MAT2705-01/02 12f Quiz 1 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).1. 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
a) Verify that this LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= satisfies the given differential equation.
b) Find the solution which satisfies the initial condition LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUY7NiRRIjJGJ0Y+Rj4=.Organize your work as though you were playing professor.2. Choose appropriately named variables and write a differential equation that models the situation:
"A spherical balloon is being blown up inside a spherical cavity of volume LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn filled with liquid in such a way that the time rate of change of the balloon's volume is inversely proportional to the balloon's volume and directly proportional to the difference between its volume and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn." [Translation: the time rate of change is proportional to the ratio of the difference in volume to the balloon volume itself.] If the balloon is being blown up as stated, what should the sign of your proportionality constant be?solution2.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==PkkkZGVxRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJWR0YkNiNJInRHRiRGLSooSSJrR0YkIiIiLCZJI1ZzR0YkRjBGKiEiIkYwRipGMw==For increasing volume, we should have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=.Right-clicking allows its solution easily, but what is this LambertW function? Just another special function like the exponential function, but less familiar. We can easily work with it the same way as the exponential function itsefl.LUknZHNvbHZlRzYiNiQvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIlZHRiQ2I0kidEdGJEYuKihJImtHRiQiIiIsJkkjVnNHRiRGMUYrISIiRjFGK0Y0PCNGKw==Let's pick some simple values for the parameters, setting the total volume and proportionality factor to 1 and starting at half the volume:QyY+SSJrRzYiIiIiISIiPkkjVnNHRiVGJkYnNiRJJGRlcUc2Ii8tSSJWR0YkNiMiIiEsJEkjVnNHRiQjIiIiIiIjLUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQ8JC8tSSJWR0YnNiMiIiEjIiIiIiIjLy1JJWRpZmZHRiU2JC1GLDYjSSJ0R0YnRjgqJiwmRjBGMEY2ISIiRjBGNkY7PCNGNg==We can plot the right hand side.LUkkcmhzRyUqcHJvdGVjdGVkRzYjLy1JIlZHNiI2I0kidEdGKSwmIiIiRi0tSSlMYW1iZXJ0V0c2JEYkSShfc3lzbGliR0YpNiMsJC1JJGV4cEdGMDYjLChGKyEiIiNGOCIiI0YtKiZeI0YtRi1JI1BpR0YkRi1GLSNGLUY6Ri0=LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCYiIiJGKi1JKUxhbWJlcnRXR0YkNiMsJC1JJGV4cEdGJDYjLChJInRHRichIiIjRjQiIiNGKiomXiNGKkYqSSNQaUdGJUYqRiojRipGNkYqL0YzOyIiISIiJg==Hmm, there is the complex number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in the expression, we can force it to rewrite it as an explicitly real function with the evaluate to a complex number commandLUkmZXZhbGNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmIiIiRiotSSlMYW1iZXJ0V0dGJDYjLCQtSSRleHBHRiQ2IywoSSJ0R0YnISIiI0Y0IiIjRioqJl4jRipGKkkjUGlHRiVGKkYqI0YqRjZGKg==LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkNyQsJi1JKUxhbWJlcnRXR0YkNiMsJC1JJGV4cEdGJDYjLCZJInRHRichIiIjRjQiIiMiIiJGNUY3RjdGN0Y3L0YzOyIiISIiJg==1.Right clicking on the blue output and menu selectingi Solve DE, for y(x),LywmKiZJInhHNiIiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkieUdGJjYjRiVGJUYnRidGLCIiJCwkKiRGJSIiJiIiIw==QyUtSSdkc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8sJiomSSJ4R0YoIiIiLUklZGlmZkdGJjYkLUkieUdGKDYjRi1GLUYuRi5GMiIiJCwkKiRGLSIiJiIiIzwjRjJGLi1JJ2V4cGFuZEdGJjYjSSIlR0YoRepeat for the diffential equation with initial condition separated by a commaNiQvLCYqJkkieEc2IiIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YnNiNGJkYmRihGKEYtIiIkLCQqJEYmIiImIiIjLy1GLjYjRihGNA==QyUtSSdkc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDwkLywmKiZJInhHRigiIiItSSVkaWZmR0YmNiQtSSJ5R0YoNiNGLkYuRi9GL0YzIiIkLCQqJEYuIiImIiIjLy1GNDYjRi9GOjwjRjNGLy1JJ2V4cGFuZEdGJjYjSSIlR0YoBy hand one substitutes the general solution into the DE everywhere and evaluate the derivative and then combine terms to cancel all but one on the LHS of the equation, finding it then equal to the RHS.QyUtSSVzdWJzRyUqcHJvdGVjdGVkRzYkLy1JInlHNiI2I0kieEdGKiwmKiRGLCIiJiMiIiIiIiUqJkkiQ0dGKkYxRiwhIiRGMS8sJiomRixGMS1JJWRpZmZHRiU2JEYoRixGMUYxRigiIiQsJEYuIiIjRjEtSSlzaW1wbGlmeUc2JEYlSShfc3lzbGliR0YqNiNJIiVHRio=The derivative isLUklZGlmZkclKnByb3RlY3RlZEc2JCwmKiRJInhHNiIiIiYjIiIiIiIlKiZJIkNHRilGLEYoISIkRixGKA==Doubling this and substituting into the LHS clearly eliminates the extra terms.
Maple has a DE checker command that will do this for us. It simplifies the difference LHS-RHS of the DE to zero if the relation solves the DE.LUkob2RldGVzdEc2IjYkLy1JInlHRiQ2I0kieEdGJCwmKiRGKiIiJiMiIiIiIiUqJkkiQ0dGJEYvRiohIiRGLy8sJiomRipGLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGJ0YqRi9GL0YnIiIkLCRGLCIiIw==JSFH