Exam 14: Functions of Several Variables

Use a double integral to find the area of the region bounded by the graphs of $y = x ^ { 2 } + 2 x + 1$ and $y = ( x + 1 )$ .

If $f ( x , y ) = \ln \left( x y ^ { 4 } + 7 \right),$ find $\frac { \partial f } { \partial x } \text { and } \frac { \partial f } { \partial y }$

Use Lagrange multipliers to find the given extremum. In each case, assume that $x , y,$ and $z$ are positive. Maximize $f ( x , y , z ) = x + y + z$ Constraints $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$

Use a double integral to find the volume of the solid bounded by the graphs of the equations $z = x ^ { 4 } , z = 0 , x = 0 , x = 3 , y = 0 , y = 3$ .

For $f ( x , y )$ , find all values of x and y such that $f _ { x } ( x , y ) = 0$ and $f _ { y } ( x , y ) = 0$ simultaneously. $f ( x , y ) = 16 x ^ { 3 } - 4 x y + 16 y ^ { 3 }$

The population density (in people per square mile) for a coastal town can be modeled by $f ( x , y ) = \frac { 110,000 } { ( 2 + x + y ) ^ { 3 } }$ where x and y are measured in miles. What is the population inside the rectangular area defined by the vertices $( 0,0 ) , ( 2,0 ) , ( 0,2 ),$ and $( 2,2 )$ ? Round to the nearest integer.

Use Lagrange multipliers to find the given extremum. Assume that $x$ and $y$ are positive. Minimize $f ( x , y ) = e ^ { xy }$ Constraint $x ^ { 2 } + y ^ { 2 } - 8 = 0$

The Cobb-Douglas production function for an automobile manufacturer is $f ( x , y ) = 100 x ^ { 0.6 } y ^ { 0.4 }$ where x is the number of units of labor and y is the number of units of capital. Estimate the average production level if the number of units of labor x varies between 250 and 300 and the number of units of capital y varies between 250 and 300.

Find the standard equation of the sphere whose center is $( - 6 , - 5,7 )$ and whose radius is 4.

Describe the trace of the surface given by the function below in the xy-plane. $x ^ { 2 } - y - z ^ { 2 } = 0$

Describe the level curves for the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ for the c-values given by $c = 0,2,4,6,8$ .

Evaluate the following integral. $\int _ { 3 x } ^ { x ^ { 3 } } \frac { 4 y } { x } d y$

Evaluate the double integral $\int _ { 0 } ^ { 2 } \int _ { y } ^ { 2 y } \left( 1 + x ^ { 2 } + y ^ { 2 } \right) d x d y$ . Round your answer to two decimal places, where applicable.

Use Lagrange multipliers to maximize the function $f ( x , y ) = \sqrt { 16 - x ^ { 2 } - y ^ { 2 } }$ subject to the following constraint: $x + y - 4 = 0$ Assume that x, y, and z are positive.

Use a symbolic integration utility to evaluate the double integral. $\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { x } e ^ { x y } d y d x$

Use Lagrange multipliers to find the minimum distance from the circle $x ^ { 2 } + ( y - 4 ) ^ { 2 } = 49$ to the point $( - 10 , - 5 )$ Round your answer to the nearest tenth.

Find the critical points of the function $f ( x , y , z ) = - 3 - ( ( x + 8 ) ( y - 7 ) ( z - 3 ) ) ^ { 2 }$ , and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

Find the domain and range of the function. $f ( x , y ) = e ^ { \frac { x } { y } }$

Find the center and radius of the sphere whose equation is $2 x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } - 8 x + 12 y - 12 z + 43 = 0$ . Round your answer to two decimal places, where applicable.

A manufacturer has an order for 1100 units of fine paper that can be produced at two locations. Let $x _ { 1 }$ and $x _ { 2 }$ be the numbers of units produced at the two plants. Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by $C = 0.2 x _ { 1 } ^ { 2 } + 25 x _ { 1 } + 0.05 x _ { 2 } ^ { 2 } + 12 x _ { 2 }$ .