No matter how close we take t2 to t1 ie no matter how

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Unformatted text preview: RTICAL SPEED. We do not get the ver tical speed at time instant t 1. W e ask tw o fundamental questions . two questions. Question 1: How can we go from inter v al [ t 1, t 2], no matter i nter how small, to reach the instant t 1 ? i nstant Question 2: How can we find the RATE OF CHANGE at instant t 1 from i nstant the AVERAGE RATE OF CHANGE over the inter v al [t1, t2] ? i nter With Calculus, for functions that are WELL BEHAVED: SINGLE VALUED, CONTINUOUS and DIFFERENTIABLE, we are able to find the ver tical speed at any chosen instant, i.e. from the function that expresses the CHANGE instant we can find the INSTANTANEOUS RATE OF CHANGE. 3 From the Analysis point of view we would like to introduce some special notation to Analysis express what we are doing. We may say: ∆ t = t2 − t1 . So t2 = t1 + ∆ t ∆ y = y(t2) − y(t1) = y(t1 + ∆ t) − y(t1) where the Greek letter ∆ denotes a difference that is measurable. VERAGE step: 1. AVERAGE step: So the AVERAGE VERTICAL SPEED may be denoted by : ∆ y y(t2) − y(t1) y(t + ∆ t) − y(t ) = t −t = (t1 + ∆ t) − t 1 ∆t 1 2 1 1 step: 2. TENDS TO step: We then 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. When t2 gets closer and closer to t1 we get a better or more accurate AVERAGE VERTICAL SPEED. This we denote by : 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 1 2 1 1 To do this we need to develop the concept of CONTINUOUS from the conce pts that ar e described using the special v o c a b u l a r y : T E N D S T O , L I M I T, I N F I N I T E S I M A L and I N S T A N T. We shall then show that the...
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