Graphics Reference
In-Depth Information
Writedown theequation of thechord joining thepoints (
a
,
f
(
a
))
and (
a
h
,
f
(
a
h
)).
If
f
(
a
h
)
f
(
a
)
h
lim
m
,
how would you describe the line
y
f
(
a
)
m
(
x
a
)?
On the strength of the ideas in qn 4, we will define the
derivative
of
a function at a point
a
of its domain.
If
a
is a cluster point of the domain of the real function
f
and
f
(
a
h
)
f
(
a
)
h
lim
h
m
, for some real number
m
,
0
then
m
is called the
derivative
of
f
at
a
, usually denoted by
f
(
a
), and
f
is said to be
differentiable
at
a
.
The definition is designed to give a formal definition of the slope of
thetangent to
y
f
(
x
)at
x
a
, and hence to make it possible to define
analytically what is meant by a tangent to a curve. On a distance
—
time
graph the slope of a chord gives the average velocity between two
points, while the slope of a tangent gives the velocity at a point.
5 Although the motivation for our study of derivatives has been the
geometric notion of tangent, there is still one circumstance when a
tangent to the graph of a function may exist without a derivative of
thefunction at thepoint in qustion. Examinethedefinition of
derivative carefully in order to identify the circumstance in
question.
6 If
f
is a constant function, what is
f
(
a
)?
The attempt to establish a converse to qn 6 exposes some
unexpected subtleties, and will be examined in qn 9.17 using the Mean
Value Theorem.
7 If
f
(
x
)
mx
c
, what is
f
(
a
)?
You will havenoticed in your calculations for qns 6 and 7, how
critical it is that in finding thelimit of a function as
h
0 wepay no
regard to the value of that function when
h
0. In fact the'slopeof the
chord' function is not defined when
h
0. An equation like
h
/
h
1is
only valid when
h
0.
