Consider a rectangle in the xy-plane, with corners at (0, 0), (a, 0), (0, b), and (a, b). If (a, b) lies on the graph of the equation y = 30 - x, find a and b such that the area of the rectangle is maximized. What economic interpretations can be given to your answer if the equation y = 30 - x represents a demand curve and y is the price corresponding to the demand x?
> Property of functions is described next. Draw some conclusion about the graph of the function. h’(3) = 4, h’’(3) = 1
> If g(3) = 2 and g’(3) = 4, find f (3) and f ‘(3), where f (x) = 2 * [g(x)]3.
> Property of functions is described next. Draw some conclusion about the graph of the function. g(1) = 5, g’(1) = -1
> Property of functions is described next. Draw some conclusion about the graph of the function. f (1) = 2, f ‘(1) > 0
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’ (x) is minimized. y a b c d e y = f(x)
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’ (x) is maximized. y a b c d e y = f(x)
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’’(x) is negative. y a b c d e y = f(x)
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’’ (x) is positive. y a b c d e y = f(x)
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’(x) is negative. y a b c d e y = f(x)
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’(x) is positive. y a b c d e y = f(x)
> Draw the graph of a function f (x) for which the function and its first derivative have the stated property for all x. f (x) decreasing and f (x) increasing
> What is the difference between an x-intercept and a zero of a function?
> If f (5) = 2, f ‘(5) = 3, g(5) = 4, and g’(5) = 1, find h(5) and h’(5), where h(x) = 3f (x) + 2g(x).
> What does it mean to say that the graph of f (x) has an inflection point at x = 2?
> Give three characterizations of what it means for the graph of f (x) to be concave up at x = 2. Concave down.
> What is the difference between having a relative maximum at x = 2 and having an absolute maximum at x = 2?
> State as many terms used to describe graphs of functions as you can recall.
> How are the cost, revenue, and profit functions related?
> Outline the procedure for solving an optimization problem.
> What is a constraint equation?
> What is an objective equation?
> Outline a procedure for sketching the graph of a function.
> Outline a method for locating the inflection points of a function.
> Draw the graph of f (x) = 2x + 18/x - 10 in the window [0, 16] by [0, 16]. In what ways is this graph like the graph of a parabola that opens upward? In what ways is it different?
> Outline a method for locating the relative extreme points of a function.
> Give two connections between the graphs of f (x) and f (x).
> State the first-derivative rule. The second-derivative rule.
> What is an asymptote? Give an example.
> How do you determine the y-intercept of a function?
> A one-product firm estimates that its daily total cost function (in suitable units) is C(x) = x3 - 6x2 + 13x + 15 and its total revenue function is R(x) = 28x. Find the value of x that maximizes the daily profit.
> The revenue function for a particular product is R(x) = x (4 - .0001x). Find the largest possible revenue.
> The revenue function for a one-product firm is R(x) = 200 – 1600/(x + 8) - x. Find the value of x that results in maximum revenue.
> If a total cost function is C(x) = .0001x3 - .06x2 + 12x + 100, is the marginal cost increasing, decreasing, or not changing at x = 100? Find the minimum marginal cost.
> Given the cost function C(x) = x3 - 6x2 + 13x + 15, find the minimum marginal cost.
> Figure 3 contains the curves y = f (x), y = g(x), and y = h(x) and the tangent lines to y = f (x) and y = g(x) at x = 1, with h(x) = f (x) + g(x). Find h(1) and h’(1). Figure 3: Y y = y = y = h(x) y = f(x) - 4x + 2.6 26x + 1.1 Fig
> Differentiate. y = x/2 – 2/x
> The cost function for a manufacturer is C(x) dollars, where x is the number of units of goods produced and C, C , and C are the functions given in Fig. 15. Figure 15: (a) What is the cost of manufacturing 60 units of goods? (b) What is the marginal c
> The revenue for a manufacturer is R(x) thousand dollars, where x is the number of units of goods produced (and sold) and R and R are the functions given in Figs. 14(a) and 14(b). Figure 14: Y 80 70 60 50 40 30 20 10 Y 3.2 2.4 1.6 .8 -.8 -1.6 -2.4 -
> Let P(x) be the annual profit for a certain product, where x is the amount of money spent on advertising. (See Fig. 13.) (a) Interpret P(0) (b) Describe how the marginal profit changes as the amount of money spent on advertising increases. (c) Explain th
> A savings and loan association estimates that the amount of money on deposit will be 1 million times the percentage rate of interest. For instance, a 4% interest rate will generate $4 million in deposits. If the savings and loan association can loan all
> The demand equation for a company is p = 200 - 3x, and the cost function is C(x) = 75 + 80x - x2, 0 ≤ x ≤ 40. (a) Determine the value of x and the corresponding price that maximize the profit. (b) If the government imposes a tax on the company of $4 per
> The monthly demand equation for an electric utility company is estimated to be p = 60 - (10-5)x, where p is measured in dollars and x is measured in thousands of kilowatt-hours. The utility has fixed costs of 7 million dollars per month and variable co
> A certain toll road averages 36,000 cars per day when charging $1 per car. A survey concludes that increasing the toll will result in 300 fewer cars for each cent of increase. What toll should be charged to maximize the revenue?
> In the planning of a sidewalk café, it is estimated that for 12 tables, the daily profit will be $10 per table. Because of overcrowding, for each additional table the profit per table (for every table in the café) will be reduced by $.50. How many tables
> A swimming club offers memberships at the rate of $200, provided that a minimum of 100 people join. For each member in excess of 100, the membership fee will be reduced $1 per person (for each member). At most, 160 memberships will be sold. How many memb
> An artist is planning to sell signed prints of her latest work. If 50 prints are offered for sale, she can charge $400 each. However, if she makes more than 50 prints, she must lower the price of all the prints by $5 for each print in excess of the 50. H
> Draw the graph of f (x) = 1/6 x3 – 5/2 x2 + 13x - 20 in the window [0, 10] by [-20, 30]. Algebraically determine the coordinates of the inflection point. Zoom in and zoom out to convince yourself that there are no relative extreme points anywhere.
> The average ticket price for a concert at the opera house was $50. The average attendance was 4000. When the ticket price was raised to $52, attendance declined to an average of 3800 persons per performance. What should the ticket price be to maximize th
> Until recently hamburgers at the city sports arena cost $4 each. The food concessionaire sold an average of 10,000 hamburgers on a game night. When the price was raised to $4.40, hamburger sales dropped off to an average of 8000 per night. (a) Assuming a
> Some years ago, it was estimated that the demand for steel approximately satisfied the equation p = 256 - 50x, and the total cost of producing x units of steel was C(x) = 182 + 56x. (The quantity x was measured in millions of tons and the price and total
> The demand equation for a product is p = 2 - .001x. Find the value of x and the corresponding price, p, that maximize the revenue.
> The demand equation for a certain commodity is p = 1/12 x2 - 10x + 300, 0 ≤ x ≤ 60. Find the value of x and the corresponding price p that maximize the revenue.
> A small tie shop sells ties for $3.50 each. The daily cost function is estimated to be C(x) dollars, where x is the number of ties sold on a typical day and C(x) = .0006x3 - .03x2 + 2x + 20. Find the value of x that will maximize the store’s daily profit
> A furniture store expects to sell 640 sofas at a steady rate next year. The manager of the store plans to order these sofas from the manufacturer by placing several orders of the same size spaced equally throughout the year. The ordering cost for each de
> A pharmacist wants to establish an optimal inventory policy for a new antibiotic that requires refrigeration in storage. The pharmacist expects to sell 800 packages of this antibiotic at a steady rate during the next year. She plans to place several orde
> Refer to Fig. 6. Suppose that (i) the ordering cost for each delivery of dried cherries is $50, and (ii) it costs $7 to carry 1 pound of dried cherries in inventory for 1 year. (a) What is the inventory cost (carrying cost plus ordering cost) if carrying
> Figure 2 contains the curves y = f (x) and y = g(x) and the tangent line to y = f (x) at x = 1, with g(x) = 3 # f (x). Find g(1) and g’(1). Figure 2: y y = .6x + 1 y = f(x) y = g(x) 1 Figure 2 Graphs of f(x) and g(x) = 3f(x). X
> Figure 6 shows the inventory levels of dried Rainier cherries at a natural food store in Seattle and the order–reorder periods over 1 year. Refer to the figure to answer the following questions. (a) What is the average amount of cherrie
> Coffee consumption in the United States is greater on a per capita basis than anywhere else in the world. However, due to price fluctuations of coffee beans and worries over the health effects of caffeine, coffee consumption has varied considerably over
> A pizza box is formed from a 20-cm by 40-cm rectangular piece of cardboard by cutting out six squares of equal size, three from each long side of the rectangle, and then folding the cardboard in the obvious manner to create a box. See Fig. 11. Let x be t
> If f (x) is defined on the interval 0 ≤ x ≤ 5 and f (x) is negative for all x, for what value of x will f (x) have its greatest value? Explain why.
> An open rectangular box of volume 400 cubic inches has a square base and a partition down the middle. See Fig. 10. Find the dimensions of the box for which the amount of material needed to construct the box is as small as possible. Figure 10 Open re
> Advertising for a certain product is terminated, and t weeks later, the weekly sales are f (t) cases, where f (t) = 1000(t + 8)-1 - 4000(t + 8)-2. At what time is the weekly sales amount falling the fastest?
> Consider a parabolic arch whose shape may be represented by the graph of y = 9 - x2, where the base of the arch lies on the x-axis from x = -3 to x = 3. Find the dimensions of the rectangular window of maximum area that can be constructed inside the arch
> The daily output of a coal mine after t hours of operation is approximately 40t + t2 – 1/15 t3 tons, 0 ≤ t ≤ 12. Find the maximum rate of output (in tons of coal per hour).
> Let f (t) be the amount of oxygen (in suitable units) in a lake t days after sewage is dumped into the lake, and suppose that f (t) is given approximately by At what time is the oxygen content increasing the fastest? f(t)=1- 10 t + 10 + 100 (t+10)²
> A closed rectangular box is to be constructed with a base that is twice as long as it is wide. If the total surface area must be 27 square feet, find the dimensions of the box that will maximize the volume.
> Draw the graph of f (x) = 1/6 x3 - x2 + 3x + 3 in the window [-2, 6] by [-10, 20]. It has an inflection point when x = 2, but no relative extreme points. Enlarge the window a few times to convince yourself that there are no relative extreme points anywhe
> An open rectangular box is to be constructed by cutting square corners out of a 16- by 16-inch piece of cardboard and folding up the flaps. [See Fig. 9.] Find the value of x for which the volume of the box will be as large as possible. Figure 9: 16
> An athletic field [Fig. 8] consists of a rectangular region with a semicircular region at each end. The perimeter will be used for a 440-yard track. Find the value of x for which the area of the rectangular region is as large as possible. h
> A certain airline requires that rectangular packages carried on an airplane by passengers be such that the sum of the three dimensions (i.e., length, width, and height), is at most 120 centimeters. Find the dimensions of the squareended rectangular packa
> A supermarket is to be designed as a rectangular building with a floor area of 12,000 square feet. The front of the building will be mostly glass and will cost $70 per running foot for materials. The other three walls will be constructed of brick and cem
> A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 150 cubic feet. The concrete for the base costs $4 per square foot, the material for the roof costs $2 per square foot, and the material for the sides costs
> Design an open rectangular box with square ends, having volume 36 cubic inches, that minimizes the amount of material required for construction.
> Shakespear’s Pizza sells 1000 large vegi pizzas per week for $18 a pizza. When the owner offers a $5 discount, the weekly sales increase to 1500. (a) Assume a linear relation between the weekly sales A(x) and the discount x. Find A(x). (b) Find the value
> Refer to Exercise 13. If the cost of the fencing for the boundary is $5 per meter and the dividing fence costs $2 per meter, find the dimensions of the corral that minimize the cost of the fencing. Exercise 13: A rectangular corral of 54 square meters i
> A rectangular corral of 54 square meters is to be fenced off and then divided by a fence into two sections, as shown in Fig. 7(b). Find the dimensions of the corral so that the amount of fencing required is minimized. Figure 7: 100 feet (a) x W LE
> Rework Exercise 11 for the case where only 200 feet of fencing is added to the stone wall. Exercise 11: Starting with a 100-foot-long stone wall, a farmer would like to construct a rectangular enclosure by adding 400 feet of fencing, as shown in Fig. 7
> Using the sum rule and the constant-multiple rule, show that for any functions f (x) and g(x) d/dx [f (x) - g(x)] = d/dx f (x) – d/dx g(x).
> Starting with a 100-foot-long stone wall, a farmer would like to construct a rectangular enclosure by adding 400 feet of fencing, as shown in Fig. 7(a). Find the values of x and w that result in the greatest possible area. Figure 7: 100 feet (a) x
> Refer to the inventory problem of Example 2. If the distributor offers a discount of $1 per case for orders of 600 or more cases, should the manager change the quantity ordered?
> A store manager wants to establish an optimal inventory policy for an item. Sales are expected to be at a steady rate and should total Q items sold during the year. Each time an order is placed a cost of h dollars is incurred. Carrying costs for the year
> A bookstore is attempting to determine the most economical order quantity for a popular book. The store sells 8000 copies of this book per year. The store figures that it costs $40 to process each new order for books. The carrying cost (due primarily to
> Foggy Optics, Inc., makes laboratory microscopes. Setting up each production run costs $2500. Insurance costs, based on the average number of microscopes in the warehouse, amount to $20 per microscope per year. Storage costs, based on the maximum number
> The Great American Tire Co. expects to sell 600,000 tires of a particular size and grade during the next year. Sales tend to be roughly the same from month to month. Setting up each production run costs the company $15,000. Carrying costs, based on the a
> A California distributor of sporting equipment expects to sell 10,000 cases of tennis balls during the coming year at a steady rate. Yearly carrying costs (to be computed on the average number of cases in stock during the year) are $10 per case, and the
> For what t does the function f (t) = t2 - 24t have its minimum value?
> Find the minimum value of f (t) = t3 - 6t2 + 40, t Ú 0, and give the value of t where this minimum occurs.
> Find the maximum value of the function f (x) = 12x - x2, and give the value of x where this maximum occurs.
> Differentiate the function f (x) = (3x2 + x - 2)2 in two ways. (a) Use the general power rule. (b) Multiply 3x2 + x - 2 by itself and then differentiate the resulting polynomial
> For what x does the function g(x) = 10 + 40x - x2 have its maximum value?
> Find the value of x for which the rectangle inscribed in the semicircle of radius 3 in Fig. 19 has the greatest area. Figure 19: -3 ม (r, y) 3
> Find the point on the line y = -2x + 5 that is closest to the origin. City A 6 miles Road X Road 11 miles City B 11-x 4 miles
> Find the point on the graph of y = √x that is closest to the point (2, 0). See Fig. 17. Figure 17: 2 0 Y 1 2 3
> A rectangular page is to contain 50 square inches of print. The page has to have a 1-inch margin on top and at the bottom and a 12-inch margin on each side (see Fig. 16). Find the dimensions of the page that minimize the amount of paper used. NIL in
> A cable is to be installed from one corner, C, of a rectangular factory to a machine, M, on the floor. The cable will run along one edge of the floor from C to a point, P, at a cost of $6 per meter, and then from P to M in a straight line buried under th
> A ship uses 5x2 dollars of fuel per hour when traveling at a speed of x miles per hour. The other expenses of operating the ship amount to $2000 per hour. What speed minimizes the cost of a 500-mile trip?
> In Example 3 we can solve the constraint equation (2) for x instead of w to get x = 20 – ½ w. Substituting this for x in (1), we get Sketch the graph of the equation and show that the maximum occurs when w = 20 and x =
> A large soup can is to be designed so that the can will hold 16p cubic inches (about 28 ounces) of soup. [See Fig. 14(b).] Find the values of x and h for which the amount of metal needed is as small as possible. Figure 14: x 2x (a) h x h (b) 2m x S
> Figure 14(a) shows a Norman window, which consists of a rectangle capped by a semicircular region. Find the value of x such that the perimeter of the window will be 14 feet and the area of the window will be as large as possible. Figure 14: 2x (a)
