
convergent.
If (
b
) is a sequence in [
a
,
b
] defined by
b
f
(
b
) and (
a
)
A
,
and (
b
)
B
, explain why
A
and
B
liein [
a
,
b
], why
f
(
A
)
A
and
f
(
B
)
B
, and provethat
A
B
.
f
1, makes the function
f
an example of a
contraction mapping
of thedomain, which, as wehaveshown, has a
uniquefixed point, satisfying
f
(
x
)
Thecondition
(
x
)
x
. This result is particularly useful
for calculating better approximations to the roots of given equations
and provides a reason for the convergence of sequences defined by an
iterative formula.
19 Provethat theequation
x
cos
x
has exactly one root in the
interval [0,
].
20 Prove that the function defined by
f
(
x
)
(
x
2) is a contraction
mapping on theintrval [0,
). Deduce that the sequence defined
in qn 3.79 is convergent when the first term is positive.
