MAT2500-01 13S Test 2 Print Name (Last, First) _________________________________|__
Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use arrows and equal signs when appropriate. Always SIMPLIFY expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation). Indicate where technology is used and what type(Maple, GC). You are encouraged to use technology to check all of your hand results.
1. 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 ; P(2,1)
a) Find a unit vector in the direction in which this function increases most rapidly at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.b) Evaluate the directional derivative of this function at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in the direction parallel to the vector 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. Is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= increasing or decreasing in this direction?
c) What is the slope of the normal line to the contour passing through LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5? What is the equation of this contour in simplest form?
2. 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) Find the two critical points of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .b) Use the second derivative test to classify both of these critical points as local minima, local maxima or saddle points. Use words to explain your reasoning.c) Does a quick contourplot of this function confirm your conclusions? Explain. (Right click on function, plots, plot builder, use a small enough window or the option contours = 50).
3) If 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, find 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 and evaluate it at the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMkYnRjRGN0ZRRjRGNEY0. [Use implicit differentiation! This is a special case of the van der Vaals generalization of the ideal gas equation, see Stewart Exercise 14.3.87.]
4. a) Find the equation of the tangent plane to the level surface of 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 at the point 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, and simplify it to the standard linear form. Identify a (simplest) normal vector 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 to this plane.b) What are the parametrized equations for the normal line to this surface at the same point using this normal? How does the function change as your parameter increases (increase/decrease? explain)?
c) Find the linear approximation function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGPS1GLDYlUSJ6RidGL0YyRkFGQUZB near the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.d) Use it to evaluate the linear approximation to 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.solution4. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEmcGxvdHNGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOUY1RlpGOQ==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 can actually solve for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to get a nicer surface plot.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRI3AyRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSNwM0YnRi9GMkY9LUYsNiVRI3A0RidGL0YyRkE=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYn1.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The gradient is proportional to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMEY0RjQvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRjQ= so the slope of the line along that direction is 1.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYn2.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY5LUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdh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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnYou can see these extrema in both the contourplot and the graph.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQQ==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.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you have completed the exam, please read and sign the dr bob integrity pledge and hand this test sheet stapled on top of your answer sheets as a cover page, with the first test page facing up:"During this examination, all work has been my own. I have not accessed any of the class web pages or any other sites during the exam. I give my word that I have not resorted to any ethically questionable means of improving my grade or anyone else's on this examination and that I have not discussed this exam with anyone other than my instructor, nor will I until after the exam period is terminated for all participants."
Signature: Date: