Understanding the Essence of Calculus

14 min readMay 29, 2022

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“ Without Mathematics, there’s nothing you can do. Everything around you is Mathematics. Everything around you is numbers.”- Shakuntala Devi (The Human Calculator)

“CALCULUS IS THE MATHEMATICS OF CHANGE”

In this Article I will not cover the traditional Calculus where we just see lot of formulas and theorems and plug those formulas in some equations to calculate the answer, rather we will dig deeper to understand what actually Calculus is ? Where we are actually using these concepts to solve problems in Data Science domain, which we earlier thought are just for getting marks in our school exams.(At least I thought of that :)

I think it is always a good way to understand, why this concept was discovered at the very first place. Why there was a need for this concept like Calculus. Lets understand from the very beginning…

1. WHAT IS CALCULUS ?

The word Calculus is a Latin word meaning “small pebbles”. Calculus is originally called infinitesimal calculus or “the calculation of infinitesimals(extremely small)”. Calculus is the study of rate of change of functions and the accumulation of extremely small quantities.

Calculus can be broadly divided into two branches :
i) Differential Calculus: This mainly concerns with the rates of changes of quantities and slopes of curves or surfaces in 2D or multidimensional space.Simply, Cuts something into small pieces to find how it changes.
ii) Integral Calculus : It is related to accumulation of quantities and the areas under and between curves. Joins (integrates) the small pieces together to find how much there is. Integration is a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation, where we reduce the functions into parts.

2. ORIGIN OF CALCULUS

There is an interesting story which involves some of the greatest mathematicians of all time which also involves lots of controversies around this concept which we will cover briefly over here so that we have a whole understanding of this topic. Lets understand from where it all Started…..

The origin of Calculus goes way back more than 2,500 years to the Ancient Greece where we begin our journey into the birth of Calculus in 400 BC Greek Mathematics “Eudoxus” used a method of Exhaustion to find the Area and Volume of shapes. He discovered that the Volume of the Cone is equal to one-third of the Volume of its corresponding Cylinder.

While “Archimedes” around 250 BC developed this idea of Exhaustion further by inventing heuristics which resemble the methods of Integral Calculus to prove the Area of circle by finding the value of π .

What is the Method of Exhaustion ?

Archimedes used this method to find an approximation of pi(π) by determining the length of the perimeter of a polygon inscribed within a circle (which is less than the circumference of the circle) and the perimeter of a polygon circumscribed outside a circle(which is greater than the circumference of the circle) and thereby come up with this approximation of π which he calculated further by continuously increasing the sides of the polygon until it closely resembles the circle which he used to derive the Area of Circle.
(Pi)π being defined as the ratio of the circumference to the diameter (C/d).He also provided the bounds 3 + 10/71 < π < 3 + 10/70, by comparing the perimeters of the circle with the perimeters of the inscribed and circumscribed 96-sided regular polygons.

That’s is why this is called the “Method of Exhaustion”.

He further used this method to find the Area of Parabola and Surface area and Volume of Sphere. He was the first to find the tangent to a curve using a method which resembles the method of Differential Calculus.

**Left(Archimedes : History of Calculus — Maths with Lisa Youtube), Right (Archimedes used method of Exhaustion by Wikipedia)**

In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics thereby stated components of calculus. Madhava and Nilakantha both discovered the other great component of calculus- infinite series.A complete theory encompassing these components is now well known in the Western world as the Taylor series or infinite series approximations.
There were many reasons why the contribution of the Kerala school has not been acknowledged a prime reason is neglect of scientific ideas emanating from the Non-European world a legacy of European colonialism and beyond.
But there is also little knowledge of the medieval form of the local language of Kerala, Malayalam, in which some of most seminal texts, such as the Yuktibhasa, from much of the documentation of this remarkable mathematics is written. A complete theory encompassing these components is now well known in the Western world as the Taylor series or infinite series approximations.However, they were not able to “combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today.

In 1630’s Cavalieri discovered the method of indivisibles which was the modern version of Archimedes Method of Exhaustion and an early step towards integral Calculus.

What is the method of Indivisibles ?

Cavalieri Principle states that If between the same parallels any two plane figures are constructed, and if in them, any straight lines being drawn equidistant from the parallels,Suppose if we have two solids of equal heights sitting ‘h’ and if at every point along ‘h’ the infinitesimally thin cross-sectional areas of solids are equal then the solids will have equal volumes.

Coming to the 17th century other famous European Mathematicians Issac Barrow (known for Fundamental Theorem of Calculus), Rene Descartes (known for cartesian coordinate system), Pierre De Fermat(known for infinitesimal Calculus), Blasie Pascal (known for Pascal’s Triangle) and John Wallis(known for infinitesimal Calculus) further pursued the emerging field and developed the concept called Derivative. Isaac Barrow, and James Gregory, the latter two proving the Second fundamental theorem of Calculus around 1670.

The Calculus Controversy

“A BATTLE between Sir Issac Newton and Gottfried Leibniz”

The two most notable Mathematician which are known today for their major contribution in Calculus are Gottfried Leibniz and Sir Issac Newton.The Calculus controversy was an argument between the mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first invented calculus.This was all started in 1666 when Newton who was a student at the University of Cambridge sat home during that time due to the wide spread of epidemic plague when he developed what we now called as Calculus to solve the problems in Physics.

First check the Issac Newton side….

**Sir Issac Newton(Image by Wikipedia, edited by Author)**

Newton described his version of differential calculus as ‘The method of Fluxions’. He wrote a paper on fluxions in 1666, but like many of his works, it was not published until decades later. His book “Principia Mathematica “ one the masterpiece of all time(Mathematical principles of natural philosophy) was published in 1687. This work includes his theories of motion and gravitation, but does not include much calculus explicitly although there is some explanation of calculus at the beginning, and Newton used calculus to formulate his theories. Nonetheless, Newton’s ‘method of fluxions’ did not explicitly appear in print until 1693.

Now let’s check the other side of story what Gottfried Leibniz contribution in Calculus

**Gottfried Leibniz( Image by Wikipedia, Edited by Author)**

Gottfried Leibniz a german philosopher and a Mathematician started his own version of calculus in 1674 ,On 1675 he make a breakthrough where he find the area under the graph “y = f(x)”. It is interesting to note that Leibniz was very conscious of the importance of good notation and put a lot of thought into the symbols he used.He made a whole new notation for his discovery get the letter “ ∫” (which is called Summa) now we use for Integration and “𝑑 “ for Differentiation. He published his first paper on Differential calculus in 1684 and published other paper on Integral Calculus in 1686 and claimed to have discovered calculus in the 1670s. From the published record, at least, Leibniz seemed to have discovered calculus first.

And the Credit for the “CALCULUS” goes to …

**Historic Battle between two great Mathematicians**

While Newton considered variables changing with time, Leibniz thought of the variables x and y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between successive values of these sequences. Leibniz knew that dy/dx gives the tangent but he did not use it as a defining property. On the other hand, Newton used quantities x’ and y’, which were finite velocities, to compute the tangent. Of course neither Leibniz nor Newton thought in terms of functions, but both always thought in terms of graphs.

While Newton and Leibniz version of calculus has problem. Newton’s Calculus version lack mathematical notation and was heavily rely on the geometric proofs for infinitesimal. Both the version of calculus were on the concept of Infinitesimal where they state that the number is not exactly zero but infinitesimal close to zero which they proved based on their philosophical ground, which was the concept of LIMIT which was discovered later in 1800s.

Newton was determined to proof that he was the sole inventor for the Calculus. Newton run campaign where he proofed that Leibniz has plagiarised Newton’s version as many of the Newton’s colleagues has close connection with Leibniz which arises the concern that may be Leibniz may have got references to Newton’s unpublished researches. Gradually during the last years of Leibniz lost control over these matters and in 1715 the Royal society proclaim Newton as the sole discoverer of Calculus. But Leibniz notation of Calculus was far better than the Newton’s calculus notation, so in 1820 it was finally accepted that both Leibniz and Newton has discovered Calculus independently.

Ultimately, Cauchy, Weierstrass, and Riemann reformulated Calculus in terms of limits rather than infinitesimals. Thus the need for these infinitely small (and nonexistent) quantities was removed, and replaced by a notion of quantities being “close” to others. The derivative and the integral were both reformulated in terms of limits.

Newton and Leibniz both created the foundations of derivative and integral.So both won the credit for the discovery of Calculus.

Enough of the history, though I have not covered all the details but still it is good to know the history behind each concept. Now let’s come back to the CONCEPT.

3. Understanding the Functions

Relation : A rule which associates each element of set(A) with at least one element in the set(B).

“A function is a relation where a set of independent values (x) has exactly one dependent value (y).”

Simply a rule which uniquely associates elements of one set (A) with the elements of another set (B) such that each element in set (A) maps to only one element in set (B).

A function consists of Domain and a Range.
DOMAIN :- Domain is the set of inputs(x) given to a function.
RANGE :- Range is the set of all output values(y).

All Functions are Relations but NOT all relation is a FUNCTION.

Methods for representing a Function

A Function can be represented in 3 ways:

→ Tabular form :- Where we define for each value of input(x) what is its corresponding value in output (y).

Tabular form for representing a function

→ Graphical form :- Here we plot for each value of x what is its corresponding value in the output.

Graphical form for representing a function

→ Algebric form:- Here we show the relation using the algebraic notation.

How to Test if a given Relation is a Function or NOT ?

Vertical Line Test- A Vertical line test is a visual way to determine if a Curve is a graph of function or not. A function can only have one output (y) for each unique input(x). If any vertical line with constant x cuts the graph exactly once, then the relation is a Function. Function is denoted as “y = f (x)” or (x, f (x)) where f is indicating the function and x is the input.

If any vertical line cuts the graph only once, then the relation is a Function (one-to-one or many-to-one).

If a line of constant x which crosses the graph of the relation more than once then the relation is NOT a Function.If more than one intersection point exists, then the intersections correspond to multiple values of y for a single value of x (one-to-many).

“FUNCTION CAN NEVER BE ONE-MANY”

Fig:- Above shows the relation is a function(left plot) and the relation is NOT a Function(right plot)

Fig:- Shows a set where left is a Function but right fig is NOT a function

Types of Functions

1. One-One Function (Injective Function)

If each element in the domain of a function has a distinct image in the co-domain, the function is said to be One-One Function. Simply A function where every “y” value has exactly one “x” value mapped onto it.

Fig:- F1 and F2 are showing One-One Function

How to Check if a function is One-One or NOT ?

If for 2 different values of “x” in the domain which are mapped to the same element “y” in the co-domain , in such case the function is NOT One-One Function.

We can check for one to one functions using the horizontal line test.When given a function, draw horizontal lines along with the coordinate system.Check if it’s possible for the horizontal lines to pass through two points.If the horizontal lines pass through only one point throughout the graph, the function is a one to one function.

Checking if the function is One-One Function or Not

2. Many- One Function

If there are “y” values that have more than one “x” value mapped onto it then it is called Many-one Function.

Fig. Modulus Function is the example of Many to One function

3. Onto Function (Surjective Function)

A function from set X to set Y is surjective if for every element y in the codomain Y of f. there is at least one element x in the domain of X of f such that f(x) = y. It is not required that x be unique: the function f may map one or more elements of X to the same element of Y.

Fig:- Onto Function(Surjective Function)

4. Into Function

Function f from set A to set B is Into function if at least set B has a element which is not connected with any of the element of set A.

5. Increasing and Decreasing function

Increasing Function:- A function is said to be an increasing function if the value of y increases with the increase in x.If there is a function y = f(x)

A function is increasing over an interval, if for every x1 and x2 in the interval, x1 < x2, f( x1) ≤ f(x2).
A function is strictly increasing over an interval, if for every x1 and x2 in the interval, x1 < x2, f( x1) < f(x2).

Decreasing Function :- A function is said to be a decreasing function if the value of y decreases with the increase in x. If there is a function y = f(x).

A function is decreasing over an interval , if for every x1 and x2 in the interval, x1 < x2, f( x1) ≥ f(x2).
A function is strictly decreasing over an interval, if for every x1 and x2 in the interval, x1 < x2, f( x1) > f(x2).

6. Odd and Even Function

Functions can also be classified as even or odd if they satisfy a particular symmetry.

Even Function :- Algebraically If a functions f(x) such that it satisfy f(x)=f(−x) for all x. Geometrically, the graph of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis then such functions are called Even Functions.

Checking Algebraically if a function is EVEN !

If we evaluate or substitute −x into f(x) and get the original or “starting” function again, this implies that f(x) is an even function.

Checking geometrically if a function is EVEN !

Odd Function: Algebraically if a functions f(x) such that it satisfy f(x)=−f(−x) for all x.Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin.

If we evaluate or substitute −x into f(x) and get the negative or opposite of the “starting” function, this implies that f(x) is an odd function.

Checking geometrically if a function is ODD !

Functions which are Neither Even nor ODD

There exists some functions f(x) such that:

If we evaluate or substitute −x into f(x) and we don’t obtain either above case, such functions f(x) implies that it is neither even nor odd. In other words, it does not fall under the classification of being even or odd.

7. Here is the list of some of the other Common Functions

I understand this has been a long journey till now let’s take a short break and enjoy the Dance moves with Maths Function revision in a Fun way…

More details regarding the Calculus will be covered in the next post.

Special Thanks!! to the below mentioned references which helped me to write this article. Feel free to read the below mentioned references to better understand this topic.