Science:Math Exam Resources/Courses/MATH103/April 2013/Question 05 (a)
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Question 05 (a) |
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Consider the iterated map . Calculate the steady states (fixed-points, equilibria) and mark them in the sketch below. Determine the stability of each equilibrium (explain analytically or graphically). Plot a cobweb starting with just above 2 (the point is indicated in the sketch).
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Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? |
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! |
Hint |
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Recall in this setting, steady states are found by setting . |
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
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Solution |
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Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. The steady states are given by solving Subtracting x from both sides yields Using the quadratic formula gives
Analytically to check stability, we would need to take the derivative of our function and then plug in the fixed points and check that the absolute value of the derivatives in the fixed points is less than 1 (this would show that the fixed point is stable). It's easier to notice that graphically, none of the fixed points are the limit of an iterative sequence and so none of the fixed points are stable. |