In modern day mathematics, calculus is probably one of the most frequently used tools and it’s development came after centuries of rigorous studies. During the ancient period any mathematical concept was termed to be calculus or in simpler words can be said that calculus was assumed to be a synonym of mathematics. However different ideas put forth by different scientists from different regions gave birth to the evolution of modern day calculus. Calculus was used by ancient people as a tool to calculate areas and volumes. For instance, early Egyptians used formulas which resemble integral calculus to calculate areas of solid shapes, Archimedes used the method of exhaustion to compute the area inside a circle and the Indians related calculus with trigonometry to give certain differentiation methods as early as the 8th century BC.
The 17th century saw the dawn of modern calculus with various mathematicians such as Blaise Pascal, Pierre De Fermat and Rene Descartes putting forth the concept of derivative. Certain concepts related to differentiation such as maxima, minima and tangential equations were created around this time especially by Fermat. Cavalieri’s theorems and methodology provided a more holistic approach to integral calculus and a refined version of the method of exhaustion. Cavalieri’s principle gave other mathematicians a foundation to work with especially after the computation of area under the x^n curve or a curve with higher degree. Also around this time Fermat created the base for integral calculus by giving the two fundamental theorems of calculus.
Majority of the modern day calculus was created by two men: Newton and Leibniz, who independently developed its foundations. Although they both were instrumental in its creation, they thought of the fundamental concepts in very different ways. 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. For Newton the calculus was geometrical while Leibniz took it towards analysis however the controversy surrounding them was both of them religiously preached infinitesimal calculus.
Although one could not argue with the success of calculus, this concept of infinitesimals bothered mathematicians. Lord Bishop Berkeley made serious criticisms of the calculus referring to infinitesimals as “the ghosts of departed quantities”. Ultimately, 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 and definite integrals.
So when do you use calculus in the real world? In fact, you can use calculus in a lot of ways and applications. Among the disciplines that utilize calculus include physics, engineering, economics, statistics, and medicine. It is used to create mathematical models in order to arrive into an optimal solution. For example, in physics, calculus is used in a lot of its concepts. Among the physical concepts that use concepts of calculus include motion, electricity, heat, light, harmonics, acoustics, astronomy, and dynamics. In fact, even advanced physics concepts including electromagnetism and Einstein’s theory of relativity use calculus. In the field of chemistry, calculus can be used to predict functions such as reaction rates and radioactive decay. Meanwhile, in biology, it is utilized to formulate rates such as birth and death rates. In economics, calculus is used to compute marginal cost and marginal revenue, enabling economists to predict maximum profit in a specific setting. In addition, it is used to check answers for different mathematical disciplines such as statistics, analytical geometry, and algebra.
signing off…….Akarsh b Vasisht