1. solve the differential equation dx/dy = 3x^2y with the initial condition y(0) = 7/3
2. Find the maximum and minimum values for f(x,y)= 2x^2-y^2-18x+2
3. Find the maximum and minimum values for f(x,y)= 2x^2-y^2-18x+2 subject to x^2+y^2 = 25.
4. At a certain factory, the daily output is q(x,y) = 60x^1/2y^1/3 units, where x is the capital investment measured in units of $1000 and y is the size of the labor force measured in worker hours. Suppose the current capital investment is $900,000 and that 1,000 worker-hours of labor are used each day. Use partial derivatives (marginal analysis) to determine the effect of the rate of daily output for an additional capital investment of $10,000 on the daily output if the size of the labor force is not changed.
5. As of today, gasoline prices are rising 7% per month. If gasoline is currently selling at $3.35 per gallon, write the differential equation describing the change in gas prices and solve it to predict the MONTHLY gas prices. Use the solution of your differential equation to predict the price of gas 6 months from now, and, if this growth rate were to continue, how fast the price is changing 6 months from now.
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