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 | 1. It is probably clear that caution is important in using technology. Equally important is redundancy. This property is sometimes thought to be a negative (wasteful, unnecessary), but its positive role is one of the lessons of this section. By redundancy, we mean investigating a problem using graphical, numerical and symbolic tools. Why is it important to use multiple methods? Answer this from a practical perspective (think of the problems in this section) and a theoretical perspective (if you have learned multiple techniques, do you understand the mathematics better?). |
| | 2. The drawback of caution and redundancy is that it they take extra time. In computing limits, when should you stop and take extra time to make sure an answer is correct, and when is it safe to go on to the next problem? Should you always look at a graph? compute function values? do symbolic work? an  proof? Prioritize the techniques in this chapter. |
| 3. The limit  will be very important in the next chapter. For a specific function and specific a, we could compute a table of values of the fraction for smaller and smaller values of h. Why should we be wary of loss-of-significance errors? |
| 4. Notice that we rationalized the numerator in example 6.7. The old rule of rationalizing the denominator is another example of rewriting an expression to try to minimize computational errors. Before computers, square roots were very difficult to compute. To see one reason why you might want the square root in the numerator, suppose that you can only get one decimal place of accuracy, so that   1.7. Compare  to  and then compare 2(1.7) to  . Which of the approximations could you do in your head? |
| In exercises 5-12, (a) use graphics and numerics to conjecture a value of the limit. (b) Find a computer or calculator graph showing a loss-of-significance error. (c) Rewrite the function to avoid the loss-of-significance error.
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| 5.  | 6.  |
| 7.  | 8.  |
| 9.  | 10.  |
| 11.  | 12. 
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| In exercises 13 and 14, compare the limits to show that small errors can have disastrous effects.
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| 13.  and 
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| 14.  and 
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| 15. Compare f (x) = sin  x and g(x) = sin 3.14 x for x = 1 (radian), x = 10 , x = 100 and x = 1000.. |
| 16. For exercise 5, follow the calculation of the function for x = 105 as it would proceed for a machine computing with a 14 - digit mantissa. Identify where the round-off error occurs. |
| In exercises 17 and 18, compare the exact answer to one obtained by a computer with a six-digit mantissa.
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| 17. (1.000003-1.000001) 107
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| 18. (1.000006-1.000001) 107
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 | 19. In this exercise, we look at one aspect of the mathematical field of chaos. First, iterate the function f (x) = starting at x0 = 0.5. That is, compute x1 = f (0.5) , then x2 = f (x1) , then x3 = f (x2) and so on. Although the sequence of numbers stays bounded, the numbers never repeat (except by the accident of round-off errors). An impressive property of chaotic functions is the butterfly effect (more properly referred to as sensitive dependence on initial conditions). The butterfly effect applies to the chaotic nature of weather, and states that the amount of air stirred by a butterfly flapping its wings in Brazil can create or disperse a tornado in Texas a few days later. Therefore, long-range weather prediction is impossible. To illustrate the butterfly effect, iterate f (x) starting at x0 = 0.5 and x0 = 0.51. How many iterations does it take before the iterations are more than 0.1 apart? Try this again with x0 = 0.5 and x0 = 0.501. Repeat this exercise for the function g(x) = x2 - 1. Even though the functions are almost identical, g(x) is not chaotic and behaves very differently. This represents an important idea in modern medical research called dynamical diseases: a small change in one of the constants in a |
| function (e.g., the rate of an electrical signal within the human heart) can produce a drastic change in the behavior of the system (e.g., the pumping of blood from the ventricles). |
| | 20. Just as we are subject to round-off error in using calculations from a computer, so are we subject to errors in a computer-generated graph. After all, the computer has to compute function values before it can decide where to plot points. On your computer or calculator, graph y = sin x2 (a disconnected graphor point plotis preferable). You should see the oscillations you expect from the sine function, but with the oscillations getting faster as x gets larger. Shift your graphing window to the right several times. At some point, the plot will become very messy and almost unreadable. Depending on your technology, you may see patterns in the plot. Are these patterns real or an illusion? To explain what is going on, recall that a computer graph is a finite set of pixels, with each pixel representing one x and one y. Suppose the computer is plotting points at x = 0 , x = 0.1 , x = 0.2 and so on. The y - values would then be sin 02 , sin 0.12 , sin 0.22 and so on. Investigate what will happen between x = 15 and x = 16. Compute all the points (15, sin 152) , (15.1, sin 15.12) and so on. If you were to graph these points, what pattern would emerge? To explain this pattern, argue that there is approximately half a period of the sine curve missing between each point plotted. Also, investigate what happens between x = 31 and x = 32. |
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