Why was calculus invented?

Question

Can anyone show me any flaw for which there was a need of inventing calculus or give me an example where normal maths fail and we need calculus?

Answer 1

Basically, it was created to compute areas of regions enclosed by curves and to determine tangent and normal lines to curves.

Answer 2

This is actually an excellent question and at the time calculus first emerged in the 17th century in the work of Leibniz and Newton (in alphabetical order, without implying any priority), many top-level scholars were sceptical and in fact issued a number of challenges. For example, Huygens adopted the position that the problems Leibniz can solve with the new calculus are no different from the problems one can solve with the old methods, and the difference is only in the language. Huygens may have been at least partly right if he was referring to earlier techniques such as Fermat's adequality which was indeed a powerful method that could solve many problems that today we typically think of as standard ware of the calculus: finding tangents to curves, maxima and minima, areas, centers of mass, etc. In fact, Lagrange considered Fermat to be the inventor of the calculus.

Half a century later, Bishop Berkeley was still vituperating against the new calculus, but the tide has clearly changed and the Fermat-Leibniz-Newton framework was an undeniable success in the hands of Euler.

Something similar is happening today, with Abraham Robinson's "legalisation" of infinitesimals in his 1966 book. Half a century later today, opposing voices are still being heard, but the tide is clearly changing, with high-profile advocates like Fields medalist Terry Tao publishing applications and new insights based on Robinson's framework on an almost monthly basis.

Appreciating the advantage of the calculus over these earlier techniques involves detailed knowledge of the calculus and it is not that easy to give an elementary problem that can be solved by the calculus but not by Fermat's adequality, for example. Using the calculus does simplify calculations significantly compared to the earlier methods.

One specific example that could be mentioned is finding the general tangent line of the cycloid curve. Fermat was able to do it by means of an intricate geometric argument relying on his technique of adequality; see this publication for details. Modern proofs using calculus are more straightforward.

Answer 3

Invention of Calculus was in order to determine area under a curve and study the rate of change.

Answer 4

What is the speed of a dropped stone after 1 second ?