- 1 4 Questions To Explore 7 Of A And B As Well As The Choice Of Initial Starting Point Xo We Wish To Investigate How Th 1 (117.05 KiB) Viewed 66 times
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QUESTIONS TO EXPLORE 7 of a and b) as well as the choice of initial starting point xo. We wish to investigate how the various choices influence the eventual behavior of the sequence. The computer is an ideal tool to use to explore such
questions. In order to keep track of patterns, we need some language to describe the behavior we see. Mathematicians distinguish be- tween convergent sequences—like those suggested by the experiments (a, b, 4; 12) = (0.5, 2, 5, 10) and (a, b, x, n) = (-3,1,0.25, 15)- and divergent sequences—like those suggested by the experiments (-2,1,1.5, 10) and (-3,1, 1, 15). Convergent sequences converge to a particular value, called the limit of the sequence. The experiment (a, b, x, n) = (0.5, 2, 5, 10) sugggests the limit is 4; and the experiment (a, b, x, n) = (-3,1, 0.25, 15) suggests the limit is 0.25. EXERCISE Write down careful definitions of the three terms convergent sequence, divergent sequence, and limit of a convergent sequence. 1.4
Questions to explore For each of the
questions below, use the computer program to find your own examples of linear functions, different from those you examine in class, that have the specified behavior and help to shed light on the posed
question:
QUESTION 1: Can you find a linear function that gives a convergent se- quence of iterates for every initial value? That is, can you find values of a and b that assure convergence regardless of the chosen xo?
QUESTION 2: Can you find a linear function that, on iteration, appears to give a divergent sequence for every initial value?
QUESTION 3: Can you find an example of a linear function whose se- quence of iterates converges to different limits for different starting values?
QUESTION 4: Can you find a linear function that will give a convergent sequence of iterates for one or more initial values and a divergent sequence for other initial values?
并O please
Question 5: What conditions on a and/or b assure that the iteration te quence of f(3) = a + b converges no matter what the choice of Bo? Guess at this from your examples (and from doing a few more examples with the computer). We'll explore this in detail in what follows.
Question 6: You can think of the set of all linear functions as points in the plane under the correspondence f(x) = ax + 4(a, b) point in R2 not in ) For example, consider Figure 1.1. With this way of exploring linear functions under iteration we can think of the (a,b) plane as being cut up into three kinds of pieces code • Type (1) points: those points that give rise to convergent sequences under iteration regardless of the value of 10, draw graph also • Type (ii) points: those points corresponding to convergent sequences for some choices of to and divergent sequences for other choices, and • Type (ii) points: points for which there is never convergence. On a piece of graph paper, draw an (a, b)-plane and, based on many examples using your computer program, indicate clearly which points you feel are of Types (i), (ii) and (iii). For instance, the example f(x) = -3x+1 we looked at earlier gave a divergent sequence for most Xo's but converged for one value. Hence this example makes (-3,1) a Type (ii) point. Similarly, looking at the function f(3) = 0.5x + 2, we are convinced that (0.5, 2) is a Type (i) point. f(x) = -3x + 1 (0.5,2) (-3,1) f(x) = 0.57 +2 х + +