A cone-shaped paper drinking cup is to be made to hold 27 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.
> Use Newton’s method to approximate the indicated root of the equation correct to six decimal places. The positive root of 3 sin x = x
> Use Newton’s method to approximate the indicated root of the equation correct to six decimal places. The negative root of ex = 4 - x2
> What is the maximum vertical distance between the line y = x + 2 and the parabola y = x2 for --1 ≤ x ≤ 2?
> (a) Explain how we know that the given equation must have a root in the given interval. (b) Use Newton’s method to approximate the root correct to six decimal places. -2x5 + 9x4 - 7x3 - 11x = 0, [3, 4]
> The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares?
> (a) Explain how we know that the given equation must have a root in the given interval. (b) Use Newton’s method to approximate the root correct to six decimal places. 3x4 - 8x3 + 2 = 0, [2, 3]
> Find two positive numbers whose product is 100 and whose sum is a minimum.
> Use Newton’s method to approximate the given number correct to eight decimal places. 8 500
> Find two numbers whose difference is 100 and whose product is a minimum.
> Use Newton’s method to approximate the given number correct to eight decimal places. 4 75
> Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a) Make a table of values, like the one at the right, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in
> Use Newton’s method with initial approximation x1 = 1 to find x2, the second approximation to the root of the equation x4 - x - 1 = 0. Explain how the method works by first graphing the function and its tangent line at (1, -1).
> Differentiate the function. F(s) = ln ln s
> Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line ,l, parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P on , so that the i
> Use Newton’s method with initial approximation x1 = -1 to find x2, the second approximation to the root of the equation x3 + x + 3 = 0. Explain how the method works by first graphing the function and its tangent line at (-1, 1).
> Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than over land because air generally rises over land and falls
> Use Newton’s method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places.) x7 + 4 = 0, x1 = -1
> Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W.
> Use Newton’s method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places.) 2x3 - 3x2 + 2 = 0 , x1 = -1
> A painting in an art gallery has height h and is hung so that its lower edge is a distance d above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer
> Where should the point P be chosen on the line segment AB so as to maximize the angle θ? В 2 Po 3 А 5
> Write the composite function in the form f ( g(x) ). [Identify the inner function u = g(x) and the outer function y = f (u).] Then find the derivative dy / dx. y − tan πx
> A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle θ. How should θ be chosen so that the gutter will carry the maximum amount of water? —
> An observer stands at a point P, one unit away from a track. Two runners start at the point S in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle of sig
> A steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?
> The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y? 12 - y 8
> Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when &Ic
> The graph shows the fuel consumption c of a car (measured in gallons per hour) as a function of the speed v of the car. At very low speeds the engine runs inefficiently, so initially c decreases as the speed increases. But at high speeds the fuel consump
> A point P needs to be located somewhere on the line AD so that the total length L of cables linking P to the points A, B. and C is minimized (see the figure). Express L as a function of x = |AP | and use the graphs of L and dL/dx to estimate the minimum
> The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be? b a a b
> Consider the situation in Exercise 51 if the cost of laying pipe under the river is considerably higher than the cost of laying pipe over land ($400,000ykm). You may suspect that in some instances, the minimum distance possible under the river should be
> A retailer has been selling 1200 tablet computers a week at $350 each. The marketing department estimates that an additional 80 tablets will sell each week for every $10 that the price is lowered. (a) Find the demand function. (b) What should the price b
> During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1, he found that the average decreased by two sales per day. (a) Find
> A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at $10, the average attendance had been 27,000. When ticket prices were lowered to $8, the average attendance rose to 33,000. (a) Find the demand function, assuming that
> (a) Show that if the profit P(x) is a maximum, then the marginal revenue equals the marginal cost. (b) If C(x) = 16,000 + 500x - 1.6x2 + 0.004x3 is the cost function and p(x) = 1700 - 7x is the demand function, find the production level that will maximiz
> (a) If C(x) is the cost of producing x units of a commodity, then the average cost per unit is c(x) = C(x)/x. Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b) If C(x) = 16,000 + 200x + 4x3/2, in dollars, fin
> What is the smallest possible area of the triangle that is cut off by the first quadrant and whose hypotenuse is tangent to the parabola y = 4 - x2 at some point?
> What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y = 3/x at some point?
> At which points on the curve y = 1 + 40x3 - 3x5 does the tangent line have the largest slope?
> Let a and b be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point (a, b).
> Find an equation of the line through the point (3, 5) that cuts off the least area from the first quadrant.
> The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 f
> Suppose the refinery in Exercise 51 is located 1 km north of the river. Where should P be located? Exercise 51: An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to stor
> An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is $400,000/k
> A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time (see the figure). She can walk at the rate of 4 mi/h and row a boat
> Solve the problem in Example 4 if the river is 5 km wide and point B is only 5 km downstream from A. Example 4: A man launches his boat from point A on a bank of a straight river, 3 km wide, and wants to reach point B, 8 km downstream on the opposite ba
> A boat leaves a dock at 2:00 pm and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 3:00 pm. At what time were the two boats closest together?
> Differentiate the function. F(t) = (ln t)2 sin t
> A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h = 1/3 H.
> A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup. A B R C
> A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
> If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of 32 inches, what diameter pizza will reward you with the largest slice?
> Answer Exercise 37 if one piece is bent into a square and the other into a circle. Exercise 37: A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be
> A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) A minimum?
> A poster is to have an area of 180 in2 with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions will give the largest printed area?
> The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm2, find the dimensions of the poster with the smallest area.
> A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 1.1.62.) If the perimeter of the window is 30 ft, find the dimensions of the window so that
> A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible surface area of such a cylinder.
> A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.
> A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder.
> If the two equal sides of an isosceles triangle have length a, find the length of the third side that maximizes the area of the triangle.
> Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r.
> Find the area of the largest trapezoid that can be inscribed in a circle of radius 1 and whose base is a diameter of the circle.
> Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle.
> Differentiate the function. g(x) = ln(xe-2x)
> Find the area of the largest rectangle that can be inscribed in the ellipse x2/a2 + y2/b2 = 1.
> Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.
> Find, correct to two decimal places, the coordinates of the point on the curve y = sin x that is closest to the point (4, 2).
> Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from the point (1, 0).
> Find the point on the line y = 2x + 3 that is closest to the origin.
> (a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square. (b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square.
> If the farmer in Exercise 18 wants to enclose 8000 square feet of land, what dimensions will minimize the cost of the fence? Exercise 18: A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed
> A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fe
> Do Exercise 16 assuming the container has a lid that is made from the same material as the sides. Exercise 16: A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the bas
> A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for t
> If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
> A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.
> A farmer wants to fence in an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?
> Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a)
> Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw s
> Find the dimensions of a rectangle with area 1000 m2 whose perimeter is as small as possible.
> Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.
> Find expressions for the first five derivatives of f (x) = x2ex. Do you see a pattern in these expressions? Guess a formula for f(n) (x) and prove it using mathematical induction.
> What is the minimum vertical distance between the parabolas y = x2 + 1 and y = x - x2?
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = 6 sin x
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f (x) = x6 - 5
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f (x) = 22x6 +
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f (x) = x5 - 5
> (a) Investigate the family of polynomials given by the equation f (x) = 2x3 + cx2 - 2x. For what values of c does the curve have maximum and minimum points? (b) Show that the minimum and maximum points of every curve in the family lie on the curve y = x
> (a) Investigate the family of polynomials given by the equation f (x) = cx4 - 2x2 + 1. For what values of c does the curve have minimum points? (b) Show that the minimum and maximum points of every curve in the family lie on the parabola y = 1 - x2. Illu
> Investigate the family of curves given by the equation f (x) = x4 + cx2 + x. Start by determining the transitional value of c at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. Ther
> Differentiate the function. f(x) = log 10 (1 + cos x)
> Investigate the family of curves given by f (x) = xe-cx, where c is a real number. Start by computing the limits as x ( ±∞. Identify any transitional values of c where the basic shape changes. What happens to the maximum or minimum points and inflection
> The family of functions f (t) = C(e-at – e-bt), where a, b, and C are positive numbers and b . a, has been used to model the concentration of a drug injected into the bloodstream at time t = 0. Graph several members of this family. What do they have in c
> Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should als
> Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should als
> Find dy/dx by implicit differentiation. x2 - 4xy + y2 = 4
> In Example 4 we considered a member of the family of functions f (x) = sin(x + sin cx) that occur in FM synthesis. Here we investigate the function with c = 3. Start by graphing f in the viewing rectangle [0, π] by [-1.2, 1.2]. How many local maximum po
> A company operates 16 oil wells in a designated area. Each pump, on average, extracts 240 barrels of oil daily. The company can add more wells but every added well reduces the average daily ouput of each of the wells by 8 barrels. How many wells should t
> Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral.
