If these "differential approximations" are confusing to you, don't give up.
It took me the looongest time to understand them.
A globe with a diameter of 30 cm is to be painted
with a uniform layer of paint which is 4 mm thick.
Use the method of differentiation to find the approximate volume of paint needed.
We have a sphere with radius 15 cm.
It has a volume according to the formula: .
By adding a layer of paint, the radius is increased by
The question is: by how much is the volume increased?
We could do the actual arithmetic,
. . but we can approximate the answer with differentials.
We have: .
Take differentials: .
If we understand what each part means, we're all set!
We know: .
The volume of the paint needed is approximately 1131 cm³