What is calculus and does it serve any purpose in the real world?
Can you think of your life where nothing changed? Those moments are rare, as everything in our life is subject to transformation. The amount of food we eat every day, the number of steps we take, and even the time the Sun rises. Calculus is a branch of mathematics that allows us to study continuous change. Most people see calculus only as a bunch of equations that involve a lot of calculations, but it is actually the set of principles that we apply in our lives every day.
The application of calculus can be seen in physics, medicine, engineering, and economics. It played a key role in the development of navigation in the 17th and 18th centuries, and today it has a vital role in space travel, and in the development of various other advanced technologies. The fields and disciplines that use calculus include thermodynamics, electricity, acoustics, geography, computer vision, economics, robotics, demographics, ship design, and engineering.
Let's have a closer look at the history of calculus.
What is calculus, and what are its types?
Calculus uses mathematical operations to study and analyze rates of change and find patterns between equations. It is a significant branch of mathematics. However, before you dive deep into calculus, you need to understand the meaning behind three terms: Function, derivative and integral.
A function defines the connection between two variables (such as distance and time, temperature and volume, etc.) in an equation. A function 𝑓 consists of an input or set of inputs, an output or set of outputs, and a rule for assigning each input to exactly one output.
The derivative is defined as the rate of change of a function with respect to a variable (i.e. output value with respect to input value). An integral is a mathematical object that can be interpreted as an area or a generalization of area.
Using calculus, scientists, astronomers, physicists, mathematicians, and chemists can develop mathematical equations that allow them to chart the orbit of the planets and stars, determine the positions and path of protons and electrons, know how medicines interact with humans cells, and find answers to many other mathematical questions.
There are two main types of calculus; differential and integral. Here is an overview of these:
What is differential calculus?
The branch of calculus that aims to find the rate of change of a function with respect to the variable on which it depends is called differential calculus. It is commonly used to find the slope of a line (how steep it is) at a given point on a curve.
While finding a slope on a straight line is relatively easy, on a curve, the slope has different values at different points, as the line bends. One way of finding the slope is to mathematically cut the curve into very tiny pieces so that each piece resembles a straight line. The slope of this straight line will then be the same as the slope of the curve at that point - this is called the tangent.
For functions that act on real numbers, the derivative (d) is the slope of the tangent line at a point on a graph. The derivative is often written as dy/dx (the difference in y divided by the difference in x).
A range of points, or limit, is chosen on either side of the range or point we are trying to find. Then, the tangent is calculated over the limit. The slope approaches a particular value as the tangents approached the real slope of the curve. The particular value it approached is the actual slope.
This process of working out a slope using limits is called differentiation, or finding the derivative.
What is integral calculus?
While differential calculus is concerned primarily with the slope of a curve. In contrast, integral calculus is concerned with calculating the area underneath a graph of a function.
For example, if we are calculating the distance a car travels and know the speed of the car at different points in time, we can draw a graph of this speed, and the distance the car travels will be the area under the graph.
This is done by dividing the graph into many very small pieces, and "drawing" rectangles under each piece. The area of a rectangle is easy to calculate, so the total area of all the rectangles can then be calculated. If the rectangles are made thin enough, then the value of the total area will approach the area underneath the graph. This value of the area is called the integral of the function.
Fundamental theorem of calculus
The theorem is actually a two-part concept that connects the differentiation of a function to the integration of a function. The theorem demonstrates that differentiation and integration are inverse processes.
The first part of the theorem establishes the procedure for computing a definite integral (an integral where you are given the upper and lower limits of integration).
It says that if we first integrate f and then differentiate the result, we get back to the original function f. The indefinite integral (F) of a function (f) can be derived from the integration of function f. Therefore, it demonstrates that indefinite integrals exist for continuous functions.
The second part of the fundamental theorem is a way to calculate a definite integral in terms of indefinite integrals. (An indefinite integral, also called an anti-derivative, is an integral without upper and lower limits).
For a function that is continuous over an interval, the theorem allows us to create a new function, F(x , by integrating over that interval. When we do this, F(x) is the anti-derivative of , and is the derivative of F(x) Further, F(x) is the accumulation of the area under the curve over that interval.
The second fundamental theorem of calculus establishes a relationship between a function and its anti-derivative. It allows us to find the derivative of a curve and evaluate it at certain values of the variable at times when using an anti-derivative is otherwise difficult.
Who invented calculus?
Greek mathematicians took the very first steps towards the development of calculus. In the late fifth century, the astronomer Eudoxus of Cnidus proposed a concept called the method of exhaustion. He claimed that by using this method, the area of any shape could be calculated by drawing a sequence of polygons inside the shape, as the area covered by all the polygons would be almost similar to the total area bounded by the shape.
Later, the Greek mathematician Archimedes used the method of exhaustion to calculate the circumference of a circle and used his findings to define the mathematical constant π (Pi). Interestingly, he didn’t stop there and employed the same method to figure out the volume of a sphere, the area of an ellipse, the area of a spiral rotation, the volume of a cylinder, and various other geometrical values.
Between the third and fifth centuries AD, Chinese mathematicians Liu Hui and Zu Gengzhi also came up with their own version of the method of exhaustion and used it to calculate the area of the circle and volume of a sphere, respectively.
Some studies point out that ancient Indian scholars knew about calculus long before it started being practiced by modern mathematicians. For instance, historical evidence suggests that in the 15th century, two Indian astronomers and mathematicians, Madhava of Sangamagrāma and Nilkantha Somayaji, developed theories encompassing different elements of present-day calculus.
Another Indian astronomer, Bhaskara II, or Bhaskaracharya, is believed to have mentioned concepts and principles similar to differential and integral calculus in his book Siddhānta Shiromani, originally written in 1150 AD. Interestingly, the book also discusses algebra, trigonometry, and various other mathematical concepts in detail.
In the modern era, German astronomer Johannes Kepler kicked off the further development of calculus. In his work titled Stereometrica Doliorum, he suggested a method to calculate the area of an ellipse.
In the years that followed, numerous other scientists and mathematicians, such as Isaac Barrow, Evangelista Torricelli, Rene Descartes, Pierre Fermat, etc., made notable contributions in the field of calculus. However, the final elucidation of calculus is attributed to both Sir Isaac Newton and Gottfried Leibniz, who independently developed its foundations and explained its underlying principles in depth.
Although they were both instrumental in developing calculus, they thought of the fundamental concepts in very different ways.
Newton viewed calculus as the scientific description of the generation of motion and magnitudes, while Leibniz focused on tangents and the notion of calculus as an explanation of the change.
Both thought in terms of graphs rather than functions. For Newton, the calculus was geometrical while Leibniz took it towards analysis, and also developed a notational system for calculus.
Applications of calculus in different fields
Calculus has made our lives comfortable and productive in many ways. Here are some applications that prove the same:
- Atmospheric scientists can predict weather and climate-related changes more accurately by using factors like wind speed, moisture level, temperature, pressure, etc. Although sensors serve as essential tools for predicting the weather, the basics of weather forecasting lie in differential equations that take into account the slightest changes in the above-mentioned weather-related variables and give out values that reflect future weather conditions.
- Since building construction requires an in-depth understanding of weight, area, height, material density, and various other variables. Civil engineers and architects employ differential and integral calculus to deal with the complex mathematics associated with the construction of buildings, bridges, and various other types of structures.
- For the proper and smooth operation of robots that primarily need to be moving most of the time, the variables linked to their functions, such as velocity, distance, acceleration, etc., must be coordinated well. Such coordination is achieved through computer programs and built-in mechanisms that are governed by calculus-based equations.
- Biologists use calculus methods to accurately predict the rate of growth of bacteria and other microbes. It is also used in patient diagnosis for calculating cardiac output, blood pressure, cell and tumor growth. Epidemiologists use calculus to study the spread of infectious diseases.
Other calculus applications involve its use by experts in the field of bioinformatics to study the effects of medicines on human cells over a period of time. The side effects of drugs are also analyzed through integral calculus. Moreover, calculus is also helpful for accurately predicting population, population density, annual rainfall, and numerous other parameters.
At the time of its elucidation, calculus was believed to be not more than an ordinary mathematical concept that explained change with respect to time. However, over the years, the concept evolved and gradually became an essential tool for finding answers related to any number of important changes and developments that take place.