LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnMAT2705-01/04 07S 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 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 may use technology to find the integer roots of the characteristic equation. [No mercy for mistakes here.]
1. 2 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 .
a) Identify the values of the characteristic decay time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==, natural frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==, and their product, the quality factor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric=for this damped harmonic oscillator system.b) Find the general solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEifkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRlcvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric=by hand and identify its characteristic decay time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== , quasifrequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== and quasiperiod 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.c) Find the solution which satisfies the initial conditions, showing all work.
d) Rewrite your sinusoidal factor of the solution (everything but the exponential factor) of the IVP in phase-shifted cosine form 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and state the initial amplitude A and phase shift \316\264.e) Use technology to plot your result of part c) for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjktRjY2MFEjLi5GJ0Y5RjtGPkZARkJGREZGRkgvRktRKHBvc3RmaXhGJy9GTlEwbWVkaXVtbWF0aHNwYWNlRicvRlFRJDBlbUYnRlJGVS1GWTYkUSI1RidGOS1GNjYwUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSFGJy9GTkZeb0Zdb0ZSRlUtRiw2JVEnJiM5NjQ7RicvRjBGPUY5 [not from part d)!], including the envelope functions 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 and make a rough sketch of what you see, labeling the axes with variable names and key tickmarks on your sketch. Does the envelope you found fit the solution curve as it should? [Yes or no, with explanation would be a good response.]
Optional. Think about only if you finish early: Evaluate the ratio of 5 times the decay time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== to the quasiperiod LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric=of your exponentially decaying sinusoidal solution function in order to get the number of quasiperiods that fit into 5 characteristic times. Does this agree with your plot?solution1. Divide through by leading coefficient 2 to get standard form:PkkkZGVxRzYiLywoLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkSSJER0YrIiIjNiNJInlHRiQ2I0kidEdGJCIiIiomIiInRjUtLUYvRjFGM0Y1RjUqJiIjISpGNS1GMkYzRjVGNSIiIQ==PkkmaW5pdHNHNiI2JC8tSSJ5R0YkNiMiIiEiIiIvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRikhIzo=a)QylJKHJlc3RhcnRHJSpwcm90ZWN0ZWRHIiIiLyZJJHRhdUc2IjYjIiIhLUkiL0c2JEYkSShfc3lzbGliR0YpNiMkIiInRitGJS8mSSZvbWVnYUdGKUYqLUklc3FydEdGLjYjJCIjISpGK0YlL0kiUUdGKSomLUkkcmhzR0YkNiNJIyUlR0YpRiUtRj82I0kiJUdGKUYlb)LUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJGRlcUdGJy1JInlHRic2I0kidEdGJw==QyUvSSJURzYiLUkiKkclKnByb3RlY3RlZEc2JCwkKiYiIiMiIiJJI1BpR0YoRi1GLSNGLSIiKkYtLUkmZXZhbGZHRig2I0kiJUdGJQ==c)Pkkoc29saW5pdEc2Ii1JJ2Rzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkPCRJJGRlcUdGJEkmaW5pdHNHRiQtSSJ5R0YkNiNJInRHRiQ=detailsPkkiWUc2ImYqNiNJInRHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmKihJJF9DMUdGJCIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCQqJiIiJEYuRidGLiEiIkYuLUkkc2luR0YxNiMsJComIiIqRi5GJ0YuRi5GLkYuKihJJF9DMkdGJEYuRi9GLi1JJGNvc0dGMUY7Ri5GLkYkRiRGJA==LS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiWUdGKDYjSSJ0R0YoQyU2JC8tSSJZRzYiNiMiIiEiIiIvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2I0YmRighIzpGKi1JJnNvbHZlR0YvNiQ8I0kiJUdGJzwkSSRfQzFHRidJJF9DMkdGJw==d)QysvSSZkZWx0YUc2Ii1JJ2FyY3Rhbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjIyEiJSIiJCIiIi1JJmV2YWxmR0YpNiNJIiVHRiVGLyomLUYxNiQsJCooIiQhPUYvLUkkcmhzR0YpNiNJIyUlR0YlRi9JI1BpR0YpISIiRi8iIiVGL0koZGVncmVlc0dGJUYvRi8+SSJBR0YlLUklc3FydEdGKDYjLCZGL0YvLUkiXkdGKTYkI0ZARi4iIiNGL0YvLUYxNiRGMyIiJg==d)LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlNyUtSSRyaHNHRiU2I0koc29saW5pdEdGJyomSSJBR0YnIiIiLUkkZXhwR0YkNiMsJComIiIkRjBJInRHRidGMCEiIkYwLCRGLkY4L0Y3OyIiISMiIiZGNi9JJmNvbG9yR0YnNyVJJHJlZEdGJ0kmYmxhY2tHRidGQw==Optional partQyk+SSR0YXVHNiIjIiIiIiIkRic+SSJUR0YlLUkiKkclKnByb3RlY3RlZEc2JCwkKiYiIiNGJ0kjUGlHRi1GJ0YnI0YnIiIqRictSSZldmFsZkdGLTYjSSIlR0YlRictRjY2IywkKigiIiZGJ0YkRidGKiEiIkYnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnNearly 2.4 oscillations.