A Few Explorations

A Few Explorations First I'll share a quick, slick method of deriving a fact I learned in Pre-Calculus, which is useful in function analysis and limits: that Lim SinHΘL = 1. The fact was proved to me Θ Θ®0

using the squeeze theorem, which gives the function an upper bounding function and a lower bounding one, then shows that the two bounding functions are equal at the point in question; this forces the function in question to take that value at that point. However, I came across it in a different and unexpected way. I was expecting to get some formula for Π, or at least something in that vein. My idea was this: look at the limit of the area of a regular n-gon with radius 1 as n goes to ¥; you should get Π. :

:

:

, 2.598 = A>, :

, 3 = A>, :

, 3.078 = A>, :

, 2.828 = A>, :

, 3.037 = A>, :

, 3.0901 = A>, :

, 2.938 = A>,

, 3.061 = A>,

, 3.099 = A>

You can see that the areas of these figures slowly approach the value of Π. So, how can we quantify this? aP The area of an n - gon with radius 1 is A = , 2 where a is the apothem and P i s the perimeter. Let' s take a look at one slice of the n - gon :

2

Explorations.nb

Θ

1

CosHΘL

2 SinHΘL Thus, A = nSin HΘL Cos HΘL. As n ® ¥, Θ ® 0, as the individual triangles get skinnier and skinnier. HThere are n of them.L Π We also know that Θ = , or n = Π Θ, so substituting we obtain n Π SinHΘL Cos HΘL A= , so that Θ Sin HΘL Cos HΘL Lim HAL = Π ” Lim = 1. But since Θ®0 Θ®0 Θ Lim HCos HΘLL = 1, we can divide by it to obtain the desired result : Θ®0

Lim Θ®0

Sin HΘL = 1. Θ

I also started messing around with plots of funky relations. Starting with the graph of x2 + y2 = k 2, I substituded other functions in for x and y. Here are all the ones I looked at:

Explorations.nb

k 2 >x2 +y2 4 2

: 0

>

-2 -4 -4

-2

0

2

4

:, ,

4

4

2

2

,

0

4

0 -2

-4 -4 -2 0

-4 -4 -2

4

>

2

,

-2 2

k 2 =x3 +y3

k 2 =x3 +y3

k 2 =x3 +y3

0 -2

k 2 >x4 +y4

0

2

-4 -4 -2

4

0

2

4

k 2 >x4 +y4

1.5 2 1.0 1 0.5 : , 0 0.0 -0.5 -1 -1.0 -2 -1.5 -2 -1 0 -1.5 -1.0 -0.50.0 0.5 1.0 1.5

k 2 >Log@xD2 +Log@yD2 3.0 2.5

:1.5

>

1

2

k 2 >Log@xD2 +Log@yD2

k 2 >Log@xD2 +Log@yD2

8

20

6

15

,4

, 10

2

5

>

2.0 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

0 0

2

4

6

8

0

k 2 >Cos@xD2 +Cos@yD2

k 2 >Cos@xD2 +Cos@yD2

: 0

5

, 0

, 0

-5

-5

-5

-5

0

5

-5

0

5

10

15

20

k 2 >Cos@xD2 +Cos@yD2

5

5

5

>

-5

0

5

3

4

Explorations.nb

k 2 >Arcsin@xD2 +Arcsin@yD2

k 2 >Arcsin@xD2 +Arcsin@yD2

:

k 2 >Arcsin@xD2 +Arcsin@yD2

1.0

1.0

1.0

0.5

0.5

0.5

,

0.0

,

0.0

-0.5

-0.5

-0.5 -1.0 -1.0

k2>

-0.5

1 x2

+

0.0

0.5

1

k2>

y2

: 2 4

0 -2 -4 -4 -2 0 2 4

1.0

-1.0 -1.0

1

1

x

2

+

y

-0.5

k2>

2

0.0

1 x

2

0.5

+

1.0

1 y

4

, 2

, 2

, 2

0

0

0

-2

-2

-2

-4

-4 4

1 x

4

2

-1.0 -1.0

k2>

2

4

-4 -2 0

>

0.0

2

+

-0.5

0.0

0.5

1.0

1 y2 >

-4 -4 -2 0

2

4

-4 -2 0

2

4

Some of them I couldn't make heads or tails of. We could find the slope of the contour at any point (x,y) by implicit differentiation, but that really doesn't tell us much. You might notice that a lot of these look like near-circles. What might be interesting is to find the area of one of them and compare it to the area of a circle the same "size," (same radius), which would give something of a measure of how close it is to a circle.