Graphics Reference
In-Depth Information
40 If the distance between two points (
x
,
y
) and (
a
,
b
) is defined to be
((
x
a
)
(
y
b
)
), show that this distance
x
a
y
b
.
What is the distance between the two points (
a
,
f
(
a
)) and (
b
,
f
(
b
))?
41 Why is thefunction
f
, of qn 39, continuous on [
1, 1] and
differentiable on (
1, 1)?
42 Apply the Mean Value Theorem to
f
to show that thedistance
between the points of qn 40 is equal to
b
a
(1
c
)
for some
c
with
1
a
c
b
1.
Arc length
43 For any subdivision
a
x
b
, show that the
polygonal arc length
of thefunction
f
on [
a
,
b
], defined by
x
x
...
x
((
x
x
)
(
f
(
x
)
f
(
x
))
)
can only increase if an additional point, or finite set of points, is
added to thesubdivision. Useqn 2.64, thetriangleinequality in the
plane.
44 Useqn 40 to show that any polygonal arc length for all or part of
the function of qn 39 is less than or equal to 4.
Deduce that the polygonal arc length of this function has a
supremum. The supremum for polygonal arc length on the interval
[
a
,
b
] is called the
arc length
of thefunction on [
a
,
b
].
45 By applying qn 42, show that any polygonal arc length on the
interval [
a
,
b
] for thefunction of qn 39 has thevalue
x
x
)
,
c
(1
with
x
as in qn 43, for some
c
s, with
x
c
x
.
46 If wedefinethefunction
g
:(
1, 1)
R by
1
(1
x
)
g
(
x
)
show that any polygonal arc length on the interval [
a
,
b
] for the
