Exploring the Methods of Differential Calculus through the ...

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Biographies Sir Isaac Newton was born prematurely on Christmas, 1642. He went to Cambridge University, where his focus was mainly in chemistry. He also studied algebra and analytic geometry. He worked under Isaac Barrow, who discovered infinitesimal calculus. In 1665 there was an outbreak of the Great Plague which caused Newton to return home to Woolsthrope, England (Cooke, 1997). He remained there for two years and while Cambridge was closed he studied his own mathematics and physics research. It was during this time that he created the Binomial Theorem. Newton died in 1727, at the age of 85, in England (Struik, 1948). Gottfried Wilhelm von Leibniz was born in Leipzig, Germany in 1646. He was a Roman Catholic and his religion was a large part of his life (Struik, 1948). At fifteen, Leibniz entered the University of Leipzig, where he studied law, following in the footsteps of Descartes, Fermat and Viète (Cooke, 1997). When he graduated at the age of 20 he was deemed too young to receive his doctorate in law. After college he served as a diplomat for the Elector of Mainz. He eventually became the diplomat for the Electors of Hanover, where he worked for four decades. It was during a mission to Paris in 1672 that Leibniz first became interested in mathematics. The following year he went to London, where he met members of the royal society, Henry Oldenburg and James Collins (Cooke, 1997). Oldenburg and Collins played major roles in the calculus controversy. It was said that they told Leibniz of Newton’s advances in calculus (Rickey, 1994). He discovered his calculus between 1673 and 1676. He was influenced by Huygens and the studies of Pascal and Descartes. Leibniz died in 1716, in Germany (Struik, 1948). Results There are three basic curves that could be the solution to the Brachistochrone problem: a concave up curve, a straight line, and a concave down curve. The first candidate is the concave up curve; the second candidate is the straight line; the third candidate is the concave down curve. These candidate ramps are graphed in Figure 5. Candidate Ramp One: Candidate Ramp Two: Candidate Ramp Three: (3)
Figure 5. Graphs of the three candidate ramps. In Figure 5 (and all other figures in this section), distance stands for the horizontal distance, not the distance along the ramp. This graph represents the three candidate ramps. The bottom line is the first candidate; the middle line is the second candidate, and the top line is the third candidate. All three candidates ramps start and end at the same point but don’t have any other common points. Candidate one starts off with a steep decline and then flattens out. Candidate two has a steady decline from point A to point B. Candidate three starts off with a slowly declining and then ends with a steep decline. The next step is to take the first derivative of each candidate ramp. The first derivative will be the velocity attained by the bead at that point on the ramp. A graph of the velocity curves is presented in Figure 6. Candidate Ramp One: Candidate Ramp Two: Candidate Ramp Three:
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Exploring the Methods of Differential Calculus through the ...

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