alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

S i n 2 when we compare the average speed with the

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Unformatted text preview: tep: the smaller the time interval t2 -- t1 the more accurate the calculation of the AVERAGE VERTICAL SPEED at time instant t1. So we fix t1 and let t2 get closer and closer to t1. The difference ∆ t between t1 and t2 becomes smaller and smaller. It becomes infinitely small. This kind of difference we denote using the Greek symbol δ. As t2→ t 1 we may say : t2 = t 1 + δ t . So : δ t = t2 − t1 = (t1 + δ t) − t 1 and δ y = y(t2) − y(t1) = y(t1 + δ t) − y(t1) Hence, δy δt y(t1 + δ t) − y(t1) (t1 + δ t) − t1 = step: 3. LIMIT step: Finally, when we let t2 coincide with t1, we get the INSTANTANEOUS VERTICAL SPEED at instant t1. This we denote by : instant dy LIMIT δy LIMIT δy LIMIT y(t + δ t) − y(t ) = t → t δt = δt→ 0 δt = δt→ 0 (t1 + δ t) − t 1 dt 2 1 1 1 “ra chang hange” This ver tical speed or “r a te of c hange” i n height got by taking the LIMIT “r is called the INSTANTANEOUS SPEED or INSTANTANEOUS RATE OF CHANGE in height at the time instant t 1 . We can now answer Question 2 on page 3. Limit of AVERAGE RATE OF CHANGE of y(t) over the interval [ t 1, t 2 ] as t 2 TENDS TO t 1 = INSTANTANEOUS RATE OF CHANGE of y(t) at the instant t1 This is called the FIRST DERIVATIVE of y(t) at instant t1. There are several instant other notations for the FIRST DERIVATIVE : dy . --- , y ’ (t 1), Dy(t 1) , y (t1) dt t () 1 58 We know from obser vation that the INSTANTANEOUS VERTICAL SPEED or INSTANTANEOUS RATE OF CHANGE in height is not the same at different instants t 1, t 2 , t 3 a nd t 4 i n time. L imit Using this same method of taking the δ t→ 0 we can find the INSTANTANEOUS RATE OF CHANGE in height at any par ticular instant t2 or t3 or t4, in the time particular par interval t0 to tn. h e i g h t y(t2) y(T ) = maximum / 2 y(t3) y(t1) h1 y(t4) h3 h2 h4 (0 , 0) t 0 t 1 t2 T / 2 t3 t4 tn = T time Can we find the INSTANTANEOUS RATE OF CHANGE in height at any instant t in the any an time interval t0 to tn, rather than at par ticular ins...
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