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It becomes infinitely small this kind of difference

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Unformatted text preview: tants t 1 , t 2, t 3 o r t 4 ? We particular par may drop the subscript and let t be a general instant in [t0 , tn]. general dy . --- , y ’ (t), Dy(t), y (t) dt So far we have only a NOTATION, some symbols that express the concept of what we are trying to do: find the INSTANTANEOUS RATE OF CHANGE of the particular function y(t) AT ANY INSTANT. 59 When we perform the calculation to find the INSTANTANEOUS RATE OF CHANGE of y(t) at any instant we get (refer Preface iii) : dy INSTANTANEOUS VERTICAL SPEED = --- = u . s i n θ -- g t dt Compare this expression of INSTANTANEOUS VERTICAL SPEED to the ear lier calculation of AVERAGE VERTICAL SPEED: y(t) 1gt t = u . s i n θ -- -2 When we compare the AVERAGE SPEED with the INSTANTANEOUS SPEED we make the profound distinction between an INTERVAL (no matter how small) and an INSTANT. Only with this distinction and the calculation using INSTANT are we able to get the INSTANTANEOUS SPEED. AVERAGE VERTICAL SPEED = δ t is an INTERVAL. δ x is an INTERVAL. L imit (a + δ t) is the INSTANT a. δt → 0 L imit (x + δ x) is any INSTANT x. δx → 0 Analysis interv From the Analysis point of view : an inter val δ t or δ x, no matter how small, is Anal inter a continuous set of instants or points. continuous instants points From the Algebra point of view : an inter val corresponds to a subset of the real Algebra interv Alg inter numbers R. This subset has UNCOUNTABLY many real numbers (both rational and irrational numbers) in a contiguous manner, ie; with none missing in between. contiguous Since we cannot even begin to COUNT them, there is no sense in talking about a subscript to enumerate them. We shall use the words DIFFERENTIATE or FIND THE DERIVATIVE to denote the process of finding the expression for the INSTANTANEOUS RATE OF CHANGE of a function. Let us now use this concept to define the INSTANTANEOUS RATE OF CHANGE of a general function f(x). 60 INSTANT ANTANEOUS RATE 1 3 . INSTANTANEOUS RATE OF CHANGE of f(x) f (x) f(x) f(x)2 f(x)1 x1 (0 , 0 ) x2 x What is the instantaneous ra...
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