Math 151-005 Lab 9/12/08 Michael Pelsmajer and Jiariu Yang First, read Fasshauer's "Introduction to Maple". As you read, execute each of the Maple commands. Observe and learn. (Later, you may wish to use Fasshauer's Introduction as a reference guide, since it quickly demonstrates the essentials of Maple.) Afterwards, work through the following problems in a Maple notebook. Save often! When you are done, revise your work so that it is easy for Jiariu (the TA) to find and understand your solutions. Then email the file as an attachment to me. Your work is due by 5pm on Sunday. 1. Graphing with technology... goes horribly wrong! In this exercise, you will make a plot and modify it several times. After each modification, decide whether the plot "makes sense" with what you know about graphs of functions. 1.a. Begin with a plot of the graph of the function 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 over the domain [0,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRJSZwaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRidGLy1JI21vR0YkNi1RIl1GJ0YvLyUmZmVuY2VHUSV0cnVlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGLw==. 1.b. Make a single plot of the graphs of the functions 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 and 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 over the domain [0,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRJSZwaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRidGLy1JI21vR0YkNi1RIl1GJ0YvLyUmZmVuY2VHUSV0cnVlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGLw==. Does this make sense (based on what you already knew about these functions)? Explain. 1.c. Make a single plot of the graphs of the functions 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 and 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 over the domain [0,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRJSZwaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRidGLy1JI21vR0YkNi1RIl1GJ0YvLyUmZmVuY2VHUSV0cnVlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGLw==. Does it make sense? Explain. 1.d. Make a single plot of the graphs of the functions 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUS1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRIkJGJ0Y0RjctRjs2LVEnJnNkb3Q7RidGPkZARkNGRUZHRklGS0ZNRk9GUi1GLDYlUSJ4RidGNEY3LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GPkZcb0Zfb0Y+RitGXG9GX29GPg==, where we will be using different values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy9GM1Enbm9ybWFsRic=, always over the domain [0,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRJSZwaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRidGLy1JI21vR0YkNi1RIl1GJ0YvLyUmZmVuY2VHUSV0cnVlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGLw==. Begin with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjVGJ0Y5LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjk=. Does the plot make sense? Change LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy9GM1Enbm9ybWFsRic= to be 10, and replot. Does it still make sense? Try it with 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 until something weird happens, and the plot does not make sense anymore. (On my system, the plot gets unreasonable with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzkwRidGOS8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y5. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJDIwMEYnRjkvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOQ== is no better, for example.) (Is there a moral to this story?) 1.e. Maple makes a plot by plotting many points and connecting the dots. When the function varies wildly like the examples above, plotting more points can help get a better plot. For example, this plot has a weird gap toward the right: 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

By plotting more points, we see that the function doesn't really do that. QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUtSSRzaW5HRiU2Iy1JJGV4cEdGJTYjSSJ4R0YoL0YwOyIiISIiJi9JKm51bXBvaW50c0dGKCIkKygiIiI=

Now, do the last plot for part 1.d. again but with more points, so that it looks right. 1.f. Can you modify "numpoints" so that it looks even worse? (I don't know.) Comment: "More points" is not foolproof. Many functions are too wild, and no computer will ever give you an accurate plot. You always need to think when you use computers, using your knowledge to correctly interpret what you see. 1.g Plot sin(1/x) on the interval (-1/10,1/10). Then do it again, using numpoints to make the plot look right. 1.h. Plot x sin(1/x) on the interval (-1/10,1/10). Then do it again, using numpoints to make the plot look right. 2. That's all fine when there's clearly something wrong with the plot. Unfortunately, it's not hard to come up with a very misleading function, that seems to be okay, but... LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY2LUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC1JI21vR0YkNi5RKiZjb2xvbmVxO0YnRjJGNS8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkstRjg2LlEifkYnRjJGNUY7Rj1GP0ZBRkNGRUZHL0ZKUSYwLjBlbUYnL0ZNRlItRiw2JlEieEYnRi9GMkY1LUY4Ni5RKCZzcmFycjtGJ0YyRjVGO0Y9Rj9GQUZDRkVGR0ZJRkwtSShtZmVuY2VkR0YkNiUtRiM2Ky1JI21uR0YkNiVRIjFGJ0YyRjUtRjg2LlEiL0YnRjJGNUY7Rj0vRkBRJXRydWVGJ0ZBRkNGRUZHL0ZKUSwwLjE2NjY2NjdlbUYnL0ZNRmNvLUZqbjYlUSIyRidGMkY1LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lK2V4ZWN1dGFibGVHRjFGMkY1RjJGNS1GODYuUSJeRidGMkY1RjtGPUY/RkFGQ0ZFRkcvRkpRLDAuMTExMTExMWVtRicvRk1GZ3AtRmVuNiUtRiM2Ky1GZW42JS1GIzYrLUZqbjYlUSQ1MDBGJ0YyRjUtRjg2LlEiKkYnRjJGNUY7Rj1GP0ZBRkNGRUZHRmJvRmRvRlRGaG9GW3BGXnBGYXBGMkY1RjJGNUZjcEZlb0Zob0ZbcEZecEZhcEYyRjVGMkY1Rk4tRjg2LlEiK0YnRjJGNUY7Rj1GP0ZBRkNGRUZHL0ZKUSwwLjIyMjIyMjJlbUYnL0ZNRltyRk4tRiw2JlEkc2luRidGL0YyRjUtRmVuNiUtRiM2KUZURmhvRltwRl5wRmFwRjJGNUYyRjUtRjg2LlEiO0YnRjJGNUY7L0Y+RmFvRj9GQUZDRkVGR0ZRRkxGaG9GW3BGXnBGYXBGMkY1 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 2.a From the plot, what does the limit of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= appear to be? 2.b. Use Maple to find the limit of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. (The following will produce a "help page" that tell you how.) ?limit; 2.c. Modify the above plot, changing the visible domain so that you can get a better look at the function near LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=, until you have obtained a plot that explains the apparent paradox. 3. Demonstrate the Squeeze Theorem by making a single plot of the graphs of these three functions: 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, 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 , and 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 . The plot should be over the domain (-10,10). Then, make a second plot that magnifies a portion of the graph near the origin, to better see how the three functions behave when x is near 0. Make a third plot that focuses on an even smaller region near the origin. 4. Execute the next command, and then click "Animate". Maybe click the "+" button a couple of times. Student[Calculus1][TangentTutor](x^2-1,x=1); (Isn't it great?) 4.a. Experiment with it: Try making the iteration -0.1, for example. Try changing the function, and the choice of tangent point. What did you try?