MAT2705-04/05 20f Quiz 4 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). Indicate where technology is used and what type (Maple, GC). Only use technology to CHECK hand calculations, not subsitute for them, unless specifically requested. Print out this quiz to respond to the graph instructions on paper.Suppose the number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= (with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn in months) of alligators in a swamp satisfies the differential equation
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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSNtbkdGJDYlUSIwRidGMi9GNlEnbm9ybWFsRidGMkZBRjJGQS1JI21vR0YkNi5RIj1GJ0YyRkEvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUklbXN1YkdGJDYmRistRiM2JkY9Ri9GMkY1LyUvc3Vic2NyaXB0c2hpZnRHRkAvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0YyRkE=a) What is the equilibrium population of alligators (when the derivative is zero!)b) Use the integration formula 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
to integrate this separable differential equation and simplify the final arbitrary constant in your implicit solution without attempting to solve for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn.c) Now isolate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn in your implicit general solution (solve for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn).d) Use your implicit general solution of part b) at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 to evaluate the arbitrary constant in terms of the initial value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIwRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdGQC9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ== and backsubstitute into your solution for part c) and simplify. Does your solution agree with Maple's direct solution of the above initial value problem? [If not use Maple's solution to continue this problem!]e) What is the value of the characteristic time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSJ+RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGS0YyRjQ= for this exponential behavior?f) Consider the case 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 alligators at the initial time. Simplify your solution expression. What happens to this population in the long run?g) Consider the case LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGNS8lL3N1YnNjcmlwdHNoaWZ0R0Y9L0krbXNlbWFudGljc0dGJFEnYXRvbWljRictSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR0ZCLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmZuLUY7NiRRJDE1MEYnRj5GQEY+ alligators at the initial time. Simplify your solution expression. What happens to this population in the long run? [Hint: "doomsday!" When does this occur?]h) Make separate rough hand sketches of the two solutions, labeling all key points and keeping the characteristic time in mind if relevant.Maple hints:
Just use LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjeDBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw== for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIwRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdGQC9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==, no need for subscripts. If you want to plot multiple expressions in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn, you can use this plot command: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or you can copy and paste the additional expressions onto a PlotBuilder plot.solutiona) This is a reverse-logistic curve behavior because the overall sign in the DE has been reversed. That means inside the logistic strip solutions decay to zero, while above it solutions grow without bound.The integration formula comes from the 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEkMTAwRidGMi8lK2V4ZWN1dGFibGVHRjFGMg==d)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About 110 months.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 equilibrium solution is unstable. Initial data above it has a doomsday cutoff when the model breaks down. Initial data below it exponentially decays.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 that when the initial value is greater than the equilibrium value, there is always a vertical asymptote at positive time, namely
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\342\211\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 since a Taylor series expansion about zero gives just this small ratio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiZRIk1GJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUklbXN1YkdGJDYmLUYvNiZRInhGJ0YyRjVGOC1GIzYmLUkjbW5HRiQ2JVEiMEYnRjUvRjlRJ25vcm1hbEYnRjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHRkgvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0YyRjVGOC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGVS8lKWJldmVsbGVkR0Y3RjVGSQ==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For the die off solution the 1 in the denominator is soon insignificant compared to the growing exponential so the quotient quickly approaches a purely exponential decay characterized by the characteristic time.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 the real situation modeled by this continouus DE is discrete and you cannot have a fractional cow left, roughly after about 350 days the population has died off.