Exam 4: Applications of the Derivative

If an open box has a square base and a volume of 500 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction.

Find the horizontal and vertical asymptotes of the graph.

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

Determine where the function is concave upward and where it is concave downward.

Find the dimensions of a rectangle with a perimeter of 200 ft that has the largest possible area.

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.

Show that the function is concave upward wherever it is defined.

You are given the graph of a function f. Determine the intervals where f is increasing, constant, or decreasing.

Find the relative maxima and relative minima, if any, of the function.

The demand for motorcycle tires imported by Dixie Import-Export is 40,000/year and may be assumed to be uniform throughout the year. The cost of ordering a shipment of tires is $400, and the cost of storing each tire for a year is $2. Determine how many tires should be in each shipment if the ordering and storage costs are to be minimized. (Assume that each shipment arrives just as the previous one has been sold.)

The management of the UNICO department store has decided to enclose an 300 area outside the building for displaying potted plants and flowers. One side will be formed by the external wall of the store, two sides will be constructed of pine boards, and the fourth side will be made of galvanized steel fencing material. If the pine board fencing costs $6/running foot and the steel fencing costs $2/running foot, determine the dimensions of the enclosure that can be erected at minimum cost.

Find the relative maxima and relative minima of the function.

A division of Chapman Corporation manufactures a pager. The weekly fixed cost for the division is $20,000, and the variable cost for producing x pagers/week is dollars. The company realizes a revenue of dollars from the sale of x pagers/week. Find the level of production that will yield a maximum profit for the manufacturer. (Hint: Use the quadratic formula.)

A book designer has decided that the pages of a book should have margins at the top and bottom and margins on the sides. She further stipulated that each page should have an area of (see the figure). Determine the page dimensions that will result in the maximum printed area on the page.

Find the relative maxima and relative minima, if any, of the function.

Find the absolute maximum value and the absolute minimum value, if any, of the function.

Suppose the quantity demanded per week of a certain dress is related to the unit price p by the demand equation , where p is in dollars and x is the number of dresses made. To maximize the revenue, how many dresses should be made and sold each week? (Hint: R(x) = px). Round the answer to the nearest integer.

A Norman window has the shape of a rectangle surmounted by a semicircle (see the accompanying figure). If a Norman window is to have a perimeter of 28 ft, what should its dimensions be in order to allow the maximum amount of light through the window?

Determine the relative maxima and relative minima, if any.

Determine where the function is concave downward.