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 .
Question 1: How can we go from inter v al [ t 1, t 2], no matter
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
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
we can find the INSTANTANEOUS RATE OF CHANGE. 3 From the Analysis point of view we would like to introduce some special notation to
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.
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
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 :
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 :
LIMIT y(t + δ t) − y(t )
= t → t δt = δt→ 0 δt = δt→ 0 (t1 + δ t) − t 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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- Fall '09
- Limit, Δx