A company manufactures and sells two products, I and II, that sell for $10 and $9 per unit, respectively. The cost of producing x units of product I and y units of product II is 400 + 2x + 3y + .01(3x2 + xy + 3y2). Find the values of x and y that maximize the company’s profits. [Note: Profit = (revenue) - (cost).
> Use partial derivatives to obtain the formula for the best least-squares fit to the data points. (1, 9), (2, 8), (3, 6), (4, 3)
> Use partial derivatives to obtain the formula for the best least-squares fit to the data points. (1, 8), (2, 4), (4, 3)
> Use partial derivatives to obtain the formula for the best least-squares fit to the data points. (1, 2), (2, 5), (3, 11)
> Find the formula (of the type in Check Your Understanding Problem 1) that gives the least-squares error for the points (8, 4), (9, 2), and (10, 3).
> Find the formula (of the type in Check Your Understanding Problem 1) that gives the least-squares error for the points (2, 6), (5, 10), and (9, 15).
> Solve the exercise by the method of Lagrange multipliers. Maximize x2 + xy - 3y2, subject to the constraint 2 - x - 2y = 0.
> Solve the exercise by the method of Lagrange multipliers. Maximize x2 - y2, subject to the constraint 2x + y - 3 = 0.
> Solve the exercise by the method of Lagrange multipliers. Minimize x2 + 3y2 + 10, subject to the constraint 8 - x - y = 0.
> By applying the result in Exercise 25 to the production function f (x, y) = kxayb, show that, for the values of x, y that maximize production, we have (This tells us that the ratio of capital to labor does not depend on the amount of money available, n
> Let f (x, y) be any production function where x represents labor (costing $a per unit) and y represents capital (costing $b per unit). Assuming that $c is available, show that, at the values of x, y that maximize production, ax af ay b
> Draw the level curves of heights 0, 1, and 2 for the function. f (x, y) = 2x - y
> A shelter for use at the beach has a back, two sides, and a top made of canvas. [See Fig. 4(b).] Find the dimensions that maximize the volume and require 96 square feet of canvas. Figure 4: (a) y 20 (b) x 22
> Find the dimensions of an open rectangular glass tank of volume 32 cubic feet for which the amount of material needed to construct the tank is minimized. [See Fig. 4(a).] Figure 4: (a) y 20 (b) x 22
> Use Lagrange multipliers to find the three positive numbers whose sum is 15 and whose product is as large as possible.
> The material for a closed rectangular box costs $2 per square foot for the top and $1 per square foot for the sides and bottom. Using Lagrange multipliers, find the dimensions for which the volume of the box is 12 cubic feet and the cost of the materials
> Find the values of x, y, z that minimize x2 + y2 + z2 - 3x - 5y - z, subject to the constraint 20 - 2x - y - z = 0.
> Find the values of x, y, z that maximize 3x + 5y + z - x2 - y2 - z2, subject to the constraint 6 - x - y - z = 0.
> Find the values of x, y, and z that maximize xy + 3xz + 3yz subject to the constraint 9 - xyz = 0.
> Find the values of x, y, and z that maximize xyz subject to the constraint 36 - x - 6y - 3z = 0.
> Consider the firm of Example 2, Section 7.3, that sells its goods in two countries. Suppose that the firm must set the same price in each country. That is, 97 - (x/10) = 83 - (y/20). Find the values of x and y that maximize profits under this new restric
> The production function for a firm is f (x, y) = 64 x3/4 y1/4, where x and y are the number of units of labor and capital utilized. Suppose that labor costs $96 per unit and capital costs $162 per unit and that the firm decides to produce 3456 units of g
> Let f (r, y, x) be the real estate tax function of Exercise 13. (a) Determine the real estate tax on a property valued at $100,000 with a homeowner’s exemption of $5000, assuming a tax rate of $2.20 per hundred dollars of net assessed value. (b) Determi
> A firm makes x units of product A and y units of product B and has a production possibilities curve given by the equation 4x2 + 25y2 = 50,000 for x ≥ 0, y ≥ 0. (See Exercise 23.) Suppose profits are $2 per unit for product A and $10 per unit for product
> Suppose that a firm makes two products, A and B, that use the same raw materials. Given a fixed amount of raw materials and a fixed amount of labor, the firm must decide how much of its resources should be allocated to the production of A and how much to
> Distance from a point to a parabola Find the point on the parabola y = x2 that has minimal distance from the point (16, ½). [See Fig. 2(b).] [Suggestion: If d denotes the distance from (x, y) to (16, ½), then d2 = (x - 16)2 + (y – ½)2. If d2 is minimized
> Find the dimensions of the rectangle of maximum area that can be inscribed in the unit circle. [See Fig. 2(a).] Figure 2: y = x² fi fi (a) (x, y) (b) (x, y) 1, (16,4)
> The amount of space required by a particular firm is f (x, y) = 1000 √(6x2 + y2), where x and y are, respectively, the number of units of labor and capital utilized. Suppose that labor costs $480 per unit and capital costs $40 per unit and that the firm
> Three hundred square inches of material are available to construct an open rectangular box with a square base. Find the dimensions of the box that maximize the volume.
> Four hundred eighty dollars are available to fence in a rectangular garden. The fencing for the north and south sides of the garden costs $10 per foot, and the fencing for the east and west sides costs $15 per foot. Find the dimensions of the largest pos
> Find the two positive numbers whose product is 25 and whose sum is as small as possible.
> Solve the exercise by the method of Lagrange multipliers. Find the values of x, y, and z that minimize xy + xz - 2yz subject to the constraint x + y + z = 2.
> Solve the exercise by the method of Lagrange multipliers. Find the values of x, y and z that minimize xy + xz - yz subject to the constraint x + y + z = 1.
> The value of residential property for tax purposes is usually much lower than its actual market value. If y is the market value, the assessed value for real estate taxes might be only 40% of y. Suppose that the property tax, T, in a community is given by
> Solve the exercise by the method of Lagrange multipliers. Find the values of x and y that maximize xy subject to the constraint x2 - y = 3.
> Solve the exercise by the method of Lagrange multipliers. Find the values of x and y that minimize f (x, y) = x - xy + 2y2 subject to the constraint x - y + 1 = 0.
> Solve the exercise by the method of Lagrange multipliers. Find the values of x and y that minimize 3x2 - 2xy + x - 3y + 1 subject to the constraint x - 3y = 1.
> Solve the exercise by the method of Lagrange multipliers. Find the values of x and y that minimize 18x2 + 12xy + 4y2 + 6x - 4y + 5 subject to the constraint 3x + 2y - 1 = 0.
> Solve the exercise by the method of Lagrange multipliers. Find the values of x and y that minimize 2x2 - 2xy + y2 - 2x + 1 subject to the constraint x - y = 3.
> Solve the exercise by the method of Lagrange multipliers. Find the values of x and y that minimize 2x2 + xy + y2 – y subject to the constraint x + y = 0.
> Solve the exercise by the method of Lagrange multipliers. Find the values of x and y that minimize x2 - 2xy + 2y2 subject to the constraint 2x - y + 5 = 0.
> Solve the exercise by the method of Lagrange multipliers. Find the values of x and y that minimize xy + y2 - x – 1 subject to the constraint x - 2y = 0.
> Solve the exercise by the method of Lagrange multipliers. Find the values of x, y that minimize x2 + xy + y2 - 2x - 5y, subject to the constraint 1 - x + y = 0.
> Solve the exercise by the method of Lagrange multipliers. Find the values of x, y that maximize -2x2 - 2xy – 3/2 y2 + x + 2y, subject to the constraint x + y - 5/2 = 0.
> Solve the exercise by the method of Lagrange multipliers. Minimize ½ x2 - 3xy + y2 + 12, subject to the constraint 3x - y - 1 = 0.
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = ½ x2 + y2 - 3x + 2y - 5
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = x2 - 3y2 + 4x + 6y + 8
> A company manufactures and sells two competing products, I and II, that cost $pI and $pII per unit, respectively, to produce. Let R(x, y) be the revenue from marketing x units of product I and y units of product II. Show that if the company’s profit is m
> A company manufactures and sells two products, I and II, that sell for $p1 and $p2 per unit, respectively. Let C(x, y) be the cost of producing x units of product I and y units of product II. Show that if the company’s profit is maximiz
> A monopolist manufactures and sells two competing products, I and II, that cost $30 and $20 per unit, respectively, to produce. The revenue from marketing x units of product I and y units of product II is 98x + 112y - .04xy - .1x2 - .2y2. Find the values
> Find the dimensions of the rectangular box of least surface area that has a volume of 1000 cubic inches.
> U.S. postal rules require that the length plus the girth of a package cannot exceed 84 inches. Find the dimensions of the rectangular package of greatest volume that can be mailed. [Note: From Fig. 5 we see that 84 = (length) + (girth) = l + (2x + 2y).]
> Find the possible values of x, y, z at which f (x, y, z) = 5 + 8x - 4y + x2 + y2 + z2 assumes its minimum value.
> The present value of A dollars to be paid t years in the future (assuming a 5% continuous interest rate) is P(A, t) = Ae-0.05t. Find and interpret P(100, 13.8).
> Find the possible values of x, y, z at which f (x, y, z) = 2x2 + 3y2 + z2 - 2x - y – z assumes its minimum value.
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Let f (x, y) = 10x2/5y3/5. Show that f (3a, 3b) = 3f (a, b).
> Let f (x, y) = x2 - 3xy - y2. Compute f (5, 0), f (5,-2), and f (a, b).
> The velocity of a skydiver at time t seconds is υ(t) = 45 - 45e-0.2t meters per second. Find the distance traveled by the skydiver the first 9 seconds.
> The velocity at time t seconds of a ball thrown up into the air is υ(t) = -32t + 75 feet per second. (a) Compute the displacement of the ball during the time interval 1 ≤ t ≤ 3. (b) Is the position of the ball at time t = 3 higher than its position at ti
> The velocity at time t seconds of a ball thrown up into the air is υ(t) = -32t + 75 feet per second. (a) Find the displacement of the ball during the time interval 0 ≤ t ≤ 3. (b) Given that the initial position of the ball is s(0) = 6 feet, use (a) to de
> A rock is dropped from the top of a 400-foot cliff. Its velocity at time t seconds is υ(t) = -32t feet per second. Find the displacement of the rock during the time interval 2 ≤ t ≤ 4.
> Refer to Fig. 7 and evaluate -1∫2 f (t)dt. Figure 7: 0 f(t) y=t-1 S || =²-1 2
> Refer to Fig. 6 and evaluate -1∫1 f (t)dt. Figure 6: I 7-1=k 0 I- 7+T=; DL
> Refer to Fig. 5 and evaluate 0∫3 f (x)dx. Figure 5: y y¹ 0 y=1-x² y = (1-x)(x − 3) 3 8 -
> Refer to Fig. 4 and evaluate 0∫2 f (x)dx. Figure 4: 2 4 0 ม y 1 บ y=r T
> Use formula (8) to help you answer the question. Given f (t) = -12t – 1/et, compute f (3) - f (0). Formula (8): [ F'(x)dx= F(b) - F(a).
> Use formula (8) to help you answer the question. Given f ‘(t) = -.5t + e-2t, compute f (1) - f (-1). Formula (8): [ F'(x)dx= F(b) - F(a).
> Determine the following: ∫7 dx
> Use formula (8) to help you answer the question. Given f ‘(x) = 73, compute f (4) - f (2). Formula (8): [ F'(x)dx= F(b) - F(a).
> Use formula (8) to help you answer the question. Given f ‘(x) = -2x + 3, compute f (3) - f (1). Formula (8): [ F'(x)dx= F(b) - F(a).
> Combine the integrals into one integral, then evaluate the integral. 0∫1 (7x + 4) dx + 1∫2 (7x + 5) dx
> Combine the integrals into one integral, then evaluate the integral. -1∫0 (x3 + x2) dx + 0∫1 (x3 + x2) dx
> Combine the integrals into one integral, then evaluate the integral. 0∫1 (4x - 2) dx + 3 0∫1 (x - 1) dx
> Combine the integrals into one integral, then evaluate the integral. 2 1∫2 (3x + 1/2 x2 - x3) dx + 3 1∫2 (x2 - 2x + 7) dx
> Given -0.5∫3 f (x)dx = 0 and -0.5∫3 (2g(x) + f (x))dx = -4, find -0.5∫3 g(x)dx.
> Given 1∫3f (x)dx = 3 and 1∫3 g(x)dx = -1, find 1∫3 (2f (x) - 3g(x)) dx.
> Given -1∫1 f (x)dx = 0 and -1∫10 f (x)dx = 4, find 1∫10 f (x)dx.
> Given 0∫1 f (x)dx = 3.5 and 1∫4 f (x)dx = 5, find 0∫4 f (x)dx.
> Determine the following: ∫x/3 dx
> Evaluate the given integral. 0∫ln 2 (ex + e-x)/2 dx
> Evaluate the given integral. 0∫1 (ex + e0.5x)/e2x dx
> Evaluate the given integral. -2∫-1 (1 + x)/x dx
> Evaluate the given integral. 1∫2 2/x dx
> Evaluate the given integral. -2∫2 2/e2t dt
> Evaluate the given integral. -1∫0 (3e3t + t) dt
> Evaluate the given integral. 1∫4 (x2 - √x)/x dx
> Evaluate the given integral. 1∫2 (5 - 2x3)/x6 dx
> Evaluate the given integral. 1∫8 (-x + 3√x) dx
> Evaluate the given integral. 1∫2 -3/x2 dx
> Determine the following: ∫4x3 dx
> Evaluate the given integral. 1∫9 1/√x dx
> Plot the graph of the solution of the differential equation y = e-x2, y(0) = 0. Observe that the graph approaches the value √π/2 ≈ .9 as x increases.
