Functions of Two Independent Variables Calculus IV Lab Karen Donnelly Saint Joseph's College

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRis= Load the plots library to use contourplots QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJnBsb3RzR0YlISIi Define plotopts to represent options for graphing: PkkpcGxvdG9wdHNHNiI2JS9JJWF4ZXNHRiRJJ25vcm1hbEclKnByb3RlY3RlZEcvSS10cmFuc3BhcmVuY3lHRiQkIiIkISIiL0kmc3R5bGVHRiRJLHBhdGNobm9ncmlkR0Yk

1. Consider the function 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 defined by 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. PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJComLUkkc2luR0YkNiM5JCIiIi1JJGNvc0dGJDYjOSVGMUYkRiRGJA== 2. Evaluate the function at the point (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). LUkiZkc2IjYkLCRJI1BpRyUqcHJvdGVjdGVkRyMiIiIiIiUsJEYnI0YqIiIk 3. To get the answer in floating point, use evalf. ( Note that % stands for the most recent computation.) LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI=

4. The graph of a function of the form 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 is a surface. The graph of f(x,y) for 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJSZwaTtGJy9GNUZCRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4=, 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJSZwaTtGJy9GNUZCRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4= can be obtained with the command plot3d as follows: LUkncGxvdDNkRzYiNiYtSSJmR0YkNiRJInhHRiRJInlHRiQvRik7LCRJI1BpRyUqcHJvdGVjdGVkRyEiIkYuL0YqRiwvSSVheGVzR0YkSSZib3hlZEdGJA== 5. As with the two dimension plot command, you can add a lot of options: LUkncGxvdDNkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNictSSJmR0YnNiRJInhHRidJInlHRicvRiw7LCRJI1BpR0YlISIiRjEvRi1GLy9JJWF4ZXNHRidJJmJveGVkR0YnL0kmc3R5bGVHRidJLHBhdGNobm9ncmlkR0Yn A level curve of a function f(x,y) is the set of points (x,y) plotted in the x-y plane where f(x,y) = c (a constant). There are several ways to visualize the level curves in Maple. One way is to select the graph above (by clicking on it), and then choose Style, Contour from the menus. 6. The command contourplot can be used to plots level curves (countour lines) of a function of two variables: LUksY29udG91cnBsb3RHNiI2JS1JImZHRiQ2JEkieEdGJEkieUdGJC9GKTssJEkjUGlHJSpwcm90ZWN0ZWRHISIiRi4vRipGLA== 7. We can use use Maple to assist in visualizing the level curve f(x,y) = 0.6 . First make a plot of the surface z =f(x,y) and save it as P1. Note the way it suppresses first the result with : , then plots it with P1. QyQ+SSNQMUc2Ii1JJ3Bsb3QzZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYmLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YvOywkSSNQaUdGKSEiIkY0L0YwRjIvSSxvcmllbnRhdGlvbkdGJTckIiM6IiNnRjU=SSNQMUc2Ig== 8. Next we plot the plane z= 0.6. QyQ+SSNQMkc2Ii1JJ3Bsb3QzZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYmJCIiJyEiIi9JInhHRiU7LCRJI1BpR0YpRi5GMy9JInlHRiVGMS9JJnN0eWxlR0YlSSxwYXRjaG5vZ3JpZEdGJSIiIg==SSNQMkc2Ig== 9. Now we display the surface z = f(x,y) along with the plane z = 0.6. The plane intersects the surface in three curves (level curve(s) corresponding to the value f(x,y) = .6). LUkqZGlzcGxheTNkRzYiNiY8JEkjUDFHRiRJI1AyR0YkL0kqdGlja21hcmtzR0YkNyUiIiVGLEYsL0ksb3JpZW50YXRpb25HRiQ3JCIjOiIjZy9JJWF4ZXNHRiRJJmJveGVkR0Yk

10. The following is the function from your text: 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 PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwkKiY5JCIiIiwoKiRGLiIiI0YvKiQ5JUYyRi9GL0YvISIiISIlRiRGJEYk LUkncGxvdDNkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiYtSSJmR0YnNiRJInhHRidJInlHRicvRiw7ISIlIiIlL0YtRi8vSSVheGVzR0YnSSZib3hlZEdGJw==

These are the exercises 45 - 48 from Larson, Section 13.1 page 893 PkkkZjQ1RzYiZio2JEkieEdGJEkieUdGJEYkNiRJKW9wZXJhdG9yR0YkSSZhcnJvd0dGJEYkLUkkZXhwR0YkNiMsKCIiIkYwKiQ5JCIiIyEiIiokOSVGM0Y0RiRGJEYk LUkncGxvdDNkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiYtSSRmNDVHRic2JEkieEdGJ0kieUdGJy9GLDshIiQiIiQvRi1GL0kpcGxvdG9wdHNHRic= LUksY29udG91cnBsb3RHNiI2Ji1JJGY0NUdGJDYkSSJ4R0YkSSJ5R0YkL0YpOyEiJCIiJC9GKkYsSSlwbG90b3B0c0dGJA== PkkkZjQ2RzYiZio2JEkieEdGJEkieUdGJEYkNiRJKW9wZXJhdG9yR0YkSSZhcnJvd0dGJEYkLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsKCIiIkYzKiQ5JCIiIyEiIiokOSVGNkYzRiRGJEYk LUkncGxvdDNkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiYtSSRmNDZHRic2JEkieEdGJ0kieUdGJy9GLDshIiIiIiIvRi1GL0kpcGxvdG9wdHNHRic= LUksY29udG91cnBsb3RHNiI2Ji1JJGY0NkdGJDYkSSJ4R0YkSSJ5R0YkL0YpOyEiIiIiIi9GKkYsSSlwbG90b3B0c0dGJA== PkkkZjQ3RzYiZio2JEkieEdGJEkieUdGJEYkNiRJKW9wZXJhdG9yR0YkSSZhcnJvd0dGJEYkLUkjbG5HRiQ2Iy1JJGFic0clKnByb3RlY3RlZEc2IywmOSUiIiIqJDkkIiIjISIiRiRGJEYk LUkncGxvdDNkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiYtSSRmNDdHRic2JEkieEdGJ0kieUdGJy9GLDshIiciIicvRi07ISIkIiIkSSlwbG90b3B0c0dGJw== LUksY29udG91cnBsb3RHNiI2Ji1JJGY0N0dGJDYkSSJ4R0YkSSJ5R0YkL0YpOyEiJyIiJy9GKjshIiQiIiRJKXBsb3RvcHRzR0Yk PkkkZjQ4RzYiZio2JEkieEdGJEkieUdGJEYkNiRJKW9wZXJhdG9yR0YkSSZhcnJvd0dGJEYkLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJiokOSQiIiMjIiIiIiIlKiQ5JUY1I0Y3RjVGJEYkRiQ= LUkncGxvdDNkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiYtSSRmNDhHRic2JEkieEdGJ0kieUdGJy9GLDshIikiIikvRi07ISIkIiIkSSlwbG90b3B0c0dGJw== LUksY29udG91cnBsb3RHNiI2Ji1JJGY0OEdGJDYkSSJ4R0YkSSJ5R0YkL0YpOyEiJSIiJS9GKjshIiQiIiRJKXBsb3RvcHRzR0Yk

These are exercises 29, 30, 31, Larson, Section 13.2, page 903 Complete Exerecise 29, page 903, section 13.2, which is started for you Plot the function defined by 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Then explore the values of the function along the two different paths y = 0 and y = x as 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 Make a conjecture about the limit and then verify analytically. PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCooOSQiIiI5JUYuLCYqJEYtIiIjRi4qJEYvRjJGLiEiIkYkRiRGJA== LUkncGxvdDNkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNigtSSJmR0YnNiRJInhHRidJInlHRicvRiw7ISIjIiIjL0YtRi8vSSVheGVzR0YnSSdub3JtYWxHRiUvSSVncmlkR0YnNyQiI0lGOS9JJnN0eWxlR0YnSSxwYXRjaG5vZ3JpZEdGJw== JSFH Along the path LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIjBGJ0ZARkBGK0ZARitGQA==: NictSSJmRzYiNiQiIiIiIiEtRiQ2JCQiIiYhIiJGKC1GJDYkJEYnRi1GKC1GJDYkJEYnISIjRigtRiQ2JCRGJyEiJEYo Along the path LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSJ4RidGNkY5RkBGK0ZALUY9Ni1RIjpGJ0ZARkJGRUZHRklGS0ZNRk9GUUZURkA= QyQtSSJmRzYiNiQiIiIkRiciIiFGJw==QyQtSSJmRzYiNiQkIiImISIiRiciIiI=NiUtSSJmRzYiNiQkIiIiISIiRictRiQ2JCRGKCEiI0YsLUYkNiQkRighIiRGMA== Discussion: The limit does not exist since along the path y = 0, the function has the value 0 and along the path y = x, the function has the value 0.5. Complete Exerecise 30, page 903 section 13.2 using the same form of commands as above for the function 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. PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJComOSUiIiIsJiokOSQiIiNGLiokRi1GMkYuISIiRiRGJEYk LUkncGxvdDNkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNictSSJmR0YnNiRJInhHRidJInlHRicvRiw7ISIjIiIjL0YtRi9JKXBsb3RvcHRzR0YnL0klZ3JpZEdGJzckIiNJRjc= NictSSJmRzYiNiQiIiIiIiEtRiQ2JCQiIiYhIiJGKC1GJDYkJEYnRi1GKC1GJDYkJEYnISIjRigtRiQ2JCRGJyEiJEYo NictSSJmRzYiNiQiIiIkRiciIiEtRiQ2JCQiIiYhIiJGLC1GJDYkJEYnRi5GMS1GJDYkJEYnISIjRjQtRiQ2JCRGJyEiJEY4 Discussion -- The limit does not exist since the along the path y=0, the function has the value 0 and along the path y=x the function has values which are increasing without bound (apparently). Complete Exerecise 31, page 903, section 13.2 Using the same form of commands as above for the function 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. This time use the paths 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 and 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. . PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwkKig5JCIiIjklIiIjLCYqJEYuRjFGLyokRjAiIiVGLyEiIkY2RiRGJEYk LUkncGxvdDNkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNictSSJmR0YnNiRJInhHRidJInlHRicvRiw7ISIlIiIlL0YtOyEiJCIiJEkpcGxvdG9wdHNHRicvSSVncmlkR0YnNyQiI0lGOg== NictSSJmRzYiNiQiIiJGJy1GJDYkJCIjRCEiIyQiIiYhIiItRiQ2JCRGJ0YsJEYnRi8tRiQ2JCRGJyEiJUYyLUYkNiQkRichIickRichIiQ= NictSSJmRzYiNiQhIiIiIiItRiQ2JCQhI0QhIiMkIiImRictRiQ2JCRGJ0YtJEYoRictRiQ2JCRGJyEiJSRGKEYtLUYkNiQkRichIickRighIiQ= Discussion: The limit does not exist since along the two different paths we get values for the function of -0.5 and 0.5 respectively.

plot3d From plots package contourplot