So, after taking a lecture on the use of difference quotients for derivatives, I happened upon something interesting.

 

The usual equation for the different quotient is the following(applied for derivative use):

 

[math]\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}[/math]

 

Now, I know that getting the derivative of the function 3x+1 and its inverse gets the following result from testing before:

 

[math]f(x)=(3x+1)(\frac{x-1}{3})[/math]

 

 

Of course, seeing this connection to a pattern I found in hailstone sequences I left it at that. However, I then began investigating the affects of the variable h on the result. Here is what I found.

 

 

I thought this was an interesting connection. Here is an example of a hailstone sequence:

 

{3, 10, 5, 16, 8, 4, 2, 1}

 

All the numbers in red and the ones that continue on are a result of 6x-2. I will continue to experiment with this. Maybe the solution for the Collatz conjecture involves looking further into the concepts of Calculus?

 

EDIT: Therefore, if my speculative idea in fact has some truth, maybe the difference quotient can be redefined also for these particular instances?

 

[math]F(x)=\lim_{h\rightarrow 0}\frac{f(x+d_{i}h)-f(d_{e}x)}{h}[/math]

 

When d_i and d_e are in the formula [math]d_{i}x+d_{e}[/math]

Edited September 13, 201411 yr by Unity+

What does anyone think of this? Do you think it will lead somewhere?