Exam 18: Multivariable Calculus

Use the method of Lagrange multipliers to determine the critical points of f(x,y)= 4 + 2 + 3 subject to the constraint x+ 2y = 9.

The production function for a company's product is P = 100L + 50k - - ,where P is the output that results from L units of labor and k units of capital.The unit costs of labor and capital are 6 and 3,respectively.If the company wants the total cost of inputs to be 30,determine the greatest output possible subject to this budget constraint.

An empirical formula relating the surface area A (in square inches)of an average human body to the weight (in pounds)and the height h (in inches)of the person is A(w,h)= 15.64 .Find and .

If z = 4xy ln (3x + 9y)find

A manufacturer produces products A and B for which the average costs of production are constant at 3 and 5 (dollars per unit),respectively.The quantities , of A and B that can be sold each week are given by the joint-demand functions = 10 - + and = 12 + - 3 where and are the prices (in dollars per unit)of A and B,respectively.Determine the prices of A and B at which the manufacturer can maximize profit.

A sporting goods store determines that the optimal quantity of athletic shoes (in pairs)to order each month is given by the Wilson lot size formula: Q(C,M,s)= ,where C is the cost (in dollars)of placing an order,M is the number of pairs sold each month,and s is the monthly storage cost (in dollars)per pair of shoes.Find .Then find and interpret .

Find and where f(x,y,z)= .

A company manufactures two products,X and Y,and the joint-cost function for these products is given by where c is the total cost of producing x units of X and y units of Y.Determine the marginal cost with respect to x when x = 450 and y = 550.

A sporting goods store determines that the optimal quantity of athletic shoes (in pairs)to order each month is given by the Wilson lot size formula: Q(C,M,s)= ,where C is the cost (in dollars)of placing an order,M is the number of pairs sold each month,and s is the monthly storage cost (in dollars)per pair of shoes.Find .Then find and interpret .

If z = 4xy ln (3x + 9y)find

If f(x,y,z)= + Z + xy,then (1,2,3)=

The Cobb-Douglas production function for a company is given by P(k,l)= 163 where P is the monthly production value when k is the number of units of capital and l is the number of units of labor.Suppose that capital costs $105 per unit,labor costs $70 per unit,and the total cost of capital and labor is limited to $152,250.Use Lagrange Multipliers to write the system of equations you would use to find the number of units of capital and labor that maximize production.

If w = z( + 3 ,find: (a) (b) (c) (d) (e)

A television manufacturing company makes two types of TV's.The cost of producing x units of type A and y units of type B is given by the function C(x,y)= 100 + + 64 - 96xy.How many units of type A and type B televisions should the company produce to minimize its cost?

The Cobb-Douglas production function for a company is given by P(k,l)= 70 where P is the monthly production value when k is the number of units of capital and l is the number of units of labor.Suppose that capital costs $450 per unit,labor costs $75 per unit,and the total cost of capital and labor is limited to $60,000.Use Lagrange multipliers to write the system of equations you would use to find the number of units of capital and labor that maximize production.

For the production function P = 5.4 ,find the marginal productivity functions and .

A sporting goods store determines that the optimal quantity of athletic shoes (in pairs)to order each month is given by the Wilson lot size formula: Q(C,M,s)= ,where C is the cost (in dollars)of placing an order,M is the number of pairs sold each month,and s is the monthly storage cost (in dollars)per pair of shoes.Find .Then find and interpret .

Let f(x,y,z)= + + y ln( - 1).Find .

If f(x,y)= find

Determine the critical points of f(x,y)= + xy + - y and also determine by the second-derivative test whether each point corresponds to a relative maximum,to a relative minimum,to neither,or whether the test gives no information.