Open Box I: A rectangular box with a square base and no top (Figure 8-3j) is to be

Differentiate: y = (3x + 5)1

Integrate: (x + 6)1 dx

Differentiate: y = x2/3

Integrate: x2/3 dx

Integrate: x2 dx

Integrate: x0 dx

cot x dx = ?

Sketch the graph: y = x1/3.

Sketch the graph of for the cubic function in Figure 8-3e. Figure 8-3e

(d/dx)(cos1 x) = ? A. sin1 x B. sin1 x C. (cos x)2

Divided Stock Pen Problem: Suppose you are building a rectangular stock pen (Figure 8-3f) using 600 ft of fencing. You will use part of this fencing to build a fence across the middle of the rectangle (see the diagram). Find the length and width of the rectangle that give the maximum total area. Justify your answer. Figure 8-3f

Motel Problem: A six-room motel is to be built with the floor plan shown in Figure 8-3g. Each room will have 350 ft2 of floor space. Figure 8-3g a. What dimensions should be used for the rooms in order to minimize the total length of the walls? Justify your answer. b. How would your answer to part a change if the motel had ten rooms? If it had just three rooms?

Two-Field Problem: Ella Mentary has 600 ft of fencing to enclose two fields. One field will be a rectangle twice as long as it is wide, and the other will be a square (Figure 8-3h). The square field must contain at least 100 ft2. The rectangular one must contain at least 800 ft2. Figure 8-3h a. If x is the width of the rectangular field, what is the domain of x? b. Plot the graph of the total area contained in the two fields as a function of x. c. What is the greatest area that can be contained in the two fields? Justify your answer

Two-Corral Problem: You work on Bill Spenders Ranch. Bill tells you to build a circular fence around the lake and to use the remainder of your 1000 yards of fencing to build a square corral (Figure 8-3i). To keep the fence out of the water, the diameter of the circular enclosure must be at least 50 yards. Figure 8-3i a. If you must use all 1000 yards of fencing, how can you build the fences to enclose the minimum total area? Justify your answer. b. What would you tell Bill if he asked you to build the fences to enclose the maximum total area?

Two-Corral Problem: You work on Bill Spenders Ranch. Bill tells you to build a circular fence around the lake and to use the remainder of your 1000 yards of fencing to build a square corral (Figure 8-3i). To keep the fence out of the water, the diameter of the circular enclosure must be at least 50 yards. Figure 8-3i a. If you must use all 1000 yards of fencing, how can you build the fences to enclose the minimum total area? Justify your answer. b. What would you tell Bill if he asked you to build the fences to enclose the maximum total area?

Open Box I: A rectangular box with a square base and no top (Figure 8-3j) is to be constructed using a total of 120 cm2 of cardboard.a. Find the dimensions of the box of maximum volume. b. Make a conjecture about the depth of the maximum-volume box in relation to the base length if the box has a fixed surface area.

Open Box III: You are building a glass fish tank that will hold 72 ft3 of water. You want its base and sides to be rectangular and the top, of course, to be open. You want to construct thetank so that its width is 5 ft but the length and depth are variable. Building materials for the tank cost $10 per square foot for the base and $5 per square foot for the sides. What is the cost of the least expensive tank? Justify your answer.

Open Box IV (Project): Figure 8-3l shows an open-top box with rectangular base x-by-y units and rectangular sides. The depth of the box is z units. Figure 8-3l a. Hold the depth constant. Show that the maximum volume of the box for a given amount of material occurs when x = y. b. Set y = x, but let both vary as the depth varies. Find the values of x and z that give the maximum volume for a given amount of material. c. In what ratio are the values of x and z for the maximum-volume box in part b? Is the maximum-volume box tall and narrow or short and wide? Based on geometry, why is your answer reasonable?

Shortest-Distance Problem: In Figure 8-3m, what point on the graph of y = ex is closest to the origin? Justify your answer. Figure 8-3m

Track and Field Problem: A track of perimeter 400 m is to be laid out on the practice field (Figure 8-3n). Each semicircular end must have a radius of at least 20 m, and each straight section must be at least 100 m. How should the track be laid out so that it encompasses the least area? Justify your answer. Figure 8-3n

Ladder Problem: A ladder is to reach over a fence 8 ft high to a wall that is 1 ft behind the fence (Figure 8-3o). What is the length of the shortest ladder that you can use? Justify your answer. Figure 8-3o

Ladder in the Hall Problem: A nonfolding ladder is to be taken around a corner where two hallways intersect at right angles (Figure 8-3p). One hall is 7 ft wide, and the other is 5 ft wide. What is the maximum length the ladder can be so that it will pass around such a corner, given that you must carry the ladder parallel to the floor? Figure 8-3p

Rotated Rectangle Problem: A rectangle of perimeter 1200 mm is rotated in space using one of its legs as the axis (Figure 8-3q). The volume enclosed by the resulting cylinder depends on the proportions of the rectangle. Find the dimensions of the rectangle that maximize the cylinders volume.

Rotated Rectangle Generalization Problem: The rectangle of maximum area for a given perimeter P is a square. Does rotating a square about one of its sides (as in Problem 13) produce the maximum-volume cylinder? If so, prove it. If not, what proportions do produce the maximum-volume cylinder?

Tin Can Problem: A popular size of tin can with normal proportions has diameter 7.3 cm and height 10.6 cm (Figure 8-3r). Figure 8-3r a. What is its volume? b. The volume is to be kept the same, but the proportions are to be changed. Write an equation expressing total surface of the can (lateral surface plus two ends) as a function of radius and height. Transform the equation so that the volume is in terms of radius alone. c. Find the radius and height of the can that minimize its surface area. Is the can tall and narrow or short and wide? What is the ratio of diameter to height? Justify your answers. d. Does the normal can use close to the minimum amount of metal? What percentage of the metal in the normal can could you save by using cans with the minimum dimensions? e. If the United States uses 20 million of these cans a day and the metal in a normal can is worth $0.06, how much money could be saved in a year by using minimum-area cans?

Tin Can Generalization Project: The tin can of minimum cost in Problem 15 is not necessarily the one with minimum surface area. In this problem you will investigate the effects of wasted metal in the manufacturing process and of overlapping metal in the seams. a. Assume that the metal for the ends of the can in Problem 15 costs k times as much per square centimeter as the metal for the cylindrical walls. Find the value of k that makes the minimum-cost can have the proportions of the normal can. Is it reasonable in the real world for the ends to cost this much more (or less) per square centimeter than the walls? Explain. b. Assume that the ends of the normal tin can in Problem 15 are cut from squares and that the remaining metal from the squares is wasted. What value of k in part a minimizes the cost of the normal can under this assumption? Is the can that uses the minimum amount of metal under this assumption closer to the proportions of the normal can or farther away? c. The specifications require that the ends of the can be made from metal disks that overhang by 0.6 cm all the way around. This provides enough overlap to fabricate the constructed cans top and bottom joints. There must also be an extra 0.5 cm of metal in the cans circumference for the overlap in the vertical seam. How do these specifications affect the dimensions of the minimum-area can in Problem 15?

Cup Problem: You have been hired by the Yankee Cup Company. They currently make a cylindrical paper cup of diameter 5 cm and height 7 cm. Your job is to find ways to save paper by making cups that hold the same amount of liquid but have different proportions. a. Find the dimensions of a same-volume cylindrical cup that uses a minimum amount of paper. b. What is the ratio of diameter to height for the minimum-area cup? c. Paper costs $2.00 per square meter. Yankee makes 300 million of this type of cup per year. Write a proposal to your boss telling her whether you think it would be worthwhile to change the dimensions of Yankees cup to those of the minimum cup. Be sure to show that the area of the proposed cup really is a minimum. d. Show that in general if a cup of given volume V has minimum total surface area, then the radius is equal to the height

Duct Problem: A duct made of sheet metal connects one rectangular opening in an air-conditioning system to another rectangular opening (Figure 8-3s). The rectangle on the left is at x = 0 in. and the one on the right is at x = 100 in. Cross sections perpendicular to the x-axis are rectangles of width z = 30 + 0.2x and y = 40 0.2x. Figure 8-3s a. Find the areas of the two end rectangles. b. What is the cross-sectional area of the duct at x = 80? c. What x-values give the maximum and the minimum cross-sectional areas?

Rectangle in Sinusoid Problem: A rectangle is inscribed in the region bounded by one arch of the graph of y = cos x and the x-axis (Figure 8-3t). What value of x gives the maximum area? What is the maximum area? Figure 8-3t

Building Problem: Tom OShea plans to build a new hardware store. He buys a rectangular lot hat is 50 ft by 200 ft, the 50-ft dimension being along the street. The store will have 4000 ft2 of floor space. Construction costs $100 per linear foot for the part of the store along the street but only $80 per linear foot for the parts along the sides and back. What dimensions of the store will minimize construction costs? Justify your answer.

Triangle under Cotangent Problem: A right triangle has a vertex at the origin and a leg along the x-axis. Its other vertex touches the graph of y = cot x, as shown in Figure 8-3u. Figure 8-3u a. As the right angle approaches the origin, the height of the triangle approaches infinity, and the base length approaches zero. Find the limit of the area as the right angle approaches the origin. b. What is the maximum area the triangle can have if the domain is the half-open interval (0, /2]? Justify your answer.

Triangle under Exponential Curve Problem: A right triangle has one leg on the x-axis. The vertex at the right end of that leg is at the point (3, 0). The other vertex touches the graph of y = ex. The entire triangle is to lie in the first quadrant. Find the maximum area of this triangle. Justify your answer.

Rectangle in Parabola Problem: A rectangle is inscribed in the region bounded by the x-axis and the parabola y = 9 x2 (Figure 8-3v). a. Find the length and width of the rectangle of greatest area. Justify your answer. b. Find the length and width of the rectangle of greatest perimeter. Justify your answer. c. Does the rectangle of greatest area have the greatest perimeter?

Cylinder in Paraboloid Problem: In Figure 8-3w, the parabola y = 9 x2 is rotated about the y-axis to form a paraboloid. A cylinder is coaxially inscribed in the paraboloid. Figure 8-3w a. Find the radius and height of the cylinder of maximum volume. Justify your answer. b. Find the radius and height of the cylinder of maximum lateral area. c. Find the radius and height of the cylinder of maximum total area (including the ends). Justify your answer. d. Does the maximum-volume cylinder have the same dimensions as either of the maximum-area cylinders in part b or c? e. Does rotating the maximum-area rectangle in Problem 23a produce the maximumvolume cylinder in this problem? f. If the cylinder of maximum volume is inscribed in the paraboloid formed by rotating the parabola y = a2 x2 about the y-axis, does the ratio (cylinder radius): (paraboloid radius) depend in any way on the length of the paraboloid? (That is, does it depend on the value of the constant a?) Justify your answer.

Cylinder in Sphere Problem: A cylinder is to be inscribed in a sphere of radius 10 cm (Figure 8-3x). The bottom and top bases of the cylinder are to touch the surface of the sphere. The volume of the cylinder will depend on whether it is tall and narrow or short and wide. Figure 8-3x a. Let (x, y) be the coordinates of a point on the circle, as shown. Write an equation for the volume of the cylinder in terms of x and y. b. What radius and height of the cylinder will give it the maximum volume? What is this volume? Justify your answer. c. How are the radius and height of the maximum-volume cylinder related to each other? How is the maximum cylinder volume related to the volume of the sphere?

Conical Nose Cone Problem: In the design of a missile nose cone, it is important to minimize the surface area exposed to the atmosphere. For aerodynamic reasons, the cone must be long and slender. Suppose that a right-circular nose cone is to contain a volume of 5 ft3. Find the radius and height of the nose cone of minimum lateral surface area, subject to the restriction that the height must be at least twice the radius of the base. (The differentiation will be easier if you minimize the square of the area.)

Cylinder in Cone Problem: In Figure 8-3y, a cone of height 7 cm and base radius 5 cm has a cylinder inscribed in it, with the base of the cylinder contained in the base of the cone.Figure 8-3y a. Find the radius of the cylinder of maximum lateral area (sides only). b. Find the radius of the cylinder of maximum total area (including the top and bottom bases). Justify your answer.

General Cylinder in Cone Problem: A given cone has a cylinder inscribed in it, with its base contained in the base of the cone. a. How should the radius of the cone and cylinder be related for the lateral surface of the cylinder to maximize its area? Justify your answer. b. Find the radius of the cylinder that gives the maximum total area. c. If the cone is short and fat, the maximum total area occurs where the height of the cylinder drops to zero and all the material is used in the top and bottom bases. How must the radius and height of the cone be related for this phenomenon to happen?

Elliptical Nose Cone Problem: The nose of a new cargo plane is to be a half-ellipsoid of diameter 8 m and length 9 m (Figure 8-3z). The nose swings open so that a cylindrical cargo container can be placed inside. What is the greatest volume this container could hold? Figure 8-3z What are the radius and height of this largest container? Justify your answer.

Submarine Pressure Hull Project: According to a new design, the forward end of a submarine hull is to be constructed in the shape of a paraboloid 16 m long and 8 m in diameter (Figure 8-3aa). Since this is a doubly curved surface, it is hard to bend thick steel plates into this shape. So the paraboloid is to be made of relatively thin steel, and the pressure hull built inside as a cylinder (a singly curved surface). A frustum of a cone is also a singly curved surface, which would be about as easy to make and which might be able to contain more volume (Figure 8-3bb). How much more volume could be contained in the maximum-volume frustum than in the maximum-volume cylinder? Some things you will need to find are the equation of this particular parabola and the equation for the volume of a frustum of a cone. Figure 8-3aa Figure 8-3bb

Local Maximum Property Problem: The definition of local maximum is as follows: f(c) is a local maximum of f on the interval (a, b) if and only if f(c) f(x) for all values of x in (a, b). Figure 8-3cc illustrates this definition. Figure 8-3cc a. Prove that if f(c) is a local maximum of f on (a, b) and f is differentiable at x = c in (a, b), then c) = 0. (Consider the sign of the difference quotient when x is to the left and to the right of c, then take left and right limits.) b. Explain why the property in part a would be false without the hypothesis that f is differentiable at x = c. c. Explain why the converse of the property in part a is false.

Corral with Short Wall Project: Millie Watt is installing an electric fence around a rectangular corral along a wall. Part or all of the wall forms all or part of one side of the corral. The total length of fence (excluding the wall) is to be 1000 ft. Find the maximum area that she can enclose if a. The wall is 600 ft long (Figure 8-3dd, left). b. The wall is 400 ft long (Figure 8-3dd, center). c. The wall is 200 ft long (Figure 8-3dd, right). Figure 8-3dd

Journal Problem: Update your journal with what youve learned in this section. Include such things as The one most important thing youve learned since your last entry The critical features of the graph of a function that its two derivatives tell you How you solve real-world problems that involve a maximum or minimum value Any techniques or ideas about extreme-value problems that are still unclear to you