Homework 10
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Calculus
Homework 10
Due Date: January 9 (Wednesday)
1. Find the first partial derivatives
(a) f (x, y) =
xy
.
x2 +y 2
x+y
.
(b) z = ln x−y
(c) w =
xy
.
x+y+z
2. Find the second partial derivatives, that is, find fxx , fxy , fyy , and fyx .
p
(a) f (x, y) = 9 − x2 − y 2 .
(b) f (x, y) =
x2 −y 2
.
2xy
(c) f (x, y) = ln(x2 + y 2 + 1).
3. Consider the Cobb-Douglas production function
f (x, y) = 200x0.7 y 0.3 .
when x = 1000 and y = 500, find
(a) the marginal productivity of labor ∂f /∂x.
(b) the marginal productivity of capital ∂f /∂y.
4. The shareholder’s equity z (in millions of dollars) for Skechers from 2001 through 2009
can be modeled by
z = 0.175x − 0.772y − 275
where x is the sales (in millions of dollars) and y is the total assets (in millions of dollars).
(a) Find ∂z/∂x and ∂z/∂y.
(b) Interpret the partial derivatives in the context of the problem.
5. Find the critical points, relative extrema, and saddle points of the function.
(a) f (x, y) = x2 + y 2 + 2x − 6y + 6.
(b) f (x, y) = x2 − 3xy − y 2 .
(c) f (x, y) = 3e−(x
2 +y 2 )
.
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Calculus H415611 Fall 2012
6. A corporation manufactures a product for a high-performance automobile engine at two
locations. The cost of producing x1 units at location 1 is
C1 = 0.05x21 + 15x1 + 5400
and the cost of producing x2 units at location 2 is
C2 = 0.03x22 + 15x2 + 6100.
The demand function for the product is
p = 225 − 0.4(x1 + x2 )
and the total revenue function is
R = [225 − 0.4(x1 + x2 )](x1 + x2 ).
Find the production levels at the two locations that will maximize the profit
P = R − C1 − C2 .
7. Use Lagrange multipliers to find the given extremum. In each case, assume that x and y
are positive.
p
(a) Maximize f (x, y) = 6 − x2 = y 2 . Constraint: x + y − 2 = 0.
(b) Minimize f (x, y) = 2x + y. Constraint: xy = 32.
(c) Maximize f (x, y) = exy . Constraint: x2 + y 2 − 8 = 0.
8. A manufacturer has an order for 2000 units of all-terrain vehicles tires that can produced
at two locations. Let x1 and x2 be the numbers of units produced at the two plants. The
cost function is modeled by
C = 0.25x21 + 10x1 + 0.15x22 + 12x2 .
Find the number of units that should be produced at each location to minimize the cost.