Calculate the following iterated integrals. ∫01 (∫-11 1/3 y3 x dy) dx
> 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. f (x, y) = ½ x2 + 4xy + y3 + 8y2 + 3x + 2
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = x3 + 3x2 + 3y2 - 6y + 7
> Suppose that a topographic map is viewed as the graph of a certain function f (x, y). What are the level curves?
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = x2 + 3xy - y2 - x - 8y + 4
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = -x2 + 2y2 + 6x - 8y + 5
> The crime rate in a certain city can be approximated by a function f (x, y, z), where x is the unemployment rate, y is the number of social services available, and z is the size of the police force. Explain why ∂f/∂x > 0, ∂f/∂y < 0, and ∂f/∂z < 0.
> A dealer in a certain brand of electronic calculator finds that (within certain limits) the number of calculators she can sell per week is given by f (p, t) = - p + 6t - .02 pt, where p is the price of the calculator and t is the number of dollars spent
> Let f (x, y) = 2x3 + x2y - y2. Compute ∂2f/∂x2, ∂2f/∂y2 and ∂2f/∂x∂y (x, y) = (1, 2).
> Let f (x, y) = x5 - 2 x3y + ½ y4. Find ∂2f/∂x2, ∂2f/∂y2, ∂2f/∂x∂y, and ∂2f/∂y∂x.
> Let f (x, y, z) = (x + y) z. Evaluate ∂f/∂y at (x, y, z) = (2, 3, 4).
> Let f (x, y) = x3y + 8. Compute ∂f/∂x (1, 2) and ∂f/∂y (1, 2).
> Let f (x, y, λ) = xy + λ(5 - x - y). Find ∂f/∂x, ∂f/∂y and ∂f/∂λ.
> Let f (x, y, z) = x3 - yz2. Find ∂f/∂x, ∂f/∂y and ∂f/∂z.
> Find a function f (x, y) that has the curve y = 2/x2 as a level curve.
> Let g (x, y) = √(x2 + 2y2). Compute g (1, 1), g (0,-1), and g (a, b).
> Let f (x, y) = x/(x - 2y). Find ∂f/∂x and ∂f/∂y.
> Let f (x, y) = ex/y. Find ∂f/∂x and ∂f/∂y.
> Let f (x, y) = 3x – ½ y4 + 1. Find ∂f/∂x and ∂f/∂y.
> Let f (x, y) = 3x2 + xy + 5y2. Find ∂f/∂x and ∂f/∂y.
> What expression involving a partial derivative gives an approximation to f (a + h, b) - f (a, b)?
> Explain how to find a second partial derivative of a function of two variables.
> Explain how to find a first partial derivative of a function of two variables.
> Give an example of a level curve of a function of two variables.
> Give a formula for evaluating a double integral in terms of an iterated integral.
> Give a geometric interpretation for ∫R∫ f (x, y) dx dy, where f (x, y) ≥ 0.
> Find a function f (x, y) that has the line y = 3x - 4 as a level curve.
> What is the least-squares line approximation to a set of data points? How is the line determined?
> Outline how the method of Lagrange multipliers is used to solve an optimization problem.
> State the second-derivative test for functions of two variables.
> Explain how to find possible relative extreme points for a function of several variables.
> Give an example of a Cobb–Douglas production function. What is the marginal productivity of labor? Of capital?
> Interpret ∂f/∂y (2, 3) as a rate of change.
> Calculate the following iterated integrals. ∫-20 (∫-11 xexy dy) dx
> Calculate the following iterated integrals. ∫-11 (∫-11 xy dx) dy
> Calculate the following iterated integrals. ∫01 (∫01 ex+y dy) dx
> Calculate the volumes over the following regions R bounded above by the graph of f (x, y) = x2 + y2. R is the region bounded by the lines x = 0, x = 1 and the curves y = 0 and y = 3√x.
> Draw the level curve of the function f (x, y) = xy containing the point (1/2, 4).
> Calculate the volumes over the following regions R bounded above by the graph of f (x, y) = x2 + y2. R is the rectangle bounded by the lines x = 1, x = 3, y = 0, and y = 1.
> Let R be the rectangle consisting of all points (x, y) such that 0 ≤ x ≤ 2, 2 ≤ y ≤ 3. Calculate the following double integrals. Interpret each as a volume. ∫R∫ ey-x dx dy
> Let R be the rectangle consisting of all points (x, y) such that 0 ≤ x ≤ 2, 2 ≤ y ≤ 3. Calculate the following double integrals. Interpret each as a volume. ∫R∫ e-x-y dx dy
> Let R be the rectangle consisting of all points (x, y) such that 0 ≤ x ≤ 2, 2 ≤ y ≤ 3. Calculate the following double integrals. Interpret each as a volume. ∫R∫ (xy + y2) dx dy
> Let R be the rectangle consisting of all points (x, y) such that 0 ≤ x ≤ 2, 2 ≤ y ≤ 3. Calculate the following double integrals. Interpret each as a volume. ∫R∫ xy2 dx dy
> Calculate the following iterated integrals. ∫01 (∫0x ex+y dy) dx
> Calculate the following iterated integrals. ∫-11 (∫x2x (x + y) dy) dx
> Calculate the following iterated integrals. ∫03 (∫x2x y dy) dx
> Calculate the following iterated integrals. ∫14 (∫xx2 xy dy) dx
> Draw the level curve of the function f (x, y) = x - y containing the point (0, 0).
> Find the least-squares error E for the least-squares line fit to the five points in Fig. 6. Figure 6: y 100, 8 7 6 5 3 2 -1 y=-1.3x + 8,3 1 2 2 3 3 4 5 III
> Find the least-squares error E for the least-squares line fit to the four points in Fig. 5. Figure 5: y 8 6 1 2 3 4 y = 1.1 +3 x
> An ecologist wished to know whether certain species of aquatic insects have their ecological range limited by temperature. He collected the data in Table 8, relating the average daily temperature at different portions of a creek with the elevation (above
> Table 7 gives the number of visitors per year at Yosemite National Park. (a) Find the least-squares line for these data. (b) Estimate the number of visitors in 2017. Table 7: Yosemite National Park Visitors Year 2010 2011 2012 2013 2014 2015 Number
> Table 6 gives the U.S. minimum wage in dollars for certain years. Table 6: U.S. Federal Minimum Wage (a) Use the method of least squares to obtain the straight line that best fits these data. (b) Estimate the minimum wage for the year 2008. (c) If th
> Table 5 gives the number of students enrolled at the University of Illinois, at Urbana-Champaign (UIUC), for the fall semesters 2012–2015. Table 5: (a) Find the least-squares line for these data. (b) The university will build more st
> Table 4: U.S. Per Capita Health Care Expenditures (a) Find the least-squares line for these data. (b) Use the least-squares line to predict the per capita health care expenditures for the year 2016. (c) Use the least-squares line to predict when per ca
> Complete Table 3 and find the values of A and B for the straight line that provides the best least-squares fit to the data. Table 3: //
> Complete Table 2 and find the values of A and B for the straight line that provides the best least-squares fit to the data. Table 2: Table 2 X 1 2 3 4 Σχ = 7 6 4 3 Σy = xy Σxy = 12 Σχ -
> Use partial derivatives to obtain the formula for the best least-squares fit to the data points. (1, 5), (2, 7), (3, 6), (4, 10)
> Draw the level curves of heights 0, 1, and 2 for the function. f (x, y) = -x2 + 2y
> 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.
