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16C-FQ02 - 16C Final Fall 2002 1 Find the general solution...

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16C Final - Fall 2002 1. Find the general solution of the differential equation x 2 y 0 - 2 xy = 9 . 2. The rate of increase in sales S (in thousands of units) of palm pilots is proportional to the current level of sales and inversely proportional to the square of the time t . That is, dS dt = kS t 2 where k is a constant. (a) Is the constant, k , positive or negative? (b) The saturation point for the market is 400,000 palm pilots. That is, the limit of S as t → ∞ is 400. After 1 year, 100,000 palm pilots have been sold. Find S as a function of time in years. 3. Sketch the region of integration and evaluate the following integral. Z 1 0 Z 1 y 1 x 3 + 2 dx dy 4. Find the critical points of the function f ( x, y ) = x + y +1 / ( xy ) and deter- mine whether they are relative minima, relative maxima, or saddle points. 5. Use Lagrange multipliers to find the point on the plane x + 2 y + 3 z = 14 which is closest to the origin. ( Hint: Minimize the square of the distance in order to avoid square roots.) 6. Find the first 5 terms of the sequence of partial sums associated to the
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