Exam 4: Applications of the Derivative

What are the dimensions of a closed rectangular box that has a square cross section, a capacity of 127 in.³ and is constructed using the least amount of material? Round the answer to two decimal places. ​

Suppose the total cost function for manufacturing a certain product is dollars, where represents the number of units produced. Find the level of production that will minimize the average cost. Round the answer to the nearest integer. ​

You are given the graph of a function Determine the intervals where f is concave upward.​ ​ ​

Find the horizontal and vertical asymptotes of the graph of the function. ​ ​

You are given the graph of some function f defined on the indicated interval. Find the absolute maximum and the absolute minimum of f, if they exist. ​ f defined on ​

The height (in feet) attained by a rocket t sec into flight is given by the function ​ . ​ When is the rocket rising and when is it descending?

Find the horizontal and vertical asymptotes of the graph of the function. ​ ​

Determine the relative maxima and relative minima, if any. Otherwise, answer none. ​ ​ Relative maxima: __________ ​ Relative minima: __________

Use the information summarized in the table to select the graph of Domain: Intercepts: x-intercepts: 0, 4 Asymptotes: None Intervals where f is ↑ and ↓: ↑ on ↓ on Relative extrema: Rel. min. at Concavity: Upward on Point of inflection: ​

Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 90 in. ​ Find the dimensions of a rectangular package that has a square cross section and the largest volume that may be sent through the mail.

A rectangular box is to have a square base and a volume of 24 ft.³. If the material for the base costs 20 cent/square foot, the material for the sides costs 10 cent/square foot, and the material for the top costs 40 cent/square foot, determine the dimensions of the box that can be constructed at minimum cost. ​ ​

The level of ozone, an invisible gas that irritates and impairs breathing, present in the atmosphere on a certain May day in the city was approximated by ​ , where is measured in pollutant standard index (PSI) and t is measured in hours, with corresponding to 7 a.m. ​ Use the second derivative test to show that the function A has a relative maximum at approximately . Interpret your results.