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Calculus/Vol and Area


Hi Scotto,

The Volume of a sphere is 4/3 pi r^3 and the area is the first differential ie 4pir^2. Since the first differential of a slope is a straight line why then is the first differential of volume=area. How about if I take the first differential of the area ie 8*pi*r..does it mean anything? Thanks!

If the equation gives the volume, the derivative of this usually gives the surface area.
For a sphere, 4πr/3.  The derivative, 4π, is the surface area.  I'm not sure what the next derivative is used for in volume.

As far as distance, the distance fallen due to gravity (in meters) is 9.81t/2.
The derivative is this with respect to time t is 9.81t, and that is the velocity.
the derivative of velocity is 9.81, and that is the acceleration.

Falling from rest, after 2 second has passed, the distance is 9.81*2/2 = 19.62.
The velocity is 9.81*2 = 19.62.  The second derivative, 9.81, is the acceleration.
This means acceleration is always the same.  That is, of course, neglecting wind resistance.


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