Exam 7: Functions of Several Variables

A company has a Cobb-Douglas production function $f(x, y)=20 x ^{0.33} y^{0.67}$ where x is the utilization of labor and y is the utilization of capital. Determine the number of units of product produced when 1728 units of labor and 27,000 units of capital are used.

A company makes cylindrical cans of radius r and height h at a cost of a cents per unit area for the top and bottom and b cents per unit area for the side. Express the cost of producing a can as a function of the two variables r, and h. Enter your answer in the form: $2 \pi r ( \mathrm { cr } \pm \mathrm { dh } )$

Let g(x, y) = $\frac { x - 6 y } { x ^ { 2 } + y ^ { 2 } }$ . Compute g(3, 4).

The demand for a certain energy-efficient home is given by f( $\mathrm { P } 1$ , $\mathrm { P } 2$ ), where p1 is the price of the home and p2 is the price of electricity. Which of the following explains why $\frac { \partial \mathrm { f } } { \partial \mathrm { p } 2 }$ > 0?

Which straight line best fits the data points (0, 1), (1, 3), (2, 7)? Use partial derivatives.

Let f(x, y, z) be the amount of heat lost each day by a rectangular building x feet wide, y feet long, and z feet high. Suppose that $\frac { \partial f } { \partial z }$ (50, 80, 15) = 45. Which one of the following conclusions can be drawn?

Let f(x, y) = $x ^ { 2 }$ $e ^ { x y }$ . Find $\frac { \partial f } { \partial x }$ . Is $\left( y x ^ { 2 } - 2 x \right) e ^ { x y }$ the correct answer?

Let $f ( x , y , z ) = ( \sqrt { x y z } + \sqrt { - 2 z } + \sqrt { x - z } ) .$ Compute f(1, -2, -1). Enter your answer as a $\sqrt { b }$ .

Minimize the function f(x, y, z) = $x ^ { 2 }$ + $y ^ { 2 }$ + $z ^ { 2 }$ subject to the constraint $x + y + z = 2$ Enter your answer an just a reduced fraction of form $\frac { a } { b }$ .

The productivity of a certain country is P(x, y) = 1600 $x ^ { 1 / 4 }$ $y ^ { 3 / 4 }$ units, where x and y are the amounts of labor and capital utilized. What is the marginal productivity of capital when x = 16 and y = 625?

Calculate the iterated integral $\int _ { 1 } ^ { 2 } \left( \int _ { 2 } ^ { 3 } x y d y \right)$ dx. Enter just a reduced fraction of form $\frac { a } { b }$ .

Let f(x, y, z) = $x ^ { 2 }$ y + $\frac { x } { z }$ . Find $\frac { \partial f } { \partial z }$ . Enter your answer as a polynomial in x in standard form (unlabeled).

Find the formula that gives the least squares error for the points (-1, 5), (2, 2), (5, -1).

Suppose the partial derivatives of a Lagrange function F(x, y, λ) are $\frac { \partial F } { \partial x }$ = 2 - 8λx, $\frac { \partial \mathrm { F } } { \partial \mathrm { y } }$ = 1 -2λy, $\frac { \partial F } { \partial \lambda } = 32 - 4 x ^ { 2 } - y ^ { 2 }$ What values of x and y minimize F(x, y, λ)? (Assume x and y are positive.)

A rectangular box of length x, width y, and height z with no top is to be constructed having a volume of 32 cubic inches. Determine the dimensions that will require the least amount of material to construct the box.

Find all points (x, y) where f(x, y) = 2 $x ^ { 2 }$ + 2 $y ^ { 3 }$ - x - 6y + 14 has a possible relative maximum or minimum. Enter your answer exactly as just (a, b), (c, d) with b > d and where a, b, c, d are either integers or reduced fractions of form $\frac { e } { f }$ .

Let f(x, y) = $x ^ { 2 }$ + 2xy + $5 y ^ { 2 }$ + 2x + 10y - 3. At which point(s) does f(x, y) have possible maximum/minimum values?

A closed rectangular box with square ends is to be designed so that the surface area of the box is minimized. [Note: Surface area = 2 $x ^ { 2 }$ + 4xy.] It is required that the volume be 32 cubic inches. Which of the following is the Lagrange function F(x, y, λ) for this problem?

Find all points (x, y) where $f ( x , y ) = x ^ { 2 } + x y + y ^ { 2 } - 3 x + 2$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.