How to mimic 4/pi*ArcTan(x)+1 without trig

  • Context: Undergrad 
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Discussion Overview

The discussion revolves around finding a mathematical formula that mimics the behavior of the function \( y = \frac{4}{\pi} \text{ArcTan}(x) + 1 \) without using trigonometric functions. Participants explore various non-trig alternatives that meet specific criteria for limits and points.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant seeks a formula with specific properties, including limits and distinct y-values at given x-values.
  • Another participant proposes a non-trig solution: \( y = \frac{\sqrt{16 x^2 + 9} - 3}{2 x} + 1 \).
  • A third participant expresses appreciation for the proposed solution.
  • Another alternative is suggested: \( y = \frac{2x}{\sqrt{x^2 + 3}} + 1 \), which is noted to be closer to the original function.
  • One participant comments on the improved slope of the second proposed formula at \( x = 0 \) compared to the first.
  • Another participant highlights that the first proposed formula had a singularity at \( x = 0 \), despite having the correct limit.

Areas of Agreement / Disagreement

Participants express varying degrees of satisfaction with the proposed formulas, but there is no consensus on a single best solution. Multiple competing views remain regarding the effectiveness of the different non-trig approaches.

Contextual Notes

Some proposed formulas may have limitations, such as singularities or differing slopes at specific points, which are acknowledged but not resolved in the discussion.

Who May Find This Useful

This discussion may be of interest to those exploring mathematical modeling, particularly in contexts where trigonometric functions are to be avoided, or for individuals analyzing data that requires specific functional properties.

thenewmans
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I need a basic math formula with the following properties:
* limit y between -1 and 3.
* (x, y) hits (-1, 0), (0, 1) and (1, 2).
* Each y value occurs only once.

I managed to do this with y=4/pi*ArcTan(x)+1. But I'd like to do this without trig. I got close with y=x*2/SQRT(1+x^2)+1. But it's not right. I keep thinking it's something simple and obvious. Can you help me?

NOTE: This is not homework. I'm graphing and analyzing some proportion data in excel for myself.
 
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Greetings thenewmans! :smile:

Here's a non-trig solution:
[tex]y=\frac {\sqrt{16 x^2+9}-3} {2 x}+1[/tex]
 
Perfect! Wow! Thank you!
 
Just for fun, here's another one that is closer to what you came up with. :smile:
[tex]y = \frac {2x}{\sqrt{x^2+3}}+1[/tex]
 
OK, just tried it and it's even better because the slope at x=0 is closer to 1 (45 degrees). Thanks again!
 
It's also better because the first one had a singularity at x=0, even though its limit was correct. ;)
 

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