Exam 4: Applications of the Derivative

The figure depicts a racetrack with ends that are semicircular in shape. The length of the track is . Find and so that the area enclosed by the rectangular region of the racetrack is as large as possible.

Determine where the function is concave downward.

A wooden beam has a rectangular cross section of height in. and width in. (see the figure). The strength of the beam is directly proportional to its width and the square of its height. What are the dimensions of the cross section of the strongest beam that can be cut from a round log of diameter 24 in.? Hint: , where is a constant of proportionality.

A manufacturer of tennis rackets finds that the total cost C(x) (in dollars) of manufacturing x rackets/day is given by Each racket can be sold at a price of p dollars, where p is related to x by the demand equation If all rackets that are manufactured can be sold, find the daily level of production that will yield a maximum profit for the manufacturer.

An apple orchard has an average yield of 48 bushels of apples/tree if tree density is 24 trees/acre. For each unit increase in tree density, the yield decreases by 3 bushels. How many trees should be planted in order to maximize the yield? __________ trees

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

The average cost per disc (in dollars) incurred by Herald Records in pressing x DVDs is given by the average cost function Find the horizontal asymptote of . __________ What is the limiting value of the average cost? __________

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If f is decreasing on (a, b), then for each x in (a, b).

Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 102 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. What is the volume of such a package? (Hint: The length plus the girth is (see the figure)).

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

Find the horizontal and vertical asymptotes of the graph.

Using the curve-sketching guide, select the graph of the function.

The total annual revenue of the Miramar Resorts Hotel is related to the amount of money the hotel spends on advertising its services by the function where both and are measured in thousands of dollars. Use this function to: A. Find the interval where the graph of is concave upward and the interval where the graph of is concave downward. B. Find the inflection point of . C. Determine if it would it be more beneficial for the hotel to increase its advertising budget slightly when the budget is $90,000 or when it is $110,000.

Find the relative extrema of the following function. Use the second derivative test, if applicable.

A stone is thrown straight up from the roof of a 50-ft building. The height (in feet) of the stone at any time t (in seconds), measured from the ground, is given by What is the maximum height the stone reaches?

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.) __________ pagers/week

Use the information summarized in the table to select the graph of f.

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.

Select the graph of the function using the curve-sketching guide.

The number of major crimes committed in the city between 1997 and 2004 is approximated by the function where N(t) denotes the number of crimes committed in year t ( corresponds to 1997). Enraged by the dramatic increase in the crime rate, the citizens, with the help of the local police, organized "Neighborhood Crime Watch" groups in early 2001 to combat this menace. Show that the growth in the crime rate was maximal in 2003, giving credence to the claim that the Neighborhood Crime Watch program was working.