We have seen how the concept of continuity is naturally associated with attempts to model gradual changes. For example, consider the function given by
, where change in
is proportional to the change in x. This simple looking function is often used to model many practical problems. One such case is given below:
Suppose 30 men working for 7 hours a day can complete a piece of work in 16 days. In how many days can 28 men working for 6 hours a day complete the work? It must be evident to most of the readers that the answer is days.
(While solving this we have tacitly assumed that the amount of work done is proportional to the number of men working, to the number of hours each man works per day, and also to the number of days each man works. Similarly, Boyle’s law for ideal gases states that pressure remaining constant, the increase in volume of a mass of gas is proportional to the increase in temperature of the gas).
But, there are exceptions to this as well. Galileo discovered that the distance covered by a body, falling from rest, is proportional to the square of the time for which it has fallen, and the velocity is proportional to the square root of the distance through which it has fallen. Similarly, Kepler’s law tells us that the square of the period of the planet going round the sun is proportional to the cube of the mean distance from the sun.
These and many other problems involve functions that are not linear. If for example we plot the graph of the distance covered by a particle versus time, it is a straight line only when the motion is uniform. But, we are seldom lucky to encounter only uniform motion. (Besides, uniform motion would be so monotonous. Perhaps, there would be no life at all motions if all motions were uniform. Imagine a situation in which each body is in uniform motion. A body at rest would be eternally at rest and those once in motion, would never stop.) So the simple method of proportionality becomes quite inadequate to tackle such non-linear problems. The genius of Newton lay in looking at those problems which are next best to linear, the ones that are nearly linear.
We know that the graph of a linear function is a straight line. What Newton suggested was to look at functions, small portions of whose graphs look almost like a straight line (see Fig 1).
In Fig 1, the graph certainly is not a straight line. But a small portion of it looks like a straight like a straight line. To formalize this idea, we need the concept of differentiability.
Definition.
Let I be an open interval and be a function. We say that f is locally linear or differentiable at
if there is a constant m such that
or equivalently, for x in a punctured interval around ,
where as
What this means is that for small enough ,
is nearly a constant or, equivalently,
is nearly proportional to the increment
. This is what is called the principal of proportional parts and used very often in calculations using tables, when the number for which we are looking up the table is not found there.
Thus, if a function f is differentiable at , then
exists and is called the derivative of f at and denoted by
. So we write
.
We need to look at functions which are not differentiable at some point, to fix our ideas. For example, consider the function defined by
.
This function though continuous at every point is not differentiable at $latex x=0$. In fact, . What all this means is that if one looks at the graph of
, it has a sharp corner at the origin.
No matter how small a part of the graph containing the point is taken, it never looks like a line segment. The reader can test for the non-differentiability of
at
.
This leads us to the notion of the direction of the graph at a point: Suppose is a function differentiable at
, and let P and Q be the points
and
respectively in the graph of f. (see Fig 2).
The chord PQ has the slope . As x comes close to
, the chord tends to the tangent to the curve at
. So,
really represents the slope of the tangent at
(see Fig 3).
Similarly, if is the position of a moving point in a straight line at time t, then
is its average velocity in the interval of time
. Its limit as t goes to
, if it exists, will be its instantaneous velocity at the instant of time
. We have
is instantaneous velocity at
.
If the limit of does not exist as x tends to
, the curve
cannot have a tangent at
, as we saw in the case of
at
; the graph abruptly changes its direction. If we look at the motion of a particle which is moving with uniform velocity till time
and is abruptly brought to rest at that instant, then its graph would look as in Fig 4a.
This is also what we think happens when a perfectly elastic ball impinges on another ball of the same mass at rest, or when a perfectly elastic ball moving at a constant speed impinges on a hard surface (see fig 4b). We see that there is a sharp turn in the space time graph of such a motion at time . Recalling the interpretation of
as its instantaneous velocity at
, we see that in the situation described above, instantaneous velocity at
is not a meaningful concept.
We have already seen that continuous functions need not be differentiable at some points of their domain. Actually there are continuous functions which are not differentiable anywhere also. On the other hand, as the following result shows, every differentiable function is always continuous.
Theorem:
If a function is differentiable at , then it is continuous there.
Proof:
If f is differentiable at , then let
. Setting
, we see that
. Thus, we have
Now,
This shows that f is continuous at .
QED.
Continuity of f at tells us that
tends to zero as
tends to zero. But, in the case of differentiability,
tends to zero at least as fast as
. The portion
goes to zero no doubt but the remainder
goes to zero at a rate faster than that of
. This is how differentiation was conceived by Newton and Leibniz. They introduced a concept called an infinitesimal. Their idea was that when
is an infinitesimal, then so is
, which is of the same order of infinitesimal as
.The idea of infinitesimals served them well but had a little problem in its definition. They were introduced seemed to run against the Archimedean property. The definition of infinitesimals can be made rigorous But, we do not go into it here. However, we can still usefully deal with concepts and notation like:
(a) as
if there exists a K such that
for x sufficiently near
.
(b) as
if
.
Informally, means
is of smaller order than
as
. In this notation, f is differentiable at
if there is an l such that
.
We shall return to this point again. Let us first give examples of derivatives of some functions.
Examples.
(The proof are left as exercises).
(a) ,
, n a positive integer.
(b) (
, where n Is a negative integer),
(c) ,
(d) ,