Functions of Two Independent Variables Calculus IV Lab Week Four LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRis= LUkld2l0aEc2IjYjSSZwbG90c0dGJA==

1. Consider the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GIzYmLUYsNiVRImZGJ0Y2RjktRj02LVEwJkFwcGx5RnVuY3Rpb247RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVRmluLUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEieEYnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRmhuL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0Y2RjlGQEZARkBGK0ZARitGQEYrRkA= as follows: 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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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiUtSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSy1GLDYlUSUmcGk7RicvJSdpdGFsaWNHRjxGN0Y3LUY0Ni1RJSZsZTtGJ0Y3RjpGPUY/RkFGQ0ZFRkcvRkpRLDAuMjc3Nzc3OGVtRicvRk1GVy1GLDYlUSJ4RicvRlJRJXRydWVGJy9GOFEnaXRhbGljRidGN0YrRjc=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJSZwaTtGJy9GNUZCRj5GPkYrRj4=, 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJSZwaTtGJy9GNUZCRj5GPkYrRj4= can be obtained with the command plot3d as follows: LUkncGxvdDNkRzYiNiYtSSJmR0YkNiRJInhHRiRJInlHRiQvRik7LCRJI1BpRyUqcHJvdGVjdGVkRyEiIkYuL0YqRiwvSSVheGVzR0YkSSZib3hlZEdGJA== As with the two dimension plot command, you can add a lot of options: LUkncGxvdDNkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNictSSJmR0YnNiRJInhHRidJInlHRicvRiw7LCRJI1BpR0YlISIiRjEvRi1GLy9JJWF4ZXNHRidJJmJveGVkR0YnL0kmc3R5bGVHRidJLHBhdGNobm9ncmlkR0Yn 5. 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 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 . First make a plot of the surface 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 and save it as P1. Note the way it suppresses first the result with : , then plots it with P1. QyQ+SSNQMUc2Ii1JJ3Bsb3QzZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYmLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YvOywkSSNQaUdGKSEiIkY0L0YwRjIvSSxvcmllbnRhdGlvbkdGJTckIiM6IiNnIiIiSSNQMUc2Ig== 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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Complete Exerecise 29, page 903, section 13.2, which is started for you 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 JSFH Along the path y =0: 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 Along the path y = x: JSFH Discussion Complete Exerecise 31, page 903 section 13.2 using the same form of commands as above. JSFH Discussion Complete Exerecise 32, page 903, section 13.2 using the same form of commands as above. JSFH Discussion: