derivative

An enlightening discussion about pi and tau.

derivative

by imagladry » Thu Jul 07, 2011 2:15 pm

Any even notice the first derivative of Area with respect to the radius is the circumference?

A = pi R^2

DA = 2 pi R DR

c = 2 pi R

That not as pretty if using Tau.
imagladry
Kindergarten
 
Posts: 1
Joined: Thu Jul 07, 2011 2:08 pm

Re: derivative

by SpikedMath » Thu Jul 07, 2011 2:54 pm

This actually still works with tau.

[tex]A=\frac{1}{2}\tau r^2[/tex]
[tex]\frac{dA}{dr}=\tau r[/tex]

and the Circumference is [tex]C=\tau r[/tex]

It should work regardless of which circle constant is used :P
Math - It's in you to give.
User avatar
SpikedMath
Site Admin
Site Admin
 
Posts: 133
Joined: Mon Feb 07, 2011 1:31 am
Location: Canada

Re: derivative

by rdococ » Sat Oct 12, 2013 6:50 pm

SpikedMath wrote:This actually still works with tau.

[tex]A=\frac{1}{2}\tau r^2[/tex]
[tex]\frac{dA}{dr}=\tau r[/tex]

and the Circumference is [tex]C=\tau r[/tex]

It should work regardless of which circle constant is used :P

If you take the inverse of the derivative on the circumference, you will realise why [tex]\tau[/tex] is the better one to use in the case of area.

[tex]\int x y \ dx = {x y^2 \over 2}[/tex]

Filling for [tex]x = \tau[/tex] and [tex]y = r[/tex],

[tex]A = \int C(r) \ dr = \int \tau r \ dr = {\tau r^2 \over 2} = \pi r^2[/tex]

[tex]\pi[/tex] does not take the meat pie when it comes to the area containing all that juicy meat. Indeed, [tex]\tau[/tex] is victorious.
rdococ
Elementary School
Elementary School
 
Posts: 13
Joined: Tue Sep 10, 2013 11:23 am

Re: derivative

by 1=0 » Sun Mar 19, 2017 11:31 am

Area comes from circumference, not the other way around, in the same way that two dimensions come from one dimension. Archimedes' method can be used to find circumference, and area is found by integrating circumference with respect to radius, yielding 1/2 C r or by splitting the polygons used in Archimedes' method into TRIANGLES, which have an area of 1/2 bh. Notice that finding the areas of the polygons using this method results in 1/2 Pa, which for a circle is 1/2 Cr. This also shows again that area comes from perimeter, since you have to find the perimeter of a polygon first before you measure its area.
τ>π
1=0
Mathlete
Mathlete
 
Posts: 50
Joined: Sun Mar 19, 2017 11:01 am


Return to The Pi Manifesto



Who is online

Users browsing this forum: No registered users and 2 guests

cron
Fatal: Not able to open ./cache/data_global.php