What is the meaning of the second derivative of the volume of a sphere with repect to it's radius?

Question

In general, what is the meaning of the second derivative of a given three dimentional figure? For a sphere, the second derivative is 8piR and the first is 4piR^2. The first derivative tells the surface area, but what does 8piR mean?

Answer 1

It is the rate at which the surface area grows with growing R (or shrinks with shrinking R).

Regarding Helmut's answer:
If instead of using x for the cube, you use the radius of the inscribed sphere, thus x = 2r, you get
V = (2r)^3 = 8r^3
dV/dr = 24r^2 = A

Why is dV/dx half the surface area, while dV/dr is the whole surface area? I have not made a calculational mistake here. The reason is, increasing the r of the cube can be thought of as pushing out all six sides of the cube simultaneously, away from the center, which is where one end of r is fixed. The volume changes by the area of all six faces together, times dr. So dV/dr is the area of all six faces together, or just A.

However, x is the length of one edge of the cube. It is measured from corner to adjacent corner. As we increase x by dx, we move one corner of the cube by dx, but hold the adjacent corner fixed in place. So, only three faces of the cube move as x grows by dx. The volume changes (dV) by the area of just the three moving faces together, times dx. So dV/dx is the area of three faces together, or A/2.

Another way to think of it is that all six faces move by half of dx. So dV is the surface area A times half of dx. So dV/dx is half of A.

Things may get a little skewed with shapes that you can't inscribe a sphere inside, such that it just touches each face. But when you can, it seems to me that dV/dr is always A.


Regarding Tom's answer, just below mine: the original poster is correct. It is the surface area, not 4 times the surface area.

Answer 2

I think it is a coincidence that the first derivative of the volume of a sphere turns out to be the surface area of that sphere. Or maybe there is some underlying principle that relates to a sphere being a so-called 'perfect' object. That, I cannot answer, but all I can say, is that the equation, V = 4*pi*r^3 / 3, can be thought of as a graph on Cartesian coordinates, in the form y = 4*pi*x^3 / 3. The derivative of this is dy/dx = 4*pi*x^2, which is the slope of the graph of y. The second derivative is d²y/dx² = 8*pi*x, which is the slope of the graph of dy/dx. Maybe that's all there is to it. It certainly was an interesting question to ponder.
P.S. : If the sphere was 'perfect', perhaps one would expect the second derivative to be the circumference or some other simple linear measure such as diameter or radius, but it looks like we'll have to face reality and know that perfection is unattainable.

Answer 3

Surface area S scales as r^2. The volume V scales as ∫ r^2 dr = r^3/3, so that V = S r/3. For V=r this reduces to 1 = S/3, or S=3. *************** Remo, no implicit assumptions about desserts and appetizers. Just symmetry, and Euclidean metric of course. More rigorously, use the Stokes theorem: ∫ r dS = ∫ ∇ r dV, r is the radius vector, ∇ r = D, D is the dimension of space. Due to symmetry this gives |r| S = D V. If |r|=V then S=D. Looks more scientific, but the essense is the same.

Answer 4

Actually dude you have got everything wrng. For your info first derivative of volume of sphere is going to be 12piR^2. And this doesn`t tell surface area but tells the rate at which the volume changes with respect to the radius.

ASsssssssss for your question

Second derivitive tells the slope of first derivative wrt the radius.

Answer 5

For a sphere,
4πR^2 = A
8πR = 4C

for a cube,
V = x^3
dV/dx = 3x^2 = A/2
d²V/dx² = 6x =3P/2

This leads to the conclusion that dV/dr = A for a sphere is coincidental, and not a general relationship.

Answer 6

Here's an easy way to think of it:

It's the circumference of the circle you would create if took the surface area of the sphere and flattened it out.

Answer 7

just think : the first derivative tells the rate of change in a certain direction of the curvature.
the second derivative is the rate of change of the first derivative in a certain direction.

Answer 8

Well we've been doing something similar in Calc. All of ours though have been in relation to rate....dx/dt , ds/dt, etc with cones and triangles. Could you be more specific? I might not be able to help you...I'm not sure which level of Calculus you're at...

Answer 9

My best answer is that it is 4 times the circumference with respect to r.

Answer 10

I think the answer is Rory Emerald?

good luck!