Exam 4: Applications of the Derivative

If exactly 150 people sign up for a charter flight, Leisure World Travel Agency charges $250/person. However, if more than 150 people sign up for the flight (assume this is the case), then each fare is reduced by $1 for each additional person. Determine how many passengers will result in a maximum revenue for the travel agency. What is the maximum revenue? What would be the fare per passenger in this case? Hint: Let x denote the number of passengers above 150. Show that the revenue function R is given by R(x) = (150 + x)(250 - x).

A stone is thrown straight up from the roof of an 48-ft building. The distance (in feet) of the stone from the ground at any time t (in seconds) is given by . Sketch the graph of h. When is the stone rising, and when is it falling? If the stone were to miss the building, when would it hit the ground?

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

The quantity demanded each month of the Sicard wristwatch is related to the unit price by the equation , where p is measured in dollars and x is measured in units of a thousand. To yield a maximum revenue, how many watches must be sold? __________

What are the dimensions of a closed rectangular box that has a square cross section, a capacity of and is constructed using the least amount of material?

Find the horizontal and vertical asymptotes of the graph. Horizontal asymptote is __________ Vertical asymptote is __________

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 is measured in hours, with corresponding to a.m. Use the second derivative test to find the approximate value of where the function has a relative maximum. Round to the nearest integer.

A grain silo has the shape of a right circular cylinder surmounted by a hemisphere (see the figure). If the silo is to have a capacity of , find the radius and height of the silo that requires the least amount of material to construct. Hint: The volume of the silo is , and the surface area (including the floor) is .

Find the horizontal and vertical asymptotes of the graph.

Find the relative maxima and relative minima, if any, of the function. Otherwise, answer none. Relative minima: __________ Relative maxima: __________

Find the inflection points, if any, of the function.

If f is not continuous on the closed interval , then f cannot have an absolute maximum value.

Neilsen Cookie Company sells its assorted butter cookies in containers that have a net content of 1 lb. The estimated demand for the cookies is 1,000,000 1-lb containers. The setup cost for each production run is $250, and the manufacturing cost is $.50 for each container of cookies. The cost of storing each container of cookies over the year is $.20. Assuming uniformity of demand throughout the year and instantaneous production, how many containers of cookies should Neilsen produce per production run in order to minimize the production cost? Hint: Show that the total production cost is given by the function . Then minimize the function on the interval (0, 1,000,000). __________ containers of cookies per production run

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. __________ rackets/day

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

The quantity demanded each month of the Sicard wristwatch is related to the unit price by the equation , where p is measured in dollars and x is measured in units of a thousand. To yield a maximum revenue, how many watches must be sold?

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.

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

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.

The quantity demanded each month of the Walter Serkin recording of Beethoven's Moonlight Sonata, manufactured by Phonola Record Industries, is related to the price/compact disc. The equation , where p denotes the unit price in dollars and x is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging x copies of this classical recording is given by . To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is , and the profit is . __________ copies