Differentiating the Area of a Circle

[S.R]

In my high school Calculus course, I've encountered several optimization problems involving the area of a circle and I noticed the obvious fact that if you differentiate the area of a circle you obtain the expression for its circumference. This implies that the rate of change of a circle's area is equal to its circumference (which is difficult to visualize). So what does notion actually mean?

S.R

 

[tiny-tim]

Hi S.R!

It means that the whole area is made of lots of little circumferences …

if you subtract one area from a slightly larger one, you get a circumference.

(works for spheres also!)

 

[S.R]

Oh :D Essentially the area consists of concentric circles?

 

[tiny-tim]

yup!