So, the topic of Calculus

[Toastedlightly]

We know that the volume of a sphere is 4/3*Pi*r^3, and the derivative of that is 4*Pi*r^2 is the surface area. What is 8*Pi*r?

 

[Oil]

Its the derivative of 4*Pi*r^2

 

[saymyname]

4 x the circumference of a cross section of the sphere.

 

[nwfsnake]

The Genesis Wave!

 

[Syringer]

Originally posted by: saymyname
4 x the circumference of a cross section of the sphere.

Hmm been thinking of this question, and that is true but I can't figure out why it is.

 

[Triumph]

Well, nothing I guess. I mean you can keep deriving forever, but what's the point? Although I used to wonder what the derivative of acceleration was. Well, it's the time rate of change of acceleration. And what about the derivative of the derivative of acceleration? And the derivative of the derivative of the derivative of acceleration? At that point it just gets pointless.

 

[nwfsnake]

Derivative of acceleration is commonly called "jerK". I kid you not!

 

[TheoPetro]

Originally posted by: nwfsnake
Derivative of acceleration is commonly called "jerK". I kid you not!

yup its how fast the acceleration is changing with respect to time

 

[Born2bwire]

Originally posted by: Syringer

Originally posted by: saymyname
4 x the circumference of a cross section of the sphere.

Hmm been thinking of this question, and that is true but I can't figure out why it is.

Circumference of circle = int(r*d\phi, 0, 2*\pi).
Area of circle = int(r*dr*d\phi, 0, r, 0, 2*\pi).
Surface area of sphere = int(r^2*sin(\theta)*d\phi*d\theta, 0, 2*pi, 0, pi)
Volume of sphere = int(r^2*sin(\theta)*dr*d\phi*d\theta, 0, r, 0, 2*pi, 0, pi)

The surface and volume formulas are related directly by d/dr due to the fact that the only difference between the integrations is the dr. But for the circumference of the circle, you drop the d\theta which was what introduced the r*sin(\theta) term in the area and volume elements. The sin(\theta) creates a factor of two after the integration while taking the derivative of r^2 brings out the other factor of 2. Hence, you will result in a factor of 4 times the circumference when you take the derivative of the surface area.