The Mathematics of Change

The mathematics of change

Today, calculus is a major part of the mathematical education, probably because people today are more prone to changes (in all aspects) than before; and calculus is all about change.

What is calculus?

Calculus is the branch of mathematics that deals with change —— speed, velocity, acceleration etc. Also, concepts like area, volume, function are a part of calculus.

Although pre – calculus is also about area, volume etc. we find calculus more dynamic in nature than pre – calculus.

There are two major branches in Calculus – Differential calculus and Integral calculus.

History of calculus

Calculus was called “infinitesimal calculus” before.

In the ancient times ideas like area, volume etc. led to Integral calculus. In the Egyptian Moscow papyrus we get to see some formulas in the form of mere instructions with no given methodology. Some are even incorrect. The Greek mathematician Eudoxus (c 408 – 355 BC) used the method of exhaustion which later developed the concept of limit, to calculate areas and volumes. This method was later reinvented by Liu Hui, in the 3rd century AD, to find the area of a circle. In the 14th century AD, Indian mathematician Madhava Sangamagrama stated many components of calculus like the Taylor series, infinite series etc. However it was in Europe , in the 17th century AD that, the modern calculus came into existence. Sir Isaac Newton and Gottfried Wilhelm Leibniz worked on the pre existing concepts and introduced the basic principles of calculus.

Principles of calculus

Limits

If we have to find out what happens to a function when the input approaches a certain value, we will use the idea of limits.

Let us understand it with the help of an example: Choose values near 3 but below 3, and calculate f(x) = for each. See what happens as they get nearer 3.

x

2.7

2.8

2.9

2.95

2.99

f(x) =

19.68

21.95

24.389

25.67

26.73

Now let us choose those values around 3 that are above 3.

x

3.3

3.2

3.1

3.05

3.01

f(x) =

35.94

32.77

29.79

28.37

27.27

Note that as x comes closer to 3, f(x) comes closer to 27. So, we can say that, the limit of  = 27 as x approaches 3.

Sometimes we have to find out what happens to a function when the input gets very big. This is called infinite limit.

Look at the table below:

x

1.5

2

2.5

3

3.5

f(x) = 1/x

0.66

0.5

0.4

0.33

0.28

Note that as the value of x becomes bigger, f(x) decreases and seems to approach 0, though it never reaches 0. So, we can say that, the limit of 1/x approaches 0 as x becomes bigger.

Differentiation

Differentiation is an operation that helps us to find what happens to a function whose output is the rate of change of a variable with respect to another variable.

For example, we consider an object in motion. The position of the object changes with time. So, we can say that the position of the object depends on time. With the help of differentiation we can track the position of the moving object at a given time.

Integration

The process that helps to find the value of an integral is called Integration.

Integrals are of two types – the indefinite integral and the definite integral.

The process of solving for anti – derivatives is called indefinite integration. It is the opposite function of differentiation.

We say that, F is an indefinite integral of f when f is a derivative of F (the use of upper- and lower-case letters for a function and its indefinite integral is common in calculus.)

Suppose we have a function and would have to determine the area underneath its graph over an interval. The area of the region can be found out by representing it with rectangles. If we find the total area of the rectangles, we get the area of the region. This is definite integration.

The fundamental theorem of calculus

This theorem states the relationship between the two very important parts of calculus – differentiation and integration.

The fact that indefinite integration is the opposite function of differentiation Is stated in the first fundamental theorem of calculus. The second fundamental theorem of calculus deals with the definite integrals.

Uses of calculus

Calculus is widely used by scientists, engineers, economists etc.as it gives us a precise understanding of space, time etc.