alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

This is known as successive second successive n n1

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Unformatted text preview: . sin (∆ x)} − sin(x) ∆x δ f(x) {sin(x) . cos(δ x) + cos (x) . sin (δ x)} − sin(x) = δx δx When x is infinitely small, say δx , then as we said earlier, we may let : 2. TENDS TO step: sin ( δx) = δx and cos ( δx) = 1 δ f(x) = δx = {sin(x) . 1+ cos (x) . δ x} − sin(x) δx cos (x) . δ x δx Here we may simplify since δx ≠ 0 : δ f(x) = cos(x) δx 3. LIMIT step: Finally we take the LIMIT as δx → 0 . df(x) LIMIT δf LIMIT = δx→ 0 δx = δx→ 0 { cos (x)} dx f’(x) = cos(x) 76 6: differentia Example 6 Let us now differentiate the function y = sin (θ) the Vedic way. dif entiate +1 y y +1 y2 = sin θ2 y1 = sin θ1 θ1 −1 x 1 +1 x θ2 −1 −1 x2 +1 x −1 Let us now magnify the two diagrams and superimpose them. +1 y y2 = sin θ2 } y −y 2 1 = sin θ2 − sin θ1 θ 2− θ 1 y1 = sin θ1 ) x2 −1 x 1 +1 x −1 With the concept of arc length = r θ , we can see from the diagram that : x2 (θ2 − θ1) ≤ y2 − y1 ≤ x1 (θ2 − θ1) Since (θ2 − θ1) ≠ 0 we may divide throughout by (θ2 − θ1) . AVERAGE RATE OF CHANGE of y with respect to θ : 77 x2 ≤ y2 − y1 ≤ x1 θ2 − θ1 Now we may take the LIMIT as θ2 → θ1 to get the INSTANTANEOUS RATE OF CHANGE of y with respect to θ : y2 − y1 L imit x ≤ θ1 θ2 − θ1 θ2 → θ1 1 L imit x L imit 2≤ → → θ2 θ1 θ2 As θ2 → θ1 we can see that x2 → x1 : dy x1 ≤ --- at θ1 ≤ x1 dθ dy d So : --- at θ1 = --- (sin θ) at θ1 = x1 = cos θ at θ1. dθ dθ We may drop the subscript of particular angle θ1 to generalise and get : d --- (sin θ) = cos θ dθ There is no need to DIFFERENTIATE a function from FIRST PRINCIPLES each and every time. We may differentiate the functions that we encounter most frequently and build a table. Then all we have to do is look up the TABLE OF DERIVATIVES. 78 16. Rules Tables and Rules A function f(x) expresses CHANGE. The CHANGE could be in position, velocity, acceleration, temperature, length, area, volume, pressure, charge o...
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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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