Unformatted text preview: differentiate each of the various functions just once and build a Table of Derivatives.
Deriv tiv
We then look at more complicated functions made up from elementary functions. To
differentiate these combinations of other functions we have a Set of Rules .
Set
We also look at the Units of Measur e in differ entiation and integr ation . A
differentia
integ
Units Measure
dif entiation
inte
calculation usually has two parts: an operation and a units of measur e. The
opera
oper
units measure
operation with a function or expression as an input operand produces as output
function expression
another function or expression. When we evaluate this output expression we get a
function expression
value. The output expression or value has a units of measure associated with
expression value
units
it. For example, we may perform an operation to find the expression π r 2 that tells
us the area of a circle. When we evaluate this expression π r 2 , say with r = 2, we get
the value 4 π . The expression π r 2 or the value 4π has units of measure
units measure
[meter2 ] associated with it.
We assume we are dealing with WELLBEHAVED functions : SINGLE VALUED,
CONTINUOUS and DIFFERENTIABLE. We have seen the first two properties ( SINGLE
VALUED and CONTINUOUS ) and why they are necessary. We conclude this part by
looking at the DIFFERENTIABLE property.
56 INSTANT
ANTANEOUS RATE
1 2 . INSTANTANEOUS RATE OF CHANGE of
h
e
i
g
h
t y(t2)
y(t1)
h1 (0 , 0) y(t)
y (t) t1 h2 t2 T time If we measure the height at any chosen instant, say t 1 , then in terms of
the height function y(t) = u . s i n θ . t  1 g t 2 t he AVERAGE SPEED over
2
time t = 0 to t 1 is :
y (t )
distance
= u . s i n θ  1 g t
=
2
time
t
We may ask: what is the actual ver tical speed at the time instant t 1 ?
i nstant
We shall do this in three steps.
VERAGE step:
1. AVERAGE step: If we take a par ticular small time interval, say t 1 t o t 2 ,
we could be more precise. AVERAGE VERTICAL SPEED from time t1 to t2 is : CHANGE in height
C HANGE in time ∆ y y(t ) − y(t )
= ∆ t = 2t − t 1
2
1 57 step:
2. TENDS TO s...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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