In the case of a time-distance graph, for example, this slope represents the speed of the object at a particular point. Having established the derivative function for a particular curve, it is then an easy matter to calcuate the slope at any particular point on that curve, just by inserting a value for x. For example, the derivative of the curve f( x) = x 4 – 5 x 3 + sin( x 2) would be f ’( x) = 4 x 3 – 15 x 2 + 2 xcos( x 2). Other derivative functions can be stated, according to certain rules, for exponential and logarithmic functions, trigonometric functions such as sin( x), cos( x), etc, so that a derivative function can be stated for any curve without discontinuities. Generalizing, the derivative of any power function f( x) = x r is rx r-1. For instance, the derivative of a straight line of the type f( x) = 4 x is just 4 the derivative of a squared function f( x) = x 2 is 2 x the derivative of cubic function f( x) = x 3 is 3 x 2, etc. This process of calculating the slope or derivative of a curve or function is called differential calculus or differentiation (or, in Newton’s terminology, the “method of fluxions” – he called the instantaneous rate of change at a particular point on a curve the “fluxion”, and the changing values of x and y the “fluents”). Without going into too much complicated detail, Newton (and his contemporary Gottfried Leibniz independently) calculated a derivative function f ‘( x) which gives the slope at any point of a function f( x). an infinitesimal change in x), then the calculation of the slope approaches closer and closer to the exact slope at a point (see image at right). As the segment of the curve being considered approaches zero in size (i.e. Intuitively, the slope at a particular point can be approximated by taking the average slope (“rise over run”) of ever smaller segments of the curve. effectively the slope of a tangent line to the curve at that point. The initial problem Newton was confronting was that, although it was easy enough to represent and calculate the average slope of a curve (for example, the increasing speed of an object on a time-distance graph), the slope of a curve was constantly varying, and there was no method to give the exact slope at any one individual point on the curve i.e. The Average Slope of a Curveĭifferentiation (derivative) approximates the slope of a curve as the interval approaches zero Unlike the static geometry of the Greeks, calculus allowed mathematicians and engineers to make sense of the motion and dynamic change in the changing world around us, such as the orbits of planets, the motion of fluids, etc. His theory of calculus built on earlier work by his fellow Englishmen John Wallis and Isaac Barrow, as well as on work of such Continental mathematicians as René Descartes, Pierre de Fermat, Bonaventura Cavalieri, Johann van Waveren Hudde and Gilles Personne de Roberval. Over two miraculous years, during the time of the Great Plague of 1665-6, the young Newton developed a new theory of light, discovered and quantified gravitation, and pioneered a revolutionary new approach to mathematics: infinitesimal calculus. His 1687 publication, the “Philosophiae Naturalis Principia Mathematica” (usually called simply the “Principia”), is considered to be among the most influential books in the history of science, and it dominated the scientific view of the physical universe for the next three centuries.Īlthough largely synonymous in the minds of the general public today with gravity and the story of the apple tree, Newton remains a giant in the minds of mathematicians everywhere (on a par with the all-time greats like Archimedes and Gauss), and he greatly influenced the subsequent path of mathematical development. Physicist, mathematician, astronomer, natural philosopher, alchemist and theologian, Newton is considered by many to be one of the most influential men in human history. But the greatest of them all was undoubtedly Sir Isaac Newton.
SIR ISAAC NEWTON INVENTIONS FULL
In the heady atmosphere of 17th Century England, with the expansion of the British empire in full swing, grand old universities like Oxford and Cambridge were producing many great scientists and mathematicians.
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