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Urgent help required i have no idea where to start and need answers quick.

in this question f is the function.
f(x) = x^2 + (13/12)x - 1/2

a) use algebra to find the fixed points of f, and to classify them as attracting repelling or indifferent.
b) use the gradient criterion to determine an interval of attraction for one of the fixed points of f
c)find the exact values of the 2nd and 3rd terms of the sequence xn obtained by iterating f with initial term xo=0 (express answers as fractions in there lowest terms) hence state the long term behaviour of this sequence, explaining your reasoning.
Submitted: 8 years ago.Category: Math Homework
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6/27/2010
Math Tutor or Teacher: Ryan, Engineer replied 8 years ago
Ryan
Ryan, Engineer
Category: Math Homework
Satisfied Customers: 9,184
Experience: B.S. in Civil Engineering
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Hi,

Welcome to JustAnswer. Thank you for your question, Here is the solution:

A) Use algebra to find the fixed points of f, and to classify them as attracting, repelling, or indifferent.

The "fixed points" of a function are those values of x for which the function itself equals x. In other words, f(x) = x.

For the function given in your problem, you want to find values of x where:
x^2 + (13/12)x - 1/2 = x

Subtracting x from both sides gives:

x^2 + (13/12)x - x - 1/2 = 0

x^2 + (1/12)x - 1/2 = 0

Multiplying by 12 gives:

12x^2 + x - 6 = 0

Factor this into:

(4x + 3)(3x - 2) = 0

The solutions are x = -3/4 and x = 2/3.

The fixed points then are x = -3/4 and x = 2/3.

To classify these points as either attracting or repelling, you need to look at the "orbits" of the points on either side of each fixed point. You calculate the orbit of a point by performing successive iterations of f(x), each time using the previous result as the new value of x. This gives a sequence of {x, f(x), f(f(x)), f(f(f(x)))...}

If the orbits of nearby points tend toward the value of the fixed point, then the point is an attracting fixed point.

Conversely, if the orbits of nearby points tend away from the value of the fixed point, then it is a repelling fixed point.

It will be seen that x = -3/4 is an attracting fixed point, and x = 2/3 is a repelling fixed point.


B) Use the gradient criterion to determine an interval of attraction for one of the fixed points of f

The interval of attraction for the point at -3/4 is -7/4 < x < 2/3.

This can be easily seen using a spreadsheet to calculate the orbits of several values of x to either side of the fixed point. In this case, for values of x greater than -7/4, the orbits tend toward -3/4 (the value of the fixed point). But for x ≤ -7/4, the orbits tend away from -3/4, and rapidly become very large positive numbers.

Similarly, at the other end of the interval, for x less than 2/3 the orbits tend toward -3/4, while for x ≥ 2/3 they again tend toward large positive numbers.

The limits of the interval then are -7/4 < x < 2/3, where the endpoints are not included in the interval.

C) Find the exact values of the 2nd and 3rd terms of the sequence xn obtained by iterating f with initial term xo=0. State the long term behavior of this sequence, explaining your reasoning.

First term of the iteration sequence for xo = 0: f(0) = -1/2

Second term of the iteration sequence for xo = 0: f(-1/2) = -19/24

Third term of the iteration sequence for xo = 0: f(-19/24) = -421/576

The long term behavior of this sequence is to tend toward the value -3/4, which is to be expected as x = 0 is within the interval of attraction found in part B.

Please feel free to ask if you have any questions about this. I will be happy to explain further if necessary.

Thanks again,

Ryan


Edited by Ryan on 6/27/2010 at 1:49 PM EST
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Customer reply replied 8 years ago
hi, please could you explain how you got the answer to part b as i understand the rest but not this.
Math Tutor or Teacher: Ryan, Engineer replied 8 years ago
Hi,

Yes, I can do that. No problem. I'll have to work up a spreadsheet though to illustrate it.

Do you know what "orbits" means in this context? If not, then I'll start by explaining that.

Ryan
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Customer reply replied 8 years ago
explain all please. many thanks
Math Tutor or Teacher: Ryan, Engineer replied 8 years ago
Hi,

Here are two files for you. One is a Word document with an explanation of the calculations involved. The other is an Excel spreadsheet showing the results of the calculations and how I found the limits of the attraction interval for the second part of the problem.

AttractInterval

Orbits.xls

In the explanation, I went back to the first part of the problem so that I could explain "orbits". That is a key concept for the second part of the problem.

I hope this answers all your questions. If it doesn't, please let me know. I'll do what I can to explain it further. Also, please let me know if you have any difficulty with these files.

Thanks,

Ryan

Edited by Ryan on 6/30/2010 at 3:00 PM EST
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