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.
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
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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- Fall '09
- Limit, Δx