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Unformatted text preview: r e p ar t. The differentiation operation gives the EXPRESSION of the
INSTANTANEOUS RATE OF CHANGE. The integr ation operation gives us the
EXPRESSION of CHANGE.
From the calculation point of view the symbol dx in the denominator of df/dx
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,
then dx has the same units of measure . For example, dt has the standard units
of measure [second] along the time axis. Likewise, dt 2 has [sec 2 ] as units of
From the Analysis point of view the symbol dx denotes the calculation at an instant
rather than over an inter val . Later in integr ation we shall see the symbol dx in
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]
dt ACCELERATION = d SPEED
[meter / sec2 ]
dt * Note: In A LITTLE MORE CALCULUS we shall see how to differentiate a function of
more than one independent variable. This is known as par tial differentiation .
84 units of measure operation
dt a= dv
dt [meters / sec] = d[meters]
[meters / sec2] = d[meters / sec] = d2[meters]
d[sec]2 v = ∫a . d t [meters / sec] = s = ∫v . d t [meters] = Area = ∫ length . d breadth d[sec]
∫ [meters / sec2] . d d[sec]
∫ [meters / sec] . d [meters]2 = ∫[meters] . d[meters] ∫ FORCE . d s = ENERGY d ENERGY
dt ∫ POWER . d t = ENERGY This format of units of measur e calculation should serve as a template in
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 +...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
- Fall '09