Relative rate of changeProblem: Volume of a cubic box is V = L 3 . How are the relative rates of change of V and L related?This problem seems really simple, but I can't understand the concept of a relative rate of change. Here are my workings:Original equation V = L 3 I write it in form of differential equation ∂ V = 3 L 2 ∂ Lthen I divide the 2nd line by the 1st ∂ V V = 3 L 2 ∂ L L 3 ∂ V V = 3 ∂ L L Expression of the form ∂ f ( x ) f ( x ) is called a relative rate of change. And it can be thought of as percentage change. So as I see it by knowing the percentage change in L we can work out the respective percentage change in V, right? Wrong. I tried to make sense of it by putting values into the equation but no success. E.g. we increase L from 2 to 3 (by 50%) thus according to the last formula we should have a respective increase in V of 150% (50% times 3) which is not true ( 3 3 − 2 3 2 3 is a 237.5% increase). Can you help me out? I'm definitely missing something either in computation or more likely in understanding the concept.

Question

Relative rate of changeProblem: Volume of a cubic box is   V  =      L    3  . How are the relative rates of change of V and L related?This problem seems really simple, but I can&#039;t understand the concept of a relative rate of change. Here are my workings:Original equation  V  =      L    3  I write it in form of differential equation  ∂  V  =  3      L    2    ∂  Lthen I divide the 2nd line by the 1st            ∂      V        V    =            3              L        2            ∂      L              L      3                  ∂      V        V    =  3            ∂      L        L  Expression of the form             ∂      f      (      x      )              f      (      x      )       is called a relative rate of change. And it can be thought of as percentage change. So as I see it by knowing the percentage change in L we can work out the respective percentage change in V, right? Wrong. I tried to make sense of it by putting values into the equation but no success. E.g. we increase L from 2 to 3 (by 50%) thus according to the last formula we should have a respective increase in V of 150% (50% times 3) which is not true (                    3        3            −              2        3                    2      3      is a 237.5% increase). Can you help me out? I&#039;m definitely missing something either in computation or more likely in understanding the concept.

ohhappyday890b · Accepted Answer

You have set             δ      L        L    = percent which is quite large, as the equation holds in the limit as   δ  L  →  0That is, calculating the relative change in volume as you did            δ      V        V    =            (      L      +      δ      L              )        3            −              L        3                    L      3        =  3            δ      L        L    +  3            (      δ      L              )        2                    L      2        =  3            δ      L        L        (    1    +                  δ        L            L        )  So as you make   δ  L smaller, the approximation to             ∂      V        V   improves. In the limit of   δ  L  →  0, we have equality (invoking a large dose of handwavium).

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