Both men contributed a great deal to mathematics in general and calculus in particular. And since then, the concept has been developed even further. Calculus is used in all branches of math, science, engineering, biology, and more.
There is a lot that goes into the use of calculus, and there are entire industries that rely on it very heavily. For example, any sector that plots graphs and analyzes them for trends and changes will probably use calculus in one way or another. There are certain formulae in particular that demand the use of calculus when plotting graphs. And if a graph's dimensions have to be accurately estimated, calculus will be used.
It's sometimes necessary to predict how a graph's line might look in the future using various calculations, and this demands the use of calculus too. Engineering is one sector that uses calculus extensively. Mathematical models often have to be created to help with various forms of engineering planning. And the same applies to the medical industry.
Anything that deals with motion, such as vehicle development, acoustics, light and electricity will also use calculus a great deal because it is incredibly useful when analyzing any quantity that changes over time. So, it's quite clear that there are many industries and activities that need calculus to function in the right way.
It might be close to years since the idea was invented and developed, but its importance and vitality has not diminished since it was invented. There are also other advanced physics concepts that have relied on the use of calculus to make further breakthroughs.
In many cases, one theory and discovery can act as the starting point for others that come after it. For example, Albert Einstein wouldn't have been able to derive his famous and groundbreaking theory of relativity if it wasn't for calculus. Relativity is all about how space and time change with respect to one another, and as a result calculus is central to the theory.
In addition, calculus is often used when data is being collected and analyzed. The social sciences, therefore, must rely on calculus very heavily. For example, calculating things like trends in rates of birth and rates of death wouldn't be possible without the use of calculus. And economic forecasts and predictions certainly use calculus a great deal. He also showed how to compute the slope of a line tangent to a spiral, the first glimmer of differential calculus.
The beauty of calculus as we now know it comes from its simplicity. The Fundamental Theorem of Calculus enables us to solve very difficult problems by applying simple calculational procedures that are justified by the Fundamental Theorem. Archimedes did not have these, so he had to rely on basic principles and, with a great deal of ingenuity, come up with clever solutions. Archimedes found volumes of the parabaloid and other solids by using a balancing argument in which he compared the moments of different solids.
He calculated the area under a parabola not by the usual method of approximating it by a sum of squares, but by using a geometric observation that enabled him to reduce the problem to finding the sum of a geometric series.
This is one of a number of important uses of geometric series that contributed to the development of calculus. The problem that I want to focus on is that of finding the area inside a spiral. Archimedes employed what has come to be known as the "method of exhaustion.
Archimedes attributes this method to Eudoxus of Cnidus — B. While Eudoxus found the first proof, this result is even older: it was found by Democritus c. The formula for the volume of a pyramid was also discovered in ancient India, and we have record of it in the Chinese book Chiu Chang Suan Ching Nine Chapters on the Mathematical Art that may have been written as early as B. Even before anyone could prove this formula, finding it required thinking of the pyramid as made up of thin slices, three sets of which could be reconfigured to make a rectangular block.
Later in this article we'll see how this mental three-dimensional geometry led Archimedes to the formula he needed to find the area of the spiral. The total area lies between. At this point, Archimedes derived a succinct formula for the sum of the first n - 1 squares:.
The formula for the sum of squares may not have been new to Archimedes, and there is evidence that it might have been discovered about the same time in India. We do know that it was rediscovered many times. The earliest proofs, including Archimedes's proof, are all geometric. We can visualize the sum of squares as a pyramid built from cubes Figure 8. In other words, he showed that. The story is that he discovered it when his teacher ordered him to add the integers from 1 through In fact, it is an ancient formula.
It can be found, for example, in India in a Jain manuscript from B. It has a very simple geometric proof as shown in Figure 9. We still have to prove equation 2 , but once we know it is true, we can combine it with the formula for the sum of the first n - 1 integers to get. We split this rectangular pyramid into two pieces Figure These pieces fit around the green pyramid in Figure 8 so that one more inverted pyramid completes the n x n x n - 1 block Figure 12 :.
Figure The n x n x n - 1 block assembled from three sums of squares and one sum of integers. You might think that having seen how useful the sum of squares formula is, Archimedes would then have found the formula for the sum of cubes.
He didn't. Both men did quite a lot of work forming a language of numbers that could accurately describe nature. So why was this new and complicated form of mathematics invented and how does one manage to come up with such an abstract idea?
Newton was foremost a physicist, and in his day, he tackled many difficult issues in physics, the most famous of which was gravity. Newton is known for developing the laws of motion and gravitation, which undoubtedly led to his work in calculus. When trying to describe how an object falls, Newton found that the speed of the object increased every split second and that no mathematics currently used could describe the object at any moment in time.
Newton took some time to ponder the question and came back with the fact that the ellipses are actually sections of cones. Armed with calculus, he could describe exactly how those sections behaved. These types of questions and the fact that Cambridge University, where Newton studied, was closed due to numerous outbreaks of the plague, drove Newton to expand on mathematics and develop the concepts of differential and integral calculus.
Like most scientific discoveries, the discovery of calculus did not arise out of a vacuum. In fact, many mathematicians and philosophers going back to ancient times made discoveries relating to calculus.
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