MAT2705-04/05 23s Quiz 7 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 substitute for them, unless specifically requested.
Consider the underdamped oscillator IVP: 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 [independent variable: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn].a) Put this into standard form by dividing through by the leading coefficient 4 and read off the exact and numerical values of 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 and the numerical value of the quality factor 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 . (2 decimal places are sufficient.)b) Find the general solution. What are the exact and approximate values of the characteristic equation root parameters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEnJiM5Njk7RicvRjBGNEYyRjxGMkY8 and the two related time scales: 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 .c) Find the solution satisfying the initial conditions.d) Re-express your solution with its sinusoidal factor in phase-shifted cosine form. Identify the initial amplitude LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIwRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdGQC9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ== and phase shift LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSJ+RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGS0YyRjQ=chosen in the interval 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 and the numerical value of the fractional phase shift 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(2 decimal places) to understand what fraction of the period the peak of the sinusoidal factor is shifted from 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 Does the sinusoidal factor peak shift left or right from the vertical axis at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8?
e) Use this plotting template below choosing the horizontal range appropriately (t = 0..5 \317\204) to plot the solution of part c) (not d)!) with its envelope functions 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 and print it on paper and attach to this quiz. Make sure you put an asterisk before the parenthesis after the exponential.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
Optional. Plot the difference of c) and d) to make sure it is (numerically approximately) zero.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LUk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that 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plot the 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