Unformatted text preview: r e p ar t. The differentiation operation gives the EXPRESSION of the
dif entiation
integ
INSTANTANEOUS RATE OF CHANGE. The integr ation operation gives us the
inte
EXPRESSION of CHANGE.
From the calculation point of view the symbol dx in the denominator of df/dx
calculation
tells us with respect to which independent variable we differentiated the function*. If
this independent variable happens to have a units of measure associated with it,
units
then dx has the same units of measure . For example, dt has the standard units
units
of measure [second] along the time axis. Likewise, dt 2 has [sec 2 ] as units of
[sec
units
measure .
From the Analysis point of view the symbol dx denotes the calculation at an instant
Analysis
instant
rather than over an inter val . Later in integr ation we shall see the symbol dx in
interv
integ
inter
inte
a similar role.
Let us look at some examples to understand the concept. Let s = position
or distance, v = speed, a = acceleration, with distance measured in [meters]
and time t measured in [secs].
SPEED = d POSITION
[meter / sec]
sec]
dt ACCELERATION = d SPEED
[meter / sec2 ]
sec
dt * Note: In A LITTLE MORE CALCULUS we shall see how to differentiate a function of
partial differentia
entiation
more than one independent variable. This is known as par tial differentiation .
par
84 units of measure operation
v= ds
dt a= dv
d2s
=2
dt
dt [meters / sec] = d[meters]
d[sec]
[meters / sec2] = d[meters / sec] = d2[meters]
d[sec]
d
d[sec]2 v = âˆ«a . d t [meters / sec] = s = âˆ«v . d t [meters] = Area = âˆ« length . d breadth d[sec]
âˆ« [meters / sec2] . d d[sec]
âˆ« [meters / sec] . d [meters]2 = âˆ«[meters] . d[meters] âˆ« FORCE . d s = ENERGY d ENERGY
= POWER
dt âˆ« POWER . d t = ENERGY This format of units of measur e calculation should serve as a template in
units measure
any application. 85 1 8. DIFFERENTIABILITY If you take a closer look at how we took the LIMIT in the FIRST DERIVATIVE of x n , you will notice that when we took
f (x + h )  f (x)
L imit f (x + Î...
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 Fall '09
 TAMERDOÄŸAN
 Limit, Î”x

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