Graphics Reference
In-Depth Information
f
0. The derived function
f
(0)
is discontinuous at 0. Examine
thegraph of
f
on a graphics calculator to see how the
intermediate value property for derivatives holds on any interval
[0,
a
]. Near to zero, the function oscillates infinitely many times
between positive and negative values.
13
(i) If thepoint (
x
,
y
) lies on the chord joining (
a
,
f
(
a
)) to (
b
,
f
(
b
)) then
y
f
(
a
)
f
(
b
)
f
(
a
)
a
,
x
b
a
so
f
(
b
)
f
(
a
)
y
f
(
a
)
(
x
a
)
b
a
and
f
(
b
)
f
(
a
)
D
(
x
)
f
(
x
)
f
(
a
)
(
x
a
).
b
a
D
satisfies the conditions for Rolle's Theorem on [
a
,
b
].
f
(
b
)
f
(
a
)
D
(
c
)
0
f
(
c
)
.
b
a
(ii)
F
(
a
)
0 from thedefinition of
K
.
F
(
b
)
0, trivially.
Differentiability follows from qn 8.10.
F
(
c
)
0
f
(
c
)
K
.
For some
c
in (
a
,
b
),
f
(
b
)
f
(
a
)
f
(
c
)
K
.
b
a
14 If
b
a
h
, any
c
in (
a
,
b
) has theform
a
h
.
15 Let
a
x
y
b
; then, for some
c
in theintrval (
x
,
y
),
f
(
y
)
f
(
x
)
x
f
(
c
)
0.
y
So
x
y
f
(
x
)
f
(
y
).
16
f
(
x
)
1
cos
x
0 on (0, 2
), so
f
is strictly increasing by qn 15.
Since
f
(0)
0,
f
(
x
)
0 on (0, 2
).
f
(
x
)
f
(
x
)
0 on (0, 2
), so
f
is strictly increasing.
Since
f
(0)
0,
f
(
x
)
0 on (0, 2
).
f
(
x
)
f
(
x
) and
f
(
x
)
f
(
x
).
The argument may be repeated.
17 Let
a
x
b
; then
f
(
x
)
f
(
a
)
a
f
(
c
)
0 for some
c
in (
a
,
x
).
x
