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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 :
LIMIT y(t + δ t) − y(t )
= t → t δt = δt→ 0 δt = δt→ 0 (t1 + δ t) − t 1
This ver tical speed or “r a te of c hange” i n height got by taking the LIMIT
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
other notations for the FIRST DERIVATIVE :
--- , 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
interval t0 to tn.
t y(t2) y(T ) = maximum
h1 y(t4) h3
(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
time interval t0 to tn, rather than at par ticular ins...
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- Fall '09