"Help me understand what is and isn't exponential growth. If the price at which something grows is proportional to itself, then you definately could name it exponential increase. do not quote me on this, however I suppose it has something of the shape y=ex. Now take for example this. if you want to calculate the extent of a sphere, the formula is v=4/3pi r^3 . The by-product with recognize to the radius is dv/dr=4 pi r^2. This suggests the rate of change of the volume with appreciate to the radius. yet the extent of the field is proportional to the radius, which in flip defines the dimensions of the field. The rate at which the volume of the sector increases is proportional to itself in a manner. Intuitively, i might assume to discover someplace in there an exponential yet there isn't alwa

Cale Terrell

Cale Terrell

Answered question

2022-10-17

Help me understand what is and isn't exponential growth.
If the price at which something grows is proportional to itself, then you definately could name it exponential increase. do not quote me on this, however I suppose it has something of the shape y = e x
Now take for example this. if you want to calculate the extent of a sphere, the formula is v = 4 / 3 π r 3 . The by-product with recognize to the radius is d v / d r = 4 π r 2 This suggests the rate of change of the volume with appreciate to the radius.
yet the extent of the field is proportional to the radius, which in flip defines the dimensions of the field. The rate at which the volume of the sector increases is proportional to itself in a manner. Intuitively, i might assume to discover someplace in there an exponential yet there isn't always. Why is that?

Answer & Explanation

Teagan Zamora

Teagan Zamora

Beginner2022-10-18Added 18 answers

You are considering the function V(r) which gives you the volume of a sphere of radius r. The rate of change V′ of this function depends on V, but it is not proportional to V. We have V ( r ) = 4 3 π r 3 and thus V ( r ) = 4 π r 2 = 4 π ( 4 3 π ) 3 2 [ V ( r ) ] 2 3 . So V′ is proportional to V 2 3 , not to V. Which is why we shouldn't expect anything exponential here.
Getting back to the beginning, I think your core misconception is "proportional = dependent on". This is not so. Proportionality is a very specific and special kind of dependence, and there are a huge variety of ways in which two things can depend on each other without it being proportional.

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