Graphics Reference
In-Depth Information
Q
Q
1/(
x
11
(i) Thefunction
f
:
.
Show that thefunction
f
satisfies the conditions (i), (ii) and (iii)
of qn 7 on [0, 2]
is defined by
f
(
x
)
2)
. Why does the conclusion of Rolle's
Theorem fail in this case?
(ii) Thefunction
g
:
Q
20.
Show that thefunction
g
satisfies the conditions (i), (ii) and
(iii) of qn 7 on [1, 4]
Q. Why does the conclusion of Rolle's
Theorem fail in this case?
Q
Q
is defined by
g
(
x
)
x
21
x
An intermediate value theorem for derivatives
Although we have seen that differentiable functions do not
necessarily have continuous derivatives (in qn 8.34), derivatives have an
intermediate value property which we might only expect from a
continuous function. So the Intermediate Value property cannot be
used to define continuity.
12 (
Darboux
'
s Theorem
, 1875) If [
a
,
b
] is contained in the domain of a
real differentiable function
f
and
f
k
f
(
a
)
(
b
), determine whether
thefunction
g
defined by
g
(
x
)
f
(
x
)
kx
is
(i) a real function which is continuous on [
a
,
b
],
(ii) a function which is bounded and attains its bounds on [
a
,
b
],
(iii) a function which is differentiable on [
a
,
b
],
(iv) a function which is not constant on [
a
,
b
].
Determine the derivative of
g
at a point where it attains a bound,
and deduce that, for some
c
in the open interval (
a
,
b
),
f
(
c
)
k
.
Check the compatibility of this result with the differentiable
function in qn 8.22 having discontinuous derivative, using graph
drawing facilities on a computer.
The Mean Value Theorem
The next question looks at a slanted version of Rolle's Theorem,
and identifies conditions under which an arc has a tangent parallel to a
chord joining two points of thearc.
13 A function
f
:[
a
,
b
]
is continuous on the closed interval [
a
,
b
]
and differentiable on the open interval (
a
,
b
).
R
(i) Construct a function
D
which measures the distance (in a
