Exam 7: Functions of Several Variables

A firm's weekly profit (in dollars)in marketing two products is given by ​ ​ Where and represent the numbers of units of each product sold weekly.Estimate the average weekly profit when varies between 40 and 50 units and varies between 45 and 50 units.

Find the lengths of the sides of the triangle with the given vertices,and determine whether the triangle is a right triangle,an isosceles triangle,or neither. ​ ​

Use the function to find ​

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

The value of an investment of $18,000 after t years in an account for which the interest rate r is compounded continuously is given by the function dollars.Write the partial derivative ​

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

The sum of the length (denote by z)and the girth (perimeter of a cross section)of packages carried by a delivery service cannot exceed 36 inches.Find the dimensions of the rectangular package of largest volume that may be sent. ​

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

Evaluate the double integral ​

Find the the distance between the two points and ​

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Data that are modeled by have a positive correlation.

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

A rectangular box is resting on the xy-plane with one vertex at the origin.The opposite vertex lies in the plane Find the dimensions that maximize the volume. (Hint: Maximize subject to the constraint ). ​

If ,find and ​

Evaluate the double integral ​

Describe the domain and range of the function. ​ ​

Sketch the region of integration ​

Find the least squares regression line for the given points. ​ ​

Find the equation of the sphere that has the points and as endpoints of a diameter. ​