CALCULUS

In around 450BC, a Greek Mathematician, Zeno of Elea, said, "If a body moves from A to B then before it reaches B it passes through the mid-point, say B1 of AB. Now to move to B1 it must first reach the mid-point B2 of AB1 . Continue this argument to see that A must move through an infinite number of distances and so cannot move." This probably is the first record of a probem involving basics of calculus.

We all know...
Speed = Distance/ Time

Realistically, the above formula gives us the average speed. A car seldom travels at a constant speed. An apple does not fall at a constant speed. So if you wanted to find instantaneous speed, using the above formula, the answer would be zero/ zero. Just like a picture of a moving car. It seems stopped. The car was actually moving but has no speed at that instant. How come.

Newton's discovery of modern calculus, may very well have been inspired by the apple falling from a tree. As an apple falls, it moves faster and faster; that is, it has not only a velocity but an acceleration. Newton expressed this mathematically by supposing that at any stage of its motion the apple drops a small additional distance s (delta s) during a brief additional time interval t (delta t). Then the velocity is very nearly equal to the distance "s" divided by the time "t" i.e., s/t. The exact velocity v would be the limit of s/t as t gets closer and closer to zero or, as we say, approaches zero. That is,

For example a runner is running and the following table gives the positions of the runner at different times:

It is obvious from the above table that the runner covered a distance of 9 meters in three seconds. The velocity would be as follows:
v = s/t = 9/3=3 meters/ second.
But this is the average quantity calculated for the total 9 meters covered.
Our observations started from t=1, so the average velocity for the observed part of the run are as follows:
v = Δs/Δt = (9-1)/(3-1) = 4m/s
Lets say if we want to find the instantaneous velocity at t=1.00, we will have;
v_inst = Δs/Δt _{lim t=1} = (1.02-1.00)/(1.01/1.00) = 2.00 meters/ second

We see that the instantaneous velocity gives us more specific information than the average velocity. At a given point in time or as the runner approaches t=1, or a graph showing a complex curve/ behaviour, the instantaneous value can be represented by using the concepts of calculus.

The quantity ds/dt is called the derivative of s with respect to t, or the rate of change of s with respect to t. It is possible to think of ds and dt as numbers whose ratio ds/dt is equal to v; ds is called the differential of s, and dt the differential of t. Just as velocity is the rate of change, or derivative, of the distance with respect to time, so the acceleration is the rate of change, or derivative, of the velocity with respect to time. Therefore a, the acceleration, would be

where v is the increase in velocity that occurs during the interval t. Since a is the derivative of v and v is the derivative of s, a is called the second derivative of s:

To find derivatives of s with respect to t, the dependence of s on t must be known; in other words, s must be expressed as a function of t. Usually this functional dependence is stated as a formula relating s and t. That part of calculus dealing with derivatives is called differential calculus.

Given s as a function of t, the derivative (that is, v) of s can be found. Conversely, if v is known it is possible to work backward to get s. This process of finding what is called the anti-derivative of v is begun by rewriting the equation v = ds/dt as ds = vdt. The quantity s is here regarded as the anti-differential of ds, denoted by a special symbol called an integral sign:

The last equation specifies s the integral of v with respect to t. That part of calculus dealing with integrals is called integral calculus. Applications of integral calculus involve finding the limit of a sum of many small quantities, such as the rectangular slices of an irregular plane figure.

As seen above, Calculus is a mathematical study of continuous change. The primary importance of calculus in the hard sciences is that it provides a language, a conceptual framework for describing relationships that would be difficult to discuss in any other language. Some scientific principles give information relating that values of variables at a given instant, for instance Ohm's Law E=IR, or the Boyle-Charles Law for ideal gasses, pV=kT. Calculus is not relevant for these rules. But many of the most important principles in science are rules for the way variables change. As shown above, physics tells you how velocity will change in various situations -- i.e. it tells you about acceleration. This is why it's important to have a mathematical way of talking about change. That's why you see the concept of the derivative used throughout science -- in physics, chemistry, biology, economics, even psychology.

Some basic examples are given here...