Derivative of the area is the circumference -- generalization

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Discussion Overview

The discussion revolves around the concept of relating the derivative of the area of geometric shapes, specifically circles and regular polygons, to their respective circumferences or perimeters. Participants explore the generalization of this idea beyond circles to other shapes, including squares and potentially higher-dimensional objects.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants note that the derivative of the area of a circle is its circumference and suggest this can be generalized to all regular polygons.
  • One participant questions the generalization and seeks clarification on what is meant by a "nice way" to express this relationship.
  • Another participant proposes using the "radius" of a square as an independent variable to explore the relationship further.
  • A participant expresses confusion regarding the statement that the derivative of the area formula for a square equals its perimeter, pointing out that the derivative of the area (x²) is 2x, while the perimeter is 4x.
  • There is a suggestion that the original text should be revised to avoid misleading interpretations.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the generalization. While some support the idea, others challenge it, leading to an unresolved discussion regarding the relationship between area derivatives and perimeter.

Contextual Notes

There are limitations in the definitions used, particularly regarding the choice of independent variables for expressing the formulas. The discussion also highlights unresolved mathematical interpretations and potential confusion in terminology.

1MileCrash
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I thought you guys might appreciate this. A lot of people notice that the derivative of area of a circle is the circle's circumference. This can be generalized to all regular polygons in a nice way.

 
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1MileCrash said:
This can be generalized to all regular polygons in a nice way.
Hi 1MileCrash:

I don't get this. What is the nice way?
The area of a square with side length x is x2.
The derivative is 2x. The circumference = boundary length is 4x.

Regards,
Buzz
 
Buzz Bloom said:
Hi 1MileCrash:

I don't get this. What is the nice way?
The area of a square with side length x is x2.
The derivative is 2x. The circumference = boundary length is 4x.

Regards,
Buzz

You've made a choice to express the formulas in terms of the square's side length, but there's nothing wrong with expressing the formulas in terms of some other measurement of the square.

The generalization is found by changing the measurement of the polygon that we express the formulas in terms of.
 
nice comment.
try using the "radius" of the square, r = x/2 as independent variable. and for a 3-ball, consider the relationship between volume and surface area. what about a 4-ball?
 
"the derivative of the area formula for a square is equal to its perimeter."

This make little sense to me. When the side equals x, the area is x^2, and the perimeter is 4x.

The derivative of x^2 is 2x, not 4x.

EDIT: I see there were other responses like mine. The text should be changed from the generalization so as not to distract the reader.
 
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