MAT1505-03/04 15F Quiz 6 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 when appropriate). Indicate where technology is used and what type (Maple, GC).1. a) The area between the logistic curve 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 and its horizontal asymptote LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= for nonnegative LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2LVErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUYjNiQtSSNtbkdGJDYkUSIwRidGMkYyLUYjNiQtRi82LVEoJmluZmluO0YnRjJGNUY4L0Y7RjdGPS9GQEY3RkFGQ0ZFRkhGMi8lMXN1cGVyc2NyaXB0c2hpZnRHRk8vJS9zdWJzY3JpcHRzaGlmdEdGTy1JJm1mcmFjR0YkNigtRiM2Ji1JI21pR0YkNiNRIUYnLUYjNiVGam4tSSVtc3VwR0YkNiUtRi82LVEvJkV4cG9uZW50aWFsRTtGJ0YyL0Y2USZ1bnNldEYnL0Y5RmdvL0Y7RmdvL0Y+RmdvL0ZARmdvL0ZCRmdvL0ZERmdvRkUvRklRLDAuMTExMTExMWVtRictRiM2JS1GLzYtUSomdW1pbnVzMDtGJ0YyRjVGOEZVRj1GVkZBRkMvRkZRLDAuMjIyMjIyMmVtRicvRklGZnAtRltvNiVRInhGJy8lJ2l0YWxpY0dGPC9GM1EnaXRhbGljRidGMkZXRjJGam5GMi1GIzYmRmBvLUYvNi1RIitGJ0YyRjVGOEZVRj1GVkZBRkNGZXBGZ3AtRk02JFEiMUYnRjJGMi8lLmxpbmV0aGlja25lc3NHRmZxLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRltyLyUpYmV2ZWxsZWRHRjctRi82LVEifkYnRjJGNUY4RlVGPUZWRkFGQ0ZFRkgtRltvNiVRImRGJy9GXHFGN0YyRmhwRjI= . Convert this to a logarithmic integral by a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-substitution (denominator!) and carefully evaluate this integral using limits. Give its exact and approximate value to 2 decimal places.b) Use Simpson's rule with 4 divisions to approximate 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 to 2 decimal places. Show the formula you evaluate to get this result. Does this seem reasonable compared to part a)? [It wouldn't hurt to sketch the logistic curve and its asymptote for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIy4uRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZOUSYwLjBlbUYnLUZQNiRRIjZGJ0Y5Rjk= and shade in this area.]2. The total energy in a cavity of volume LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at temperature LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= of a Planck black body radiation gas (like the cosmic microwave background!) is given by
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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the speed of light, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM4NDYzO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is Planck's constant, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the temperature of the radiation, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is Boltzman's constant, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is the frequency of the radiation. This is an integral over all frequencies of the energy per unit frequency using the Bose-Einstein distribution function.a) Show that an obvious LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-substitution converts this formula 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 . What is the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=?b) Use technology to evaluate the dimensionless definite integral factor. What is its value, exactly and numerically to 2 decimal places?c) Simplify the final formula for the energy (exactly). [The power law temperature dependence is a key feature of our understanding of the early universe.]d) Plot the dimensionless integrand 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIy4uRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZOUSYwLjBlbUYnLUZQNiRRIzEyRidGOUY5. What is the value of 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? How does that make the endpoint 0 not "improper" in the context of this semi-infinite interval integral?solution[use reverse side or extra paper from bob]a) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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2)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We locate the peak in the plot below in order to evaluate the peak value and see when the function decreases to 1 percent of its peak value.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