Let f (x, y) = x/(x - 2y). Find ∂f/∂x and ∂f/∂y.
> Construct angles with the following radian measure. 3π/2, 3π/4, 5π
> Give the radian measure of each angle described.
> Give the radian measure of each angle described. 0
> Match the graph of the function to the systems of level curves shown in Figs. 8(a)–(d). Figure 8: x 2 z = x² - x² - y² 2 e (b) 2₂x0 몸 라 (d) 2.
> Give the radian measure of each angle described.
> Give the radian measure of each angle described.
> Give the radian measure of each angle described.
> Give the radian measure of each angle described.
> Give the radian measure of each angle described.
> Give the radian measure of each angle described.
> Let f (x, y, λ) = xy + λ (5 - x - y). Find f (1, 2, 3).
> If A dollars are deposited in a bank at a 6% continuous interest rate, the amount in the account after t years is f (A, t) = Ae0.06t. Find and interpret f (10, 11.5).
> Let f (x, y, z) = x2ey/z. Compute f (-1, 0, 1), f (1, 3, 3), and f (5, -2, 2).
> Let f (x, y) = x√y/(1 + x). Compute f (2, 9), f (5, 1), and f (0, 0).
> Match the graph of the function to the systems of level curves shown in Figs. 8(a)–(d). Figure 8: 2 -4 1 + x² + 2y² 2 e (b) 2₂x0 몸 라 (d) 2.
> The present value of y dollars after x years at 15% continuous interest is f (x, y) = ye-0.15x. Sketch some sample level curves. (Economists call this collection of level curves a discount system.)
> Let R be the rectangle consisting of all points (x, y), such that 0 ≤ x ≤ 4, 1 ≤ y ≤ 3, and calculate the double integral. ∫R∫ 5 dx dy
> Let R be the rectangle consisting of all points (x, y), such that 0 ≤ x ≤ 4, 1 ≤ y ≤ 3, and calculate the double integral. ∫R∫ (2x + 3y) dx dy
> Calculate the iterated integral. ∫05 (∫14 (2xy4 + 3) dy) dx
> Calculate the iterated integral. ∫01 (∫04 (x√y + y) dy) dx
> Find the straight line that best fits the following data points, where “best” is meant in the sense of least squares. (0, 1), (1, -1), (2, -3), (3, -5)
> Find the straight line that best fits the following data points, where “best” is meant in the sense of least squares. (1, 1), (3, 4), (5, 7)
> Find the straight line that best fits the following data points, where “best” is meant in the sense of least squares. (1, 1), (2, 3), (3, 6)
> The solution to Exercise 29 is x = 10, y = 20, λ = 10. If 1 additional foot of fencing becomes available, compute the new optimal dimensions and the new area. Show that the increase in area (compared with the area in Exercise 29) is approxim
> A person wants to plant a rectangular garden along one side of a house and put a fence on the other three sides. (See Fig. 1.) Using the method of Lagrange multipliers, find the dimensions of the garden of greatest area that can be enclosed with 40 feet
> A certain production process uses units of labor and capital. If the quantities of these commodities are x and y, respectively, the total cost is 100x + 200y dollars. Draw the level curves of height 600, 800, and 1000 for this function. Explain the signi
> Find the dimensions of a rectangular box of volume 1000 cubic inches for which the sum of the dimensions is minimized.
> Use the method of Lagrange multipliers to: Find the values of x, y, z that minimize 3x2 + 2y2 + z2 + 4x + y + 3z, subject to the constraint 4 - x - y - z = 0.
> Use the method of Lagrange multipliers to: Find the values of x, y that minimize -x2 - 3xy – ½ y2 + y + 10, subject to the constraint 10 - x - y = 0.
> Use the method of Lagrange multipliers to: Maximize 3x2 + 2xy - y2, subject to the constraint 5 - 2x - y = 0.
> Find the values of x, y, z at which f (x, y, z) = x2 + 4y2 + 5z2 - 6x + 8y + 3 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. 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) = 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
> Calculate the following iterated integrals. ∫01 (∫-11 1/3 y3 x 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.
