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applied calculus

Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions

For what value of a is the shaded area in Fig. 4 equal to 1? y = x² Figure 4 Y 0 a (a, a²) X
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 approximately equal to 10 (the value of λ).
Find the values of x, y, and z that maximize xy + 3xz + 3yz subject to the constraint 9 - xyz = 0.
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = x2 - 3y2 + 4x + 6y + 8
In Exercises, find an antiderivative of f (x), call it F (x), and compare the graphs of F (x) and f (x) in the given window to check that the expression for F (x) is reasonable. [That is, determine whether the two graphs are consistent. When F (x) has a relative extreme point, f (x) should be zero;
In Exercises, find an antiderivative of f (x), call it F (x), and compare the graphs of F (x) and f (x) in the given window to check that the expression for F (x) is reasonable. [That is, determine whether the two graphs are consistent. When F (x) has a relative extreme point, f (x) should be zero;
Find a function f (x) whose graph goes through the point (1, 1) and whose slope at any point (x, f (x)) is 3x2 - 2x + 1.
Suppose that water is flowing into a tank at a rate of r (t) gallons per hour, where the rate depends on the time t according to the formula(a) Consider a brief period of time, say, from t1 to t2. The length of this time period is t = t2 - t1. During this period the rate of flow does not change
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.
If money is deposited steadily in a savings account at the rate of $4500 per year, determine the balance at the end of 1 year if the account pays 9% interest compounded continuously.
Drilling of an oil well has a fixed cost of $10,000 and a marginal cost of C′(x) = 1000 + 50x dollars per foot, where x is the depth in feet. Find the expression for C(x), the total cost of drilling x feet.
Since 1987, the rate of production of natural gas in the United States has been approximately R(t) quadrillion British thermal units per year at time t, with t = 0 corresponding to 1987 and R(t) = 17.04e0.016t. Find a formula for the total U.S. production of natural gas from 1987 until time t.
What number does the sumapproach as n gets very large? [P² + (₁ + ²)² + (₁ + ²)² + (₁ + ²)² 1 3 1 +1+ n n n n +---+ (1+"=')*] - / n n
The United States has been consuming iron ore at the rate of R(t) million metric tons per year at time t, where t = 0 corresponds to 1980 and R(t) = 94e0.016t. Find a formula for the total U.S. consumption of iron ore from 1980 until time t.
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 of Visitors in
Let f (x, y) = 2x3 + x2y - y2. Computeat (x, y) = (1, 2). 2f2f 2f and 2 дх2 ду ах ду
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = x4 - 8xy + 2y2 - 3
Find the values of x and y that minimize f (x, y) = x - xy + 2y2 subject to the constraint x - y + 1 = 0.
Table 6 gives the U.S. minimum wage in dollars for certain years.(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 the trend determined by the straight line in part (a) continues, when will the minimum
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = x2 - y3 + 5x + 12y + 1
Physicians, particularly pediatricians, sometimes need to know the body surface area of a patient. For instance, they use surface area to adjust the results of certain tests of kidney performance. Tables are available that give the approximate body surface area A in square meters of a person who
In Exercises, let R be the rectangle consisting of all points (x, y), such that 0 ≤ x ≤ 4, 1 ≤ y ≤ 3, and calculate the double integral. J[5 J. R 5 dx dy
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 (x, y) = x3 - 2xy + 4y
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 (x, y) = x3 - y2 - 3x + 4y
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 (x, y) = x2 - 2xy + 3y2 + 4x - 16y + 22
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 (x, y) = x2 + 2xy + 5y2 + 2x + 10y - 3
In Exercises, let R be the rectangle consisting of all points (x, y), such that 0 ≤ x ≤ 4, 1 ≤ y ≤ 3, and calculate the double integral. ff (2.x R (2x + 3y) dx dy
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 (x, y) = -x2 - 8xy - y2
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. 17.2 f(x, y) = ¹7x² + 2xy + 5y² + 5x − 2y +
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 (x, y) = -2x2 + 2xy - y2 + 4x - 6y + 5
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.)
Computewhere f (x, y) = 60x3/4y1/4, a production function (where y is units of capital). Explain whyis always negative. 2²f ,2⁹
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 (x, y) = 2x2 + 3xy + 5y2
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,Let F (x, y, λ) = f (x, y) + λ(c - ax - by). The result follows from
Computewhere f (x, y) = 60x3/4y1/4, a production function (where x is units of labor). Explain whyis always negative. fre д.+2' ах
Calculate the iterated integral. L' (xVy + y) dy dx 1) dy ) d.
Let f (x, y) = 3x2 + 2xy + 5y, as in Example 5. Show that f (1 + h, 4) - f (1, 4) = 14h + 3h2. Thus, the error in approximating f (1 + h, 4) - f (1, 4) by 14h is 3h2. (If h = .01, for instance, the error is only .0003.)Example 5Interpret the partial derivatives of f(x, y) = 3x2 + 2xy + 5y
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. (b) 北 y 22
Calculate the iterated integral. .5 L² ( 1 4 (2xy + 3) dy dx 3) dy ) d
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 (x, y) = x2 - 2xy + 4y2
For the production function f (x, y) = 60x3/4y1/4 considered in Example 8, think of f (x, y) as the revenue when x units of labor and y units of capital are used. Under actual operating conditions, say, x = a andis referred to as the wage per unit of labor andis referred to as the wage per unit of
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).] x Figure 4 (a) y 2)
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 (x, y) = x4 - 12x2 - 4xy - y2 + 16
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 (x, y) = 3x2 + 8xy - 3y2 - 2x + 4y + 1
Richard Stone determined that the yearly consumption of food in the United States was given by f (m, p, r) = (2.186)m0.595p-0.543r0.922. Determine which partial derivatives are positive and which are negative, and give interpretations of these facts.
Using data collected from 1929 to 1941, Richard Stone determined that the yearly quantity Q of beer consumed in the United Kingdom was approximately given by the formula Q = f (m, p, r, s), where f (m, p, r, s) = (1.058)m0.136p-0.727r0.914s0.816 and m is the aggregate real income (personal
In Exercises, 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 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 (x, y) = -2x2 + 2xy - 25y2 - 2x + 8y - 1
In Exercises, 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)
The volume (V) of a certain amount of a gas is determined by the temperature (T) and the pressure (P) by the formula V = .08(T/P). Calculate and interpretwhen P = 20, T = 300. Ꮩ Ꮲ and Ꮩ . Ꭲ
The demand for a certain gas-guzzling car is given by f (p1, p2), where p1 is the price of the car and p2 is the price of gasoline. Explain why af др1 < 0 and af др2 < 0.
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 is minimized. [Refer to Fig. 4(a)] The cost
Let p1 be the average price of MP3 players, p2 the average price of audio files, f (p1, p2) the demand for MP3 players, and g (p1, p2) the demand for audio files. Explain whyand af ap2
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 (x, y) = 2xy + y2 + 2x - 1
Use Lagrange multipliers to find the three positive numbers whose sum is 15 and whose product is as large as possible.
In Exercises, 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)
Refer to Exercise 27. Let g (p1, p2) be the number of people who will take the train when p1 is the price of the bus ride and p2 is the price of the train ride. Would you expect δg/δp1 to be positive or negative? How about δg/δp2?Exercise 27In a certain suburban community, commuters have the
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 of fencing. Y Figure 1 A garden. X
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 (x, y) = 6xy - 3y2 - 2x + 4y - 1
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 (x, y) = x2 + 2xy + 10y2
Find the values of x, y, z that minimize x2 + y2 + z2 - 3x - 5y - z, subject to the constraint 20 - 2x - y - z = 0.
In a certain suburban community, commuters have the choice of getting into the city by bus or train. The demand for these modes of transportation varies with their cost. Let f (p1, p2) be the number of people who will take the bus when p1 is the price of the bus ride and p2 is the price of the
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 (x, y) = x4 - x2 - 2xy + y2 + 1
Find the values of x, y that minimizesubject to the constraint 10 - x - y = 0. -x² − 3xy - y² + y + 10,
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 dimensions of a rectangular box of volume 1000 cubic inches for which the sum of the dimensions is minimized.
The production function for a firm is f (x, y) = 64x3/4y1/4, 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 goods.(a) Determine the amounts of labor and
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 (x, y) = 8xy + 8y2 - 2x + 2y - 1
Match the graphs of the functions in Exercises to the systems of level curves shown in Figs. 8(a)–(d). Figure 8 (a) +r DL 8 (b) (c) (d)
Find the values of x, y, z that minimize 3x2 + 2y2 + z2 + 4x + y + 3z, subject to the constraint 4 - x - y - z = 0.
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 (x, y) = 3x2 + 8xy - 3y2 + 2x + 6y
Find the values of x, y, and z that maximize xyz subject to the constraint 36 - x - 6y - 3z = 0.
Let f (x, y) = xey + x4y + y3. Find 2²f af af dx² ay² dx dy' and a²f ду ах
The productivity of a country is given by f (x, y) = 300x2/3y1/3, where x and y are the amount of labor and capital.(a) Compute the marginal productivities of labor and capital when x = 125 and y = 64.(b) Use part (a) to determine the approximate effect on productivity of increasing capital from 64
Match the graphs of the functions in Exercises to the systems of level curves shown in Figs. 8(a)–(d). Figure 8 (a) +r DL 8 (b) (c) (d)
Match the graphs of the functions in Exercises to the systems of level curves shown in Figs. 8(a)–(d). Figure 8 (a) +r DL 8 (b) (c) (d)
In Exercises, both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f(x, y): X + 1 y + xy; (1, 1)
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(x, y) = -2x2 + 6xy - 17y2 - 4x + 6y
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 B. If x units of A and y units of B are
A farmer can produce f (x, y) = 200√6x2 + y2 units of produce by utilizing x units of labor and y units of capital. (The capital is used to rent or purchase land, materials, and equipment.)(a) Calculate the marginal productivities of labor and capital when x = 10 and y = 5.(b) Let h be a small
Let f (x, y) = x3y + 2xy2. Find 2²f ²f 2² f dx² dy²' dx dy' and 2²f dy dx
Maximize 3x2 + 2xy - y2, subject to the constraint 5 - 2x - y = 0.
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 (x, y) = -5x2 + 4xy - 17y2 - 6x + 6y + 2
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. Suppose profits are $2 per unit for product A and $10 per unit for product B. Find the production schedule that maximizes the total
Find the values of x, y, z at which f (x, y, z) = x2 + 4y2 + 5z2 - 6x + 8y + 3 assumes its minimum value.
Find the point on the parabola y = x2 that has minimal distance from the point (16, 1/2). y = x² Figure 2 Y (x, y) (b) (16, 2) x
LetComputeand interpret your result. f(x, y) = X y = 6*
In Exercises, 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 (x, y) = x3 + y2 - 3x - 8y + 12
Find the dimensions of the rectangle of maximum area that can be inscribed in the unit circle. y (a) (x, y) X
In Exercises, both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state.f (x, y) = yex - 3x - y + 5; (0, 3)
Let f (x, y) = xy2 + 5. Evaluateand interpret your result. af ду at (x, y) = (2,-1)
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 (x, y) = 7x2 - 5xy + y2 + x - y + 6
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 significance of these curves.
In Exercises, both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state.f (x, y) = x4 - 4xy + y4; (0, 0), (1, 1), (-1, -1)
In Exercises, 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 (x, y) = x2 + 3xy + 4y2 - 13x -
Let f (x, y) = (x + y2)3. Evaluate af af and at (x, y) = (1, 2). ах ду
In Exercises, find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f(x, y) = x² + 4xy + y² + 8y² + 3x + 2 ½ 2 3
Let f (x, y) = x2 + 2xy + y2 + 3x + 5y. Findand af 2x -(2,-3)
Suppose that a topographic map is viewed as the graph of a certain function f (x, y). What are the level curves?
In Exercises, both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state.f (x, y) = 2x2 - x4 - y2; (-1, 0), (0, 0), (1, 0)

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