Exam 7: Functions of Several Variables

A rancher plans to use an existing stone wall and the side of a barn as a boundary for two adjacent rectangular corrals.Fencing for the perimeter costs 10 per foot.To separate the corrals,a fence that costs 8 per foot will divide the region.The total area of the two corrals is to be 7,000 square feet.Use Lagrange multipliers to find the dimensions that will minimize the cost of the fencing. ​ ​

Use a double integral to find the area of the region bounded by the graphs of ​

Find the slopes of the surface in the x- and y- directions at the point ​

Examine the function given below for relative extrema and saddle points. ​ ​

The two planes and are parallel?

Evaluate the double integral Round your answer to two decimal places. ​

Because of the forces caused by its rotation,a planet is actually an oblate ellipsoid rather than a sphere.The equatorial radius is 3967 miles and the polar radius is 3951 miles.Find an equation of the ellipsoid.Assume that the center of a planet is at the origin and the xy- trace corresponds to the equator. ​

Describe the level curves of the function.Sketch the level curves for the given c-values. ​ , ​

Find the least squares regression line for the points ​

Sketch the region of integration ​

Sketch the level curves for the function below for the given c-values ​ ​

Sketch the region R whose area is given by the following double integral. ​ ​

Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false. ​ When the correlation coefficient is ,the model is a good fit.

Find the average value of over the region R: square with vertices , , ,

The Cobb-Douglas production function for an automobile manufacturer is 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 200 and 250 and the number of units of capital y varies between 200 and 250.Round your answer to two decimal places. ​

If ,find and ​

Find the critical points of the function ,and,from the form of the function,determine whether a relative maximum or a relative minimum occurs at each point. ​

Sketch the region of integration

Examine the function given below for relative extrema and saddle points. ​ ​

Find if the midpoint of the line segment joining the two points and is ​ ​