Exam 13: Functions of Several Variables

According to the Ideal Gas Law, , where P is pressure, V is volume, T is temperature (in Kelvins), and k is a constant of proportionality. A tank contains 2500 cubic inches of nitrogen at a pressure of 36 pounds per square inch and a temperature of 700 K. Write P as a function of V and T after evaluating k.

The two radii of the frustum of a right circular cone are increasing at a rate of 7 centimeters per minute, and the height is increasing at a rate of 16 centimeters per minute (see figure). Find the rate at which the surface area is changing when the two radii are 19 centimeters and 24 centimeters, and the height is 8 centimeters. [Note: The surface area does not include the top and bottom circles.] Round your answer to two decimal places.

For the function given by describe the level surface given by . ​

Find by using the limits and .

Find three positive numbers x, y, and z whose sum is 21 and the sum of the squares is a minimum.

Find the absolute extrema of on the region .

Find the limit for the function . ​

Let , where , , and . Find . ​

The temperature at the point on a metal plate is modeled by , . Find the direction of greatest increase in heat from the point . ​

For , evaluate at the point .

Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function at the critical point , if . ​

Differentiate implicitly to find , given . ​

Find an equation of the tangent plane to the surface at the point . ​

Suppose a pharmaceutical corporation has two plants that produce the same over-the-counter medicine. If and are the numbers of units produced at plant 1 and plant 2, respectively, then the total revenue for the product is given by . When and find the marginal revenue for plant 1. ​

Assume in 2009, an investment of $2200 was made in a bond earning 9% compounded annually. Assume that the buyer pays at rate R and the annual rate of inflation is I. Suppose in the year 2019, the value V of the investment in constant 2009 dollars is . Calculate the function for and . Round your answer to the nearest cent.

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

Find an equation of the tangent plane to the surface at the point .

Suppose the temperature at any point in a steel plate is where x and y are measured in meters. At the point find the rate of change of the temperature with respect to the distance moved along the plate in the direction of the x-axis. Round your answer to one decimal place. ​

Find the minimum distance from the point to the plane .

Suppose the formula for wind chill C (in degrees Fahrenheit) is given by where v is the wind speed in miles per hour and T is the temperature in degrees Fahrenheit. The wind speed is miles per hour and the temperature is Fahrenheit. Use to estimate the relative error in calculating the wind chill. Round your answer to two decimal places.