Calculus Made Easy

By

Ramesh Kumar

Introduction

Calculus in Latin means “small pebble”. Romans used pebbles for addition and subtraction on a counting board. Therefore, this word “calculus” became associated with computation. Further the word “small” has great significant. We shall understand it when we go deep into Calculus to understand how its general formulae have developed using the concept infinitesimally small quantity.

Calculus

Calculus is the branch of mathematics which best handles motion and change. It is the branch of mathematics which deals with rates of change. In fact, to be precise “instantaneous” rate of change, over infinite small intervals. It’s much simpler much easy to model changes over infinitesimal small intervals in comparison to changes over finite intervals. Before Calculus was invented mathematics was all static. We did not have the tools to understand intricacies of rates of changes, the fabulousness of instantaneous rates of changes and their co relation to their functions or “equations”.

How Calculus was Developed

Newton tried to describe the speed of a falling object. However, while doing this, he found that the speed of a falling object increases every second. During his time here was no existing mathematical explanation for this or a mathematical tool to handle it and describe it. The concept of movement and the rate of change had not yet been explored in the field of mathematics. Newton saw a void here and started working on it. Newton was highly fascinated and curious about planetary ellipses, why planets moved, orbits of planets, gravity, and gravitation force between planets, how plants stayed in orbit etc. He applied these concepts of rate of change to understand these things and calculus developed.

Leibniz also worked and led to the discovery of the calculus because of the work of his German predecessor, Johannes Kepler. Kepler was the first human being to accurately describe the motion of the planets in the solar system and described them in his 3 laws of planetary motion. However, Kepler did not have the benefit of the calculus during his time and thus had great difficulty in dealing with how to measure the rates of rates of change. Therefore, Kepler called for a “new geometry”, which he knew would be necessary for astronomy. It was this work which prompted Leibniz and put him on the path to discovering the calculus.

It is said that both Newton and Leibniz developed calculus independently.

Differential Calculus

There are mainly two branches of Calculus: Differential Calculus and Integral Calculus. Differential Calculus deals with the problems of rates of change. For example, when we have a formula for distance in terms of time that a body covers, differential calculus helps in calculation its velocity and acceleration over any instant. It also helps us to find the greatest and the smallest values a function takes.

Look at the above figure. Y=F(x) curve has been shown above. There are two points on this curve, p and Q. We wish to find out the average rate of change of function F(x) or y with respect to x at point P. for this we take two points P and another point Q nearby. At point P the value of the value along X axia is x and the value of the function f(x) for y plotted along Y axis is f(x).

Similarly for point Q these values are (x+∆x) and f(x+∆x) or ∆y

Therefore, the average rate of change of y with respect to x between points P and Q is:

0r

The above expression gives average rate of change of y with respect to x between points and .

The above is also equal to slope or tangent of secant line PQ with respect to X Axis

However, our main aim is to find rate of change of y with respect to x at point For this let us keep point P fixed and bring point Q nearer and nearer to point Q. See the figure below:

In the above figure, the secant line PQ is coming downward so that point “Q” is coming downward on the curve as closer and closer to point P. Exert your mind a bit. Consider point Q closer to pint Q. Closer than the close you can think of. Point Q is Infinitesimally closer to point P. This of any closeness it is closer than that! Note in this condition the secant line line “PQ” becomes a tangent line PT to the f(x) curve at point “P”. See the line PT is tangent to the f(x) curve in violet colour at point ‘P” This is denoted as under as under:

=

or

=

In above equation If we replace by h for simplicity of expression.

The notation is pronounced as The notation “Lim” with is read as “the limit, as approaches 0.

The above expression is the derivative of y or function f(x) with respect to x at point x. This also equal to slope of tangent line PT with respect to X Axis.