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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Differentiable 87 example 2 is fx x differentiable

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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 = ∫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 = POWER dt ∫ 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 + ...
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