Solve f(x): Find Attracting Interval w/ Gradient Criterion

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In summary, the purpose of finding an attracting interval in a function is to identify a range of values for the independent variable that will result in convergence towards a specific point on the function. This is done using the gradient criterion, which involves calculating the derivative of the function and determining where it is positive and negative. An attracting interval can be found for any continuous function, and it is the range of values where the gradient is positive, indicating that the function is increasing and approaching a specific point. This is important because it provides valuable insights into the behavior and trends of a function, and can also be used for making predictions and informing decision-making in various fields.
  • #1
feely
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1. In this question, f is the function f(x) = 1/4x^2 - 1/8x - 5/8

The first part of the question asked to find the fixed points, which I got to 3.65 and 0.85, and to classify them as attracting, repelling or indifferent, which I did, however, the second part asks

2. Use the gradient criterion to determine an interval of attraction for one of the fixed points of f.I haven't got an attempt at this to show, because I don't really understand the process, and that's what I'm hoping to get help with. I'm not looking the answer, more the process. That way, hopfully I will be able to answer it myself in future.

Thanks in advance.

Sean

PS - I really hope I have posted this in the correct place, if not, I'm very sorry.
 
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  • #2
feely said:
1. In this question, f is the function f(x) = 1/4x^2 - 1/8x - 5/8

The first part of the question asked to find the fixed points, which I got to 3.65 and 0.85, and to classify them as attracting, repelling or indifferent, which I did, however, the second part asks

2. Use the gradient criterion to determine an interval of attraction for one of the fixed points of f.


I haven't got an attempt at this to show, because I don't really understand the process, and that's what I'm hoping to get help with. I'm not looking the answer, more the process. That way, hopfully I will be able to answer it myself in future.

Thanks in advance.

Sean

PS - I really hope I have posted this in the correct place, if not, I'm very sorry.

An "interval of attraction" of a point is an interval in which the "motion" tends back to the point. In this case, the "motion" is given by the derivative. If the derivative is negative, the "motion" is decreasing, if the derivative is positive, increasing.

If, as you say, the fixed points are 0.85 and 3.65,(I didn't check that myself) then an interval of attraction for 0.85 is all x< 0.85 where the derivative is positive (so the "motion" is toward 0.85) and all 0.85< x< 3.65 where the derivative is negative. An interval of attraction for 3.65 is all 0.85< x< 3.65 where the derivative is positive and all x> 3.65 where the derivative in negative.
 
  • #3
you american really have a weird way of wording your explanations
 
  • #4
seboastien said:
you american really have a weird way of wording your explanations

I found Halls' explanation to be quite good. But then again, I am an American :rolleyes:
 

1. What is the purpose of finding an attracting interval in a function?

The purpose of finding an attracting interval is to identify a range of values for the independent variable that will result in convergence towards a specific point on the function. This can be useful in understanding the behavior of the function and making predictions about its future values.

2. How is an attracting interval identified using the gradient criterion?

The gradient criterion involves calculating the derivative of the function and determining where it is positive and negative. An attracting interval is then identified as the range of values for the independent variable where the derivative is positive.

3. Can an attracting interval be found for any type of function?

Yes, an attracting interval can be found for any continuous function. However, the method for finding it may vary depending on the complexity of the function.

4. What is the relationship between the gradient and the attracting interval?

The gradient of a function at a specific point represents the slope of the tangent line at that point. When the gradient is positive, the function is increasing, and when it is negative, the function is decreasing. The attracting interval is the range of values where the gradient is positive, indicating that the function is increasing and approaching a specific point.

5. Why is it important to find an attracting interval in a function?

Identifying an attracting interval can provide valuable insights into the behavior and trends of a function. It can also be used to make predictions and inform decision-making in various fields such as economics, physics, and engineering.

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