MAT2705-01/02 14f 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 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).1. 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
a) Verify that this LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= satisfies the given differential equation.
b) Find the solution which satisfies the initial condition LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVRjpGPg==.Organize your work as though you were playing professor.2. Choose appropriately named variables and write a differential equation that models the situation:
"When sugar is dissolved in water the time rate of change of the amount that remains undissolved in water is proportional to that amount."Knowing that this amount should decrease, what is the sign of the constant of proportionality you introduced?solution2.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==PkkkZGVxRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJBR0YkNiNJInRHRiRGLSomSSJrR0YkIiIiRipGMA==For decreasing amount, we should have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=.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.LUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQvLUklZGlmZkdGJTYkLUkiQUdGJzYjSSJ0R0YnRjAqJkkia0dGJyIiIkYtRjM8I0Yt1.Right clicking on the blue output and menu selectingi Solve DE, for y(x),Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5RzYiNiNJInhHRilGKy1JJGV4cEdGKTYjLCRGJyEiIg==LUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQvLUklZGlmZkdGJTYkLUkieUdGJzYjSSJ4R0YnRjAtSSRleHBHRiQ2IywkRi0hIiI8I0YtRepeat for the diffential equation with initial condition separated by a commaNiQvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInlHNiI2I0kieEdGKkYsLUkkZXhwRzYkRiZJKF9zeXNsaWJHRio2IywkRighIiIvLUYpNiMiIiFGNw==LUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQ8JC8tSSVkaWZmR0YlNiQtSSJ5R0YnNiNJInhHRidGMS1JJGV4cEdGJDYjLCRGLiEiIi8tRi82IyIiIUY6PCNGLg==By 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.QyUtSSVzdWJzRyUqcHJvdGVjdGVkRzYkLy1JInlHNiI2I0kieEdGKi1JI2xuRzYkRiVJKF9zeXNsaWJHRio2IywmRiwiIiJJJF9DMUdGKkYzLy1JJWRpZmZHRiU2JEYoRiwtSSRleHBHRi82IywkRighIiJGMy1JKXNpbXBsaWZ5R0YvNiNJIiVHRio=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.LUkob2RldGVzdEc2IjYkLy1JInlHRiQ2I0kieEdGJC1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJkYqIiIiSSRfQzFHRiRGMi8tSSVkaWZmR0YuNiRGJ0YqLUkkZXhwR0YtNiMsJEYnISIiNiI=