MAT2500-01/04 19S 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 EQUAL SIGNS and arrows when appropriate. Always SIMPLIFY expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation if appropriate). Indicate where technology is used and what type(Maple, GC). You are encouraged to use technology to CHECK but NOT SUBSTITUTE all of your hand results. Everything on this test is straightforward to evaluate by hand and must be shown explicitly.1. Identify each quantity you evaluate with its proper symbol and defining formula.a) Find the direction 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 in 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increases most rapidly at the point (0,1,2). b) What is the maximum rate of increase there? c) What is the directional derivative in the direction of the origin from this point?d) Use the chain rule to evaluate the derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn along the almost twisted cubic 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 as it passes through the point (0,1,2).e) Write a simplified (standard linear form) equation for the tangent plane to the level surface at this point. What is the equation of the level surface itself through this point? Evaluate the linear approximation 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 at this point.
2. A mathematical "sports dome" for Villanova has the shape of an ellipsoidally deformed hemisphere (upper half of an ellipsoid with axes aligned with a Cartesian coordinate system). The volume of this half ellipsoid is 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 in terms of the 3 semiaxes with the same (semiaxis) dimensions as Villanova's planned dome with a more rectangularly shaped base: 38'x110'x185' (in feet, according to The Villanovan). So let:
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 .Suppose a waterproofing layer is 1/8" (in inches) thick when applied to the dome. Use differentials to estimate the change in volume in cubic feet (to the nearest cubic foot) if we increase the dimensions by this amount, which is an estimate of the volume of waterproofing material needed.3. The roof of this mathematical sports dome has some internal spotlights (for night use) mounted so that they are pointing down perpendicular to the surface.
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
The point on the dome above the floor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 at 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 is approximately at 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 to high accuracy.
Write down an equation for the normal line to this point on the dome and evaluate the location 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 where it hits the floor, i.e., where the center of this spotlight hits the floor. Give your results to the nearest tenth of a foot.
[Hint: work the problem in terms of the parameters and insert the numbers at the end to simplify matters.]Optional Comment. Does your result make sense to you in comparison with the horizontal location of the spotlight, given the concave downward curvature of the dome? If so, why?solution2.The Villanovan, Feb 15 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3.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We include a minus sign so the normal line points inwards, and remove a factor of 2 from the gradient to use as the normal.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 the actual numbers directly substituted: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To the nearest foot, the location is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtbkdGJDYlUSU5Ny45RicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVElNDcuMEYnRjRGN0Y0RjdGNEY3RjRGNw==, which makes sense since the rounded dome should point more towards its center compared to the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtbkdGJDYlUSQxMDBGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RIixGJ0Y0RjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSM1MEYnRjRGN0Y0RjdGNEY3RjRGNw== directly underneath.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 is a representation of the situation: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1.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=pledgeLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=When 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: