alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Since there are uncountably many of them it does not

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Unformatted text preview: say about f ” (x) ? Is the cur ve changing shape ? It makes sense to say that some object is going up and coming down at shor ter and shor ter time inter vals. If we now note that the maximum height in succeeding time intervals (t0, t1), (t1, t2), (t2, t3), ... is decreasing, we may reasonably interpret this to mean that the object is bouncing in some kind of force (gravitational) field which is causing it to slow down. If on the other hand the time intervals were of the same duration (t0, t1) = (t1, t2) = (t2, t3) = ... and the maximum height in succeeding intervals kept decreasing, how would you interpret this ? A pendulum oscillating ? Is it not amazing that Galileo knew so much about falling objects, pendulums, projectiles and planetary motion, yet missed discovering gravity ! 124 INTEGRATION Par t 4 : INTEGRATION 125 VEDIC mathematicians treated Integration in the Ekadikha Sutra as the natural inverse of Differentiation: the ANTIDERIVATIVE. The application was esoteric in nature and more of aesthetic value ( Bijaganita = Algebra )rather than B ijaganita A lgebra geometric ( Patig anita ): finding the area under the curve. tiganita While various approximations were employed in different ancient civilizations to find the areas enclosed by cer tain special cur ves, the credit is usually given to Archimedes (c.287-212B.C.), of unsinkable fame, for the embryonic idea of the general method of INTEGRATION for finding the area under a cur ve. This concept remained dor mant till the Renaissance mathematicians Cavalieri (1591-1647) and Torricelli (1608-1647), both students of Galileo (1564-1642), made it germinate. Issac Barrow (1630-1677), a teacher of the genius Sir Isaac Newton (16421727), was the first to make the connection between the slope of the TANGENT (deri v a ti v e ) and “ar ea under the cur v e ” in 1659. This idea eriv tiv area was abstracted and developed both by Newton and Leibniz to establish fir mly the r elationship between Dif fer ential and Inte gr al Calc...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

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