Consider the problem of finding the dimensions of the rectangular garden of area 100 square meters for which the amount of fencing needed to surround the garden is as small as possible. (a) Draw a picture of a rectangle and select appropriate letters for the dimensions. (b) Determine the objective and constraint equations. (c) Find the optimal values for the dimensions.
> 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)
> Find the equation of the tangent line to the curve y = 8 / x2 + x + 2 at x = 2.
> Find two positive numbers, x and y, whose product is 100 and whose sum is as small as possible.
> Find two positive numbers, x and y, whose sum is 100 and whose product is as large as possible.
> Find the dimensions of the rectangular garden of greatest area that can be fenced off (all four sides) with 300 meters of fencing.
> A farmer has $1500 available to build an E-shaped fence along a straight river so as to create two identical rectangular pastures. (See Fig. 13.) The materials for the side parallel to the river cost $6 per foot, and the materials for the three sections
> A canvas wind shelter for the beach has a back, two square sides, and a top. Find the dimensions for which the volume will be 250 cubic feet and that requires the least possible amount of canvas.
> Find the dimensions of the closed rectangular box with square base and volume 8000 cubic centimeters that can be constructed with the least amount of material.
> A closed rectangular box with a square base and a volume of 12 cubic feet is to be constructed from two different types of materials. The top is made of a metal costing $2 per square foot and the remainder of wood costing $1 per square foot. Find the dim
> A rectangular garden of area 75 square feet is to be surrounded on three sides by a brick wall costing $10 per foot and on one side by a fence costing $5 per foot. Find the dimensions of the garden that minimize the cost of materials.
> Postal requirements specify that parcels must have length plus girth of at most 84 inches. Consider the problem of finding the dimensions of the square-ended rectangular package of greatest volume that is mailable. (a) Draw a square-ended rectangular box
> Find the slope of the tangent line to the curve y = (x2 - 15)6 at x = 4. Then write the equation of this tangent line.
> Figure 12(b) shows an open rectangular box with a square base. Consider the problem of finding the values of x and h for which the volume is 32 cubic feet and the total surface area of the box is minimal. (The surface area is the sum of the areas of the
> There are $320 available to fence in a rectangular garden. The fencing for the side of the garden facing the road costs $6 per foot and the fencing for the other three sides costs $2 per foot. [See Fig. 12(a).] Consider the problem of finding the dimensi
> Find the positive values of x, y, and z that maximize Q = xyz, if x + y = 1 and y + z = 2. What is this maximum value?
> Find the positive values of x and y that minimize S = x + y if xy = 36, and find this minimum value.
> In Exercise 7, can there be a maximum for Q = x2 + y2 if x + y = 6? Justify your answer. Exercise 7: Find the minimum of Q = x2 + y2 if x + y = 6.
> Find the minimum of Q = x2 + y2 if x + y = 6.
> Find two positive numbers x and y that maximize Q = x2y if x + y = 2.
> Find the maximum of Q = xy if x + y = 2.
> Find the x-intercepts of the given function. y = 4 - 2x - x2
> Find the x-intercepts of the given function. y = 2x2 + 5x + 2
> Write the equation of the tangent line to the curve y = x3 + 3x - 8 at (2, 6).
> Find the x-intercepts of the given function. y = x2 + 5x + 5
> Find the x-intercepts of the given function. y = x2 - 3x + 1
> The canopy height (in meters) of the tropical bunch-grass elephant millet t days after mowing (for t Ú 32) is f (t) = -3.14 + .142t - .0016t2 + .0000079t3 - .0000000133t4. (Source: Crop Science.) (a) Graph f (t) in the window [32, 250] by [-1.2, 4.5]. (b
> In a medical experiment, the body weight of a baby rat in the control group after t days was f (t) = 4.96 + .48t + .17t2 - .0048t3 grams. (Source: Growth, Development and Aging.) (a) Graph f (t) in the window [0, 20] by [-12, 50]. (b) Approximately how m
> If f (a) = 0 and f (x) is decreasing at x = a, explain why f (x) must have a local maximum at x = a.
> If f (a) = 0 and f (x) is increasing at x = a, explain why f (x) must have a local minimum at x = a.
> Find the quadratic function f (x) = ax2 + bx + c that goes through (0, 1) and has a local minimum at (1,-1).
> Find the quadratic function f (x) = ax2 + bx + c that goes through (2, 0) and has a local maximum at (0, 1).
> Determine which function is the derivative of the other. f(x) g(x) y X
> Determine which function is the derivative of the other. f(x) |(x)b y
> Find the slope of the tangent line to the curve y = x3 + 3x – 8 at (2, 6).
> Sketch the graphs of the following functions for x > 0. y = 1/√x + x/2
> Sketch the graphs of the following functions for x > 0. y = 6√x - x
> Sketch the graphs of the following functions for x > 0. y = 1/x2 + x/4 – 5/4
> Sketch the graphs of the following functions for x > 0. y = 2/x +x/2 + 2
> Sketch the graphs of the following functions for x > 0. y = 12/x + 3x + 1
> Sketch the graphs of the following functions for x > 0. y = 9/x + x + 1
> Sketch the graphs of the following functions for x > 0. y = 2/x
> Sketch the graphs of the following functions for x > 0. y = 1/x + ¼ x
> Sketch the graphs of the following functions. f (x) = (x + 2)4 - 1
> Sketch the graphs of the following functions. f (x) = (x - 3)4
> Find the slope of the graph of y = f (x) at the designated point. f (x) = x10 + 1 + √(1 – x), (0, 2)
> Differentiate. y = 1/3x3
> Sketch the graphs of the following functions. f (x) = 3x4 - 6x2 + 3
> Sketch the graphs of the following functions. f (x) = x4 - 6x2
> Sketch the graphs of the following functions. f (x) = 1/3 x3 - 2x2
> Sketch the graphs of the following functions. f (x) = 1 - 3x + 3x2 - x3
> Sketch the graphs of the following functions. f (x) = -3x3 - 6x2 - 9x - 6
> Sketch the graphs of the following functions. f (x) = 4/3 x3 - 2x2 + x
> Sketch the graphs of the following functions. f (x) = 2x3 + x - 2
> Sketch the graphs of the following functions. f (x) = 5 - 13x + 6x2 - x3
> Sketch the graphs of the following functions. f (x) = x3 + 2x2 + 4x
> Sketch the graphs of the following functions. f (x) = x3 + 3x + 1
> Find the slope of the graph of y = f (x) at the designated point. f (x) = 3x2 - 2x + 1, (1, 2)
> Sketch the graphs of the following functions. f (x) = -x3
> Sketch the graphs of the following functions. f (x) = x3 - 6x2 + 12x - 6
> Show that the function f (x) = -x3 + 2x2 - 6x + 3 is always decreasing.
> Show that the function f (x) = 1/3 x3 - 2x2 + 5x has no relative extreme points.
