1. (30 pts) The total weekly revenue (in dollars) of the Country Workshop company is 2 2 given by R ( x , y )=0.2 x 0.25 y 0.2 xy +200 x +160 y where x denotes the number of finished units and y the number of unfinished units the company manufactures and sells each week. The total weekly cost is given by C ( x , y ) =100 x +70 y +4000 dollars. Determine how many finished units and how many unfinished units the company should manufacture each week to maximize profit. What is the maximum profit realizable? 2. (30 pts)The proprietor of Carson Hardware Store has decided to set up a sinking fund for the purpose of purchasing a computer server 2 years from now. It is expected that the dA server will cost $30,000. The fund grows at the rate of dt =rA + p where A denotes the size of the fund at any time t , fund compounded continuously, and r is the annual interest rate earned by the P is the amount (in dollars) paid into the fund by the proprietor per year (assume that this is essentially continuous). If the fund earns 5% interest per year compounded continuously, then determine the size of the yearly investment the proprietor should pay into the fund. 3. Find the following tangent lines. (a) (15 pts) The tangent line to f ( x )=x ln ( x ) at 2 (b) (15 pts) The tangent line of g ( x ) =e x + x at x=1 x=0 3 4. (20 pts) Find the maximum of f ( x )=x 3 x +5 on the interval [2,3] x 5. (30 pts) Find the area under the curve f ( x )=x e between x=0 and x=2 . 6. (30 pts) Two cars start out side by side and travel along a straight road. The velocity of Car 1 is f ( t ) ft/sec over the interval [0,180] . The velocity of Car 2 is g(t) ft/sec over the interval [0,180] . Furthermore, suppose the graphs of f and g are depicted in the accompanying figure. Which car travels farther between t=0 and t=180 ? By how many feet? 7. (30 pts.) For its beef stew, the Betty More Company uses aluminum containers that have the form of right circular cylinders. Find the radius and height of a container if it has the 3 capacity of 36 and is constructed using the least amount of material