Graphics Reference
In-Depth Information
Usethemthod of
1.
Provethat
f
is not increasing in any neighbourhood of 0 by
showing that
f
(1/2
n
qn 22 to show that
f
(0)
f
(1/(2
n
) for any
n
Z
)
1)
. Show
also that
f
(1/(2
n
)
)
1.
Derived functions
x
26 For thereal function
f
given by
f
(
x
)
, which is defined for all
x
R, find thesubst of thedomain which consists
of thosepoints
of
R
at which
f
is differentiable.
If a real function
f
:
A
R is differentiable at each point of a subset
B
A
, with
B
maximal, wedefinethe
derived function
of
f
,
f
:
B
R by
f
:
x
f
(
x
), and wesay that
f
is differentiable on
B
.
27 Sketch the graph of the derived function of
f
where
f
is defined by
f
(
x
)
x
.
28 Sktch thegraph of thefunction given by
f
(
x
)
x
2
x
2 and of
its derived function. How does the point where the graph of the
derived function cuts the
x
-axis relate to the shape of the graph of
f
?
29 Define the terms
local maximum
and
local minimum
in such a way
that they describe respectively the points (
1, 5) and (1, 1) on the
graph of the real function defined by
f
(
x
)
x
3
x
3. What is
f
(
1) and
f
(1)?
30 If a real function
f
has a local maximum or a local minimum at a
point
a
of its domain, and
f
is differentiable at
a
, provethat
f
(
a
)
0 by considering limits from above and below.
31 Givean exampleof a function
f
which shows that it is possibleto
have
f
(
a
)
0 without thefunction
f
having a local maximum or a
local minimum at
a
.
