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College Mathematics for Business Economics Life Sciences and Social Sciences 12th edition Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen - Solutions
In Problems 25 - 30, fund the indicated function f of a single variable. f(y) = H(0,y) for H(x,y) = x2 - 5xy - y2 + 2
In Problem 25 - 30, find the indicated function f of a single variable. f(x) = L(x,-2) for L (x,y) = 25 - x + 5y - 6xy
In Problems 25-30, find the indicated function f of a single vari - able. f(x) = N(x,2x) for N(x,y) = 3xy + x2 - y2 + 1
Let F(x,y) = 5x - 4y + 12. Find all values of x such that F(x,0) = 0.
Let F(x,y) = xy + 2x2 + y2 - 25. Find all values of y such that F(y, y) = 0.
Let G{a,b,c) = a3 + 63 + c3 - (ab + ac + be) - 6. Find all values of bsuch that G(2, b,1) = 0.
For the function f(x,y) = x2 + 2y2, find
For the function f(x,y)= 2xy2, find
Let f{x,y) = √4 - y2. (A) Explain why the cross sections of the surface z = f{x,y)produced by cutting it with planes parallel to x = 0 are semicircles of radius 2. (B) Describe the cross sections of the surface in the planes y = 0, y = 2,and y = 3. (C) Describe the surface z = f{x,y).
Let f (x,y) = 100 + 10x + 25y - x2 - 5y2. (A) Describe the cross sections of the surface z = f(x, y) produced by cutting it with the planes y = 0, y = 1, y = 2, and y = 3. (B) Describe the cross sections of the surface in the planes x = 0, x = 1 , x = 2, and x = 3. (C) Describe the surface z =
Let f(x, y) = 4 - √x2 + y2. (A) Explain why f{a,b) = f{c,d) whenever (a, b) and (c, d) are points on the same circle with center at the origin in the xy plane. (B) Describe the cross sections of the surface z = f(x,y) produced by cutting it with the planes x = 0, y = 0, and x = y. (C) Describe
Advertising and sales. A company spends $.v thousand per week on online advertising and $y thousand per week on TV advertising. Its weekly sales are found to be given by S(x,y)=5x2y3 Find S(3,2) and S(2,3)
A company manufactures 10 - and 3 - speed bicycles. The weekly demand and cost equations are p = 230 - 9x + y q = 130 + x - 4y C(x,y) = 200 + 80x + 30y where %p is the price of a 10 - speed bicycle, $q is the price of a 3 - speed bicycle, x is the weekly demand for 10-speed bicycles, y is the
The petroleum company in Problem 53 is taken over by another company that decides to double both the units of labor and the units of capital utilized in the production of petroleum. Use the Cobb-Douglas production function given in Problem 53 to find the amount of petroleum that will be produced by
In problems 45-56 find the indicated function or value if s(x,y) = x3 in y + 4y2ex Sxx(-1,1)
Blood flow. Poiseuille's law states that the resistance R for blood flowing in a blood vessel varies directly as the length L of the vessel and inversely as the fourth power of its radius r. Stated as an equation.Find R (8.1) and R(4,0.2)
In Problems 1-10, find the indicated values of the functions f{x,y) = 2x + 7y - 5 and g(x,y ) = 88/x2 + 3y g(-2, 0)
Safety research. Under ideal conditions, if a person driving a car slams on the brakes and skids to a stop, the length of the skid marks (in feet) is given by the formula. L(w,v) = kwv2 Where k = constant w = weight of car in pounds v = speed of car in miles per hour for k = 0.000 013 3, find
In Problems 1-10, find the indicated values of the functions f{x,y) = 2x + 7y - 5 andg(0,0)
In Problems I-18, find the indicated first-order partial derivative for each function z = f(x,y). fx(4,1) if f(x,y) = 3xey
In Problems I-18, find the indicated first-order partial derivative for each function z = f(x,y). fy(2,4) if f(x,y) = x4lny
In Problems I-18, find the indicated first-order partial derivative for each function z = f(x,y). fy(3,3) if = f(x,y)e3x - y2
In Problems I-18, find the indicated first-order partial derivative for each function z = f(x,y).
In Problems 23 - 34, find the indicated second-order partial derivative for each function f(x, y). Fyx(x,y) if f(x,y) = - 2x + y + 8
In Problems 23-34, find the indicated second-order partial derivative for each function f(x, y). Fyy(x,y) if f(x,y) = x2 + 9y2 - 4
In Problems 23-34, find the indicated second-order partial derivative for each function f(x, y). Fyx(x,y) if f(x,y) = e3x+2y
In Problems 23-34, find the indicated second-order partial derivative for each function f(x, y).
In Problems 23-34, find the indicated second-order partial derivative for each function f(x, y). fyx(x, y) if f(x,y) = (3x - 8y)6
In Problems 23 - 34, find the indicated second-order partial derivative for each function f(x, y). fyy(.X, y) if f(x,y) = (1 + 2xy2)8
In Problems 45 - 56, find the indicated function or value if S(x, y) = x3In y + 4y2ex. Sx(x,y)
In Problems 45 - 56, find the indicated function or value if S(x, y) = x3In y + 4y2ex. Sx(-1,1)
In Problems 45 - 56, find the indicated function or value if S(x, y) = x3In y + 4y2ex. Sxy(x,y)
In Problems 45-56, find the indicated function or value if S(x, y) = x3In y + 4y2ex. Sxx(x,y)
In Problems 45 - 56, find the indicated function or value if S(x, y) = x3In y + 4y2ex. Sxy(-1,1)
In Problems 45 - 56, find the indicated function or value if S(x, y) = x3In y + 4y2ex. Sxx(-1,1)
ln Problems 57 - 62, S(T, r)= 50(T - 40)(5 - r) gives an ice cream shop's daily sales as a function of temperature T (in °F) and rain r (in inches). Find the indicated quantity (include the appropriate units) and explain what it means. S(80,0)
ln Problems 57 - 62, S(T, r)= 50(T - 40)(5 - r) gives an ice cream shop's daily sales as a function of temperature T (in °F) and rain r (in inches). Find the indicated quantity (include the appropriate units) and explain what it means. ST(90,1)
ln Problems 57 - 62, S(T, r)= 50(T - 40)(5 - r) gives an ice cream shop's daily sales as a function of temperature T (in °F) and rain r (in inches). Find the indicated quantity (include the appropriate units) and explain what it means. SrT (90,1)
(A) Find an example of a function f(x, y)such that(B) How many such functions are there? Explain.
In Problems 65-70, find fxx(x, y),fxy(x,y),fyx(x,y), and fyy( x, y) for each function f. f(x,y) = x3y3 + x + y2
In Problems 65-70, find fxx(x, y),fxy(x,y),fyx(x,y), and fyy( x, y) for each function f.
In Problems 65-70, find fxx(x, y),fxy(x,y),fyx(x,y), and fyy( x, y) for each function f. f(x,y) = x In (xy)
For C(x,y) = 2x2 + 2xy + 3y2 - 16x - 18y + 54 find all values of x and y such that Cx(x,y) = 0 and Cy(x,y) = 0 simultaneously.
For G(x,y) = x2In y - 3x - 2y + 1 find all values of x and y such that Gx(x,y) = 0 and Gy(.x, y) = 0 Simultaneously
Let f(x,y)= 5 - 2x + 4y - 3x2 - y2. (A) Find the maximum value of fix, y) when x = 2. (B) Explain why the answer to part (A) is not the maximum value of the function fix, y).
Let f(x,y) = ex + 2ey + 3xy2 + 1. (A)Use graphical approximation methods to find d (to three decimal places) such that f(l,d) is the minimum value of fix, y)when x = 1. (B) Find fx(l,d) and fy,(l,d)
F(x,y) = x3 - 3xy2
For f(x,y) = 2xy2, find(A)(B)
Advertising and sales. A company spends $* per week on online advertising and $y per week on TV advertising. Its weekly sales were found to be given by S(x,y) = 10x0.4y8 Find Sx,(3,000. 2,000) and Sy(3,000, 2,000), and interpret the results.
Revenue and profit functions. A company manufactures 10 - and 3 - speed bicycles. The weekly demand and cost functions are p = 230 - 9x + y q = 130 + x - 4y C(x,y)=200 + 80A- + 30y where $p is the price of a 10-speed bicycle, $q is the price of a 3-speed bicycle..Y is the weekly demand for 10 -
The productivity of an automobile-manufacturing company is given approximately by the function f(x,y) = 5√xy = 50x0.5y0.5 with the utilization of .v units of labor and y units of capital. (A) Find fx(x,y) and fy(x,y). (B) If the company is now using 250 units of labor and 125 units of capital,
Product demand. The daily demand equations for the sale of brand A coffee and brand B coffee in a supermarket are x = f(p,q) = 200 - 5p + 4q Brand A coffee x = g(P-q) = 300 + 2p - 4q Brand 3 coffee
The monthly demand equations for the sale of tennis rackets and tennis balls in a sporting good store are x = f(p, q) = 500 - 0.5p - q2 Tennis rackets y = g(P>q) = 10.000 - 8P - I00q2 Tennis balls (cans)
Poiseuille's law states that the resistance R for blood flowing in a blood vessel varies directly as the length L of the vessel and inversely as the fourth power of its radius r. Stated as an equation,Find RL(4,0.2) and Rr(4,0.2), and interpret the results.
Safety research. Under ideal conditions, if a person driving a car slams on the brakes and skids to a stop, the length of the skid marks (in feet) is given by the formula. L(w, v) = kwv2 where k = constant w = weight of car in pounds v = speed of car in miles per hour For k = 0.000 013 3, find
Use Theorem 2 to find local extrema in Problems 5-24. f(x,y) = x2 - y2 + 2x + 6y - 4
Use Theorem 2 to find local extrema in Problems 5-24. f(x,y) = - x2 + xy - 2y2 + x + 10y -5
Use Theorem 2 to find local extrema in Problems 5-24. f(x,y) = 2x2 - xy + y2 - x - 5y + 8
Use Theorem 2 to find local extrema in Problems 5-24. F(x,y) = x2y - xy2
Use Theorem 2 to find local extrema in Problems 5-24. F(x,y) = 2y3 - 6xy - x2
In Problems 1-4, find fx(x, y) and fy(x, y), and explain, using Theorem 1, why f(x,y) has no local extrema. f(x,y) = 10 - 2x - 3y + x2
Use Theorem 2 to find local extrema in Problems 5-24. F(x,y) = 16xy - x4 - 2y2
Use Theorem 2 to find local extrema in Problems 5-24. F(x,y) = 2x2 - 2x2y + 6y3
Use Theorem 2 to find local extrema in Problems 5-24. f(x,y) = x In y + x2 - 4x -5x + 3
(A) Find the local extrema of the functions f(x,y) = x+ y, g(x,y) = x2 + y2, and h (x,y) = x3 + y3. (B) Discuss the local extrema of the function k(x,y) = xn + yn, where n is a positive integar.
(A) Show that (0,0) is a critical point of the function g (x,y) = exy2 + x2y3 + 2, but that the second - derivative test local extrema fails. (B) Use cross section, as in Example 2, to decide whether g has a local maximum, a local minimum, or a saddle point (0,0).
The annual labor and automated equipment cost (in millions of dollars) for a company's production of HDTV's is given by C(x,y) = 2x2 + 2xy + 3y2 - 16x - 18y + 54 Where x is the amount spent per year on labor and y is the amount spent per year on automated equipment (both in millions of dollars).
A store sells two brands of laptop sleeves. The store pays $25 for each brand A sleeve and $30 for each brand B sleeve. A consulting firm has estimated the daily demand equations for these two competitive products to bex = 130 - 4p + q Demand equation for brand Ay = 115 + 2p - 3q Demand
Repeat Problem 33, replacing the coordinates of B with B(6, 9) and the coordinates of C with C(9,0).
A rectangular box with no top and two intersecting partitions (see the figure) must hold a volume of 72 cubic inches. Find the dimensions that will require the least amount of material.
A shipping box is to be reinforced with steel bands in all three directions, as shown in the figure. A total of 150 inches of steel tape is to be used, with 6 inches of waste because of a 2-inch overlap in each direction. Find the dimensions of the box with maximum volume that can be taped as
In Problems 1-4, find fx(x, y) and fy(x, y), and explain, using Theorem 1, why f(x,y) has no local extrema. f(x,y) = x3 - y2 + 7x + 3y + 1
Use Theorem 2 to find local extrema in Problems 5-24. f(x,y) = 3 - x2 - y2 + 6y
Use Theorem 2 to find local extrema in Problems 5-24. f(x,y) = x2 + y2 - 4x + 6y + 23
Use the method of Lagrange multipliers in Problems 7-16. Minimize the product of two numbers if their difference must be 10.
Use the method of Lagrange multipliers in Problems 7-16. Maximize f(x, y, z) = xyz subject to 2x + y + 2z = 120
Use the method of Lagrange multipliers in Problems 7-16. Maximize and minimize f(x, y, z) = 2x + 4y + 4z subject to x2 + y2 + z2 = 9
Use the method of Lagrange multipliers in Problems 7-16. Maximize and minimize f(x,y) = x + ey Subject to x2 + y2 = 1
Problems 17 and 18, use Theorem 1 to explain why no maxima or minima exist. Minimize f(x,y) = x3 + 2y3 subject to 6x - 2y = 1
Use the method of Lagrange multipliers in Problems 1-4. Minimize f(x,y) = 6xy subject to y - x = 6
Consider the problem of minimizing f(x, y)subject to g(x,y) = 0, where g(x,y) = 4x - y + 3. Explain how the minimization problem can be solved without using the method of Lagrange multipliers.
Consider the problem of minimizing f(x,y) = x2 + 2y2 subject to the constraint g(x,y) = yex2 - 1 = 0. (A) Solve the constraint equation for y, and then substitute into f(x, y) to obtain a function h(x) of the single variable x. Solve the original minimization problem by minimizing h (round answers
Budgeting for maximum production. A manufacturing firm has budgeted $60,000 per month for labor and materials. als. If $x thousand is spent on labor and $y thousand is spent on materials, and if the monthly output (in units) is given by N(x,y) = 4xy - 8x then how should the $60,000 be allocated to
Productivity. The research department of a manufacturing company arrived at the following Cobb-Douglas production function for a particular product: N(x,y)=10x0.6y0.4 In this equation, x is the number of units of labor and y is the number of units of capital required to produce N(x,y) units of the
Maximum volume. A mailing service states that a rectangular package shall have the sum of its length and girth not to exceed 120 inches (see the figure). What are the dimensions of the largest (in volume) mailing carton that can be constructed to meet these restrictions?
Diet and minimum cost. A group of guinea pigs is to receive 25,600 calories per week. Two available foods produce 200xy calories for a mixture of x kilograms of type M food and y kilograms of type N food. If type M costs $1 per kilogram and type TV costs $2 per kilogram, how much of each type of
Use the method of Lagrange multipliers in Problems 1-4. Maximize f(x,y) = 25 - x2 - y2 subject to 2x + y = 10
In Problems 5 and 6, use Theorem 1 to explain why no maxima or minima exist. Maximize f(x,y) = 6x + 5y + 24 subject to 3x + 2y = 4
Use the method of Lagrange multipliers in Problems 7-16. Find the maximum and minimum of f(x, y) = x2 - y2 subject to x2 + y2 = 25.
In Problems 7-14, find the least squares line and use it to estimate y for the indicated value of x. Round answers to two decimal places.Estimate y when x = 20.
In Problems 7-14, find the least squares line and use it to estimate y for the indicated value of x. Round answers to two decimal places.Estimate y when x = 15
In Problems 7-14, find the least squares line and use it to estimate y for the indicated value of x. Round answers to two decimal places.Estimate y when x = 10
Repeat Problem 15 for the points (-1, -2), (0, 1), (1, 2), and (2,0).
Problems 17 and 18 refer to the system of normal equations and the formulas for a and b given on page 836.Ifare the averages of the x and y coordinates, respectively, show that the point (,)satisfies the equation of the least squares line, y = ax + b.
In Problems 1-6, find the least squares line. Graph the data and the least squares line.
A) Give an example of a set of six data points such that half of the points lie above the least squares line and half lie below. (B) Give an example of a set of six data points such that just one of the points lies above the least squares line and five lie below.
(A) Find the linear, quadratic, and logarithmic functions that best fit the data points (1, 3.2), (2, 4.2), (3, 4.7), (4,5.0), and (5,5.3). (Round coefficients to two decimal places.) (B) Which of the three functions best fits the data? Ex-plain.
Cable TV revenue. Data for cable TV revenue are given in the table for the years 2003 through 2007.(A) Find the least squares line for the data, using x = 0 for 2000. (B) Use the least squares line to predict cable TV revenue in 2017.
A market research consultant for a supermarket chain chose a large city to test market a new brand of mixed nuts packaged in 8-ounce cans. After a year of varying the selling price and recording the monthly demand, the consultant arrived at the following demand table, where y is the number of cans
In biology, there is an approximate rule, called the bioclimatic rule for temperate climates. This rule states that in spring and early summer, periodic phenomena such as the blossoming of flowers, the appearance of insects, and the ripening of fruit usually come about 4 days later for each 500
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