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calculus

Mathematical Applications for the Management Life and Social Sciences 11th edition Ronald J. Harshbarger, James J. Reynolds - Solutions

If u = f (w, x, y, z) = y2 - x2z + 4x, find the following. (a) ∂u/∂w (b) ∂u/∂x (c) ∂u/∂y (d) ∂u/∂z
If u = x2 + 3xy + xz, find the following. (a) ux (b) uy (c) uz
If C(x1, x2, x3) = 4x12 + 5x1x2 + 6x22 + x3, find the following. (a) ∂C/∂x1 (b) ∂C/x2 (c) ∂C/x3
If f (x, y, z) = 2x√yz - 1 + x2z3, find the following. (a) ∂f/∂x (b) ∂f/∂y (c) ∂f/∂z
If z = x2 + 4x - 5y3, find the following. (a) zxx (b) zxy (c) zyx (d) zyy
If z = x3 - 5y2 + 4y + 1, find the following. (a) zxx (b) zxy (c) zyx (d) zyy
If z = x2y + 4xy2, find the following. (a) zxx (b) zxy (c) zyx (d) zyy
1. If z = x3 + 4x2y + 6y2, find zx and zy. 2. If z = 3xy + y2, find zx and zy.
If z = xy2 + 4xy - 5, find the following. (a) zxx (b) z= (c) zyx (d) zyy
If f (x, y) = x2 + exy, find the following. (a) ∂2f/∂x2 (b) ∂2f/∂yx (c) ∂2f/xy (d) ∂2f/∂y2
If z = xexy, find the following. (a) zxx (b) zyy (c) zxy
If f (x, y) = y2 - ln xy, find the following. (a) ∂2f /∂x2 (b) ∂2f/∂y∂x (c) ∂2f/∂x∂y (d) ∂2f/∂y2
If f (x, y) = x3 + ln (xy - 1), find the following. (a) ∂2f/x2 (b) ∂2f/∂y2 (c) ∂2f/∂x ∂y (d) ∂2f/∂y∂x
If f (x, y) = x3y + 4xy4, find ∂2/∂x2 f (x, y) |(1, -1)
If f (x, y) = x4y2 + 4xy, find ∂2/∂y2 f (x, y) | (1, 2).
If f (x, y) = 2x / x2 + y2, find the following a. ∂2f/∂x2 |(-1, 4) b. ∂2f/∂y2 |(-1, 4)
If f (x, y) = 2y2 / 3xy = 4, find the following a. ∂2f/∂x2 |(1,-2) b. ∂2f/∂y2 |(1, -2)
If z = x2y + yex2, find zyx |(1, 2).
If z = xy3 + x ln y2, find zxy |(1, 2).
1. If z = x2 - xy4y3, find zxyx. 2. If z = x3 - 4x2y + 5y3, find zyyx.
If w = 4x3y + y2z + z3, find the following. (a) wxxy (b) wxyx (c) wxyz
If w = 4xyz + x3y2z + x3, find the following. (a) wxyz (b) wxzz (c) wyyz
When a homeowner has a 25-year variable-rate mortgage loan, the monthly payment R is a function of the amount of the loan A and the current interest rate i (as a percent); that is, R = f (A, i).Interpret each of the following. (a) f (100,000, 8) =771.82 (b) ∂f/∂I (100,000, 8) = 66.25
Suppose that in a certain city, the number of people N using the mass transportation system is a function of the fare f and the daily cost of downtown parking p, so that N =N(f, p). Interpret each of the following. (a) N(5, 20) = 65,000 (b) ∂N/∂f (5, 20) = - 4000 (c) ∂N/∂p (5, 20) = 2500
In economics, the most economical quantity Q of goods (TVs, dresses, gallons of paint, etc.) for a store to order is given by Wilson's lot size formula Q = √2KM/h where K is the cost of placing the order, M is the number of items sold per week, and h is the weekly holding costs for each item (the
Suppose that the total cost (in dollars) of producing a product is C(x, y) = 25 + 2x2 + 3y2, where x is the cost per pound for material and y is the cost per hour for labor. (a) If material costs are held constant, at what rate will the total cost increase for each $1-per-hour increase in
Suppose that the number of thousands of insects killed by two brands of pesticide is given by f (x, y) = 10,000 - 6500e-0.01x - 3500e-0.02y where x is the number of liters of brand 1 and y is the number of liters of brand 2. What is the rate of change of insect deaths with respect to the number of
If f (x, y) = (x3 + 2y2)3, find ∂f/∂x and ∂f/∂y.
Suppose that the profit (in dollars) from the sale of Kisses and Kreams is given by P(x, y) = 10x + 6.4y - 0.001x2 - 0.025y2 where x is the number of pounds of Kisses and y is the number of pounds of Kreams. Find P y, and give the approximate rate of change of profit with respect to the number of
1. If U = f (x, y) is the utility function for goods X and Y, the marginal utility of X is U x and the marginal utility of Y is ∂U/∂y. If U = x2y2, find the marginal utility of (a) X. (b) Y. 2. Utility If the utility function for goods X and Y is U = xy + y2, find the marginal utility of (a)
Production Suppose that the output Q (in units) of a certain company is Q = 75K1/3L2/3, where K is the capital expenditures in thousands of dollars and L is the number of labor hours. Find Q K and ∂Q/∂L when capital expenditures are $729,000 and the labor hours total 5832. Interpret each answer.
Suppose that the production Q (in gallons of paint) of a paint manufacturer can be modeled by Q = 140K1/2L1/2, where K is the company's capital expenditures in thousands of dollars and L is the size of the labor force (in hours worked). Find ∂Q/∂K and ∂Q/∂L when capital expenditures are
(a) To see how the wind chill temperature changes with wind speed, find ∂WC/∂s. (b) Find ∂WC/∂s when the temperature is 10°F and the wind speed is 25 mph. What does this mean?
(a) To see how wind chill temperature changes with temperature, find ∂WC/∂t. (b) Find ∂WC/∂t when the temperature is 10°F and the wind speed is 25 mph. What does this mean?
If f (x, y) = (xy3 + y)2, find ∂f/∂x and ∂f/∂y.
If f (x, y) = √2x2 - 5y2, find fx and fy.
If g(x, y) = √xy - x, find gx and gy.
If C(x, y) = 600 - 4xy + 10x2y, find ∂C/∂x and ∂C/∂y.
1. The cost (in dollars) of manufacturing one item is given by C(x, y) = 30 + 3x + 5y where x is the cost of 1 hour of labor and y is the cost of 1 pound of material. (a) If the hourly cost of labor is $20, and the material costs $3 per pound, what is the cost of manufacturing one of these
Suppose the joint cost function for x units of product X and y units of product Y is given by C(x, y) = 2500 √xy + 1 dollars Find the marginal cost with respect to (a) x. (b) y.
Suppose that the joint cost function for two products is C(x, y) = 1200 ln (xy + 1) + 10,000 dollars Find the marginal cost with respect to (a) x. (b) y.
Suppose that the joint cost function for two products is C(x, y) = y ln (x +1) dollars Find the marginal cost with respect to (a) x. (b) y.
Suppose that the production function for a product is z = √4xy, where x represents the number of work-hours per month and y is the number of available machines. Determine the marginal productivity of (a) x. (b) y.
Suppose the production function for a product is z = 60x2/5y3/5 where x is the capital expenditures and y is the number of work-hours. Find the marginal productivity of (a) x. (b) y.
Suppose that the production function for a product is z = √x ln(y +1), where x represents the number of work-hours and y represents the available capital (per week). Find the marginal productivity of (a) x. (b) y.
Suppose that a company's production function for a certain product is z(x = 1)1/2 ln ( y2 + 1) where x is the number of work-hours of unskilled labor and y is the number of work-hours of skilled labor. Find the marginal productivity of (a) x. (b) y.
Find the output when x = 300 and y =500.Suppose that the number of crates of an agricultural product is given bywhere x is the number of hours of labor and y is the number of acres of the crop.
Find and interpret the marginal productivity of the number of acres of the crop (y) when x = 300 and y = 500.Suppose that the number of crates of an agricultural product is given by zwhere x is the number of hours of labor and y is the number of acres of the crop.
Find and interpret the marginal productivity of the number of hours of labor (x) when x = 300 and y = 500.Suppose that the number of crates of an agricultural product is given by zwhere x is the number of hours of labor and y is the number of acres of the crop.
If a production function is given by z = 12x3/4y1/3, find the marginal productivity of (a) x. (b) y.
Suppose the Cobb-Douglas production function for a company is given by z = 400x3/5y2/5 where x is the company's capital investment and y is the size of the labor force (in work-hours). (a) Find the marginal productivity of x. (b) If the current labor force is 1024 work-hours, substitute y = 1024 in
Suppose the Cobb-Douglas production function for a company is given by z = 300x2/3y1/3 where x is the company's capital investment and y is the size of the labor force (in work-hours). (a) Find the marginal productivity of x. (b) If the current labor force is 729 work-hours, substitute y = 729 in
In Problems 1-3, prices p1 and p2 are in dollars and q1 and q2 are numbers of units. 1. The demand functions for two products are given by q1 = 300 - 8p1 - 4p2 q2 = 400 - 5p1 - 10p2 Find the demand for each of the products if the price of the first is p1 = $10 and the price of the second is p2 =
In Problems 1-2, the demand functions for qA and qB units of two related products, A and B, are given. Complete parts (a)-(e) for each problem. Assume pA and pB are in dollars.(a) Find the marginal demand of qA with respect to pA.(b) Find the marginal demand of qA with respect to pB.(c) Find the
The markets for new cars and for used cars are related. As new car sales increase, the available supply of used cars (trade-ins) increases. This tends to decrease the price of used cars. As the prices of used cars decline, typically the demand for new cars also declines.(a) Does this analysis
The total cost of producing an item is C(x, y) = 40 + 4x + 6y + x2y / 100 dollars where x is the cost per pound of raw materials and y is the cost per hour for labor. How will an increase of (a) $1 per pound of raw materials affect the total cost? (b) $1 per hour in labor costs affect the total
The total cost of producing 1 unit of a product is given bywhere x represents the cost per pound of raw materials and y represents the hourly rate for labor. The present cost for raw materials is $10 per pound and the present hourly rate for labor is $24. How will an increase of (a) $1 per pound
The total cost of producing 1 unit of a product is given by C(x, y) = 30 + 0.5x2 + 30y - xy dollars where x is the hourly labor rate and y is the cost per pound of raw materials. The current hourly rate is $25, and the raw materials cost $6 per pound. How will an increase of (a) $1 per pound for
The joint cost (in dollars) for two products is given by C(x, y) = 30 + x2 + 3y + 2xy where x represents the quantity of product X produced and y represents the quantity of product Y produced. (a) Find and interpret the marginal cost with respect to x if 8 units of product X and 10 units of product
The joint cost (in dollars) for products X and Y is given by C(x, y) = 40 + 3x2 + y2 + xy where x represents the quantity of X and y represents the quantity of Y. (a) Find and interpret the marginal cost with respect to x if 20 units of product X and 15 units of product Y are produced. (b) Find and
If the joint cost function for two products is C(x, y) = x√y2 + 1 dollars(a) Find the marginal cost (function) with respect to x.(b) Find the marginal cost with respect to y.
In Problems 1-3, find each function's relative maxima, relative minima, and saddle points, if they exist. 1. z = 9 - x2 - y2 2. z = 16 - 4x2 - 9y2 3. z = x2 + y2 + 4
In Problems 1 and 2, use the points given in the tables to write the equation of the line that is the best fit for the points.1.2.
Suppose that the quarterly profit from the sale of Kisses and Kreams is given by P(x, y) = 10x + 6.4y - 0.001x2 - 0.025y2 dollars where x is the number of pounds of Kisses and y is the number of pounds of Kreams. Selling how many pounds of Kisses and Kreams will maximize profit? What is the maximum
The profit from the sales of two products is given by P(x, y) = 20x + 70y - x2 - y2 dollars where x is the number of units of product 1 sold and y is the number of units of product 2. Selling how much of each product will maximize profit? What is the maximum profit?
A new food is designed to add weight to mature beef cattle. The weight in pounds is given by W = 13xy (20 - x - 2y), where x is the number of units of the first ingredient and y is the number of units of the second ingredient. How many units of each ingredient will maximize the weight? What is the
The profit for a grain crop is related to fertilizer and labor. The profit per acre is P = 100x + 40y - 5x2 - 2y2 dollars where x is the number of units of fertilizer and y is the number of work-hours. What values of x and y will maximize the profit? What is the maximum profit?
Suppose that P = 3.78x2 + 1.5y2 - 0.09x3 - 0.01y3 tons is the production function for a product with x units of one input and y units of a second input. Find the values of x and y that will maximize production. What is the maximum production?
Suppose that x units of one input and y units of a second input result in P = 40x + 50y - x2 -y2 - xy units of a product. Determine the inputs x and y that will maximize P. What is the maximum production?
Profit Suppose that a manufacturer produces two brands of a product, brand 1 and brand 2. Suppose the demand for brand 1 is x = 70 - p1 thousand units and the demand for brand 2 is y = 80 - p2 thousand units, where p1 and p2 are prices in dollars. If the joint cost function is C = xy, in thousands
Suppose that a firm produces two products, A and B, that sell for $a and $b, respectively, with the total cost of producing x units of A and y units of B equal to C(x, y). Show that when the profit from these products is maximized,
Find the values for each of the dimensions of an open-top box of length x, width y, and height 500,000 (xy) (in inches) such that the box requires the least amount of material to make.
Manufacturing Find the values for each of the dimensions of a closed-top box of length x, width y, and height z (in inches) if the volume equals 27,000 cubic inches and the box requires the least amount of material to make.
A company manufactures two products, A and B. If x is the number of thousands of units of A and y is the number of thousands of units of B, then the cost and revenue in thousands of dollars are C(x, y) = 2x2 - 2xy + y2 - 7x - 10y + 11 R(x, y) = 5x + 8y Find the number of each type of product that
Let x be the number of work-hours required and let y be the amount of capital required to produce z units of a product. Show that the average production per work-hour, z x, is maximized when ∂z / ∂x = z / x Use z = f (x, y) and assume that a maximum exists.
1. (a) If Sea Islands Chicken Shack prices chicken dinners differently for eat-in and take-out customers, how many dinners per week would it expect to sell to each type of customer in order to maximize weekly profit? (b) What prices should Sea Islands charge each market segment to maximize the
The data in the table show the average earnings of year-round full-time workers by gender for several different levels of educational attainment. Average Annual Earnings Males _____________________ Females__ $21,659 ........................... $17,023 26,277
The table gives the approximate benefits for PepsiCo executives who earned an average of $250,000 per year during the last 5 years of service, based on the number of years of service, from 15 years to 45 years. (a) Use linear regression to find the linear equation that is the best fit for the
The table gives the actual or projected world population in billions for selected years from 2000 to 2050. (a) Use linear regression to find the linear equation that is the best fit for the data, with x equal to the number of years past 2000. (b) Use the equation to predict the population in
The following table shows the balance of federal direct student loans (in billions of dollars) for selected years from 2011 and projected to 2023. (a) Find the linear regression equation for the federal direct student loan balance as a function of the years past 2010. Report the model with three
Find the minimum value of z=x2+y2 subject to the condition x + y=6.
Find the maximum value of z = xy2 subject to 2x2+y2=600; x ≥ 0, y ≥ 0.
Find the minimum value of w=x2+y2+z2 subject to the constraint x + y + z =3.
Find the minimum value of w=x2+y2+z2 subject to the condition 2x-4y+z=21.
Find the maximum value of w=xz + y subject to the constraint x2+y2+z2=1.
Find the maximum value of w=x2yz subject to the constraint 4x+y+z=4, x ≥ 0, y ≥ 0, and z ≥ 0.
Suppose that the utility function for two commodities is given by U=xy2 and that the budget constraint is 3x+6y=18. What values of x and y will maximize utility?
Suppose that the budget constraint in Problem 15 is 5x+20y=90. What values of x and y will maximize U=xy2?
Suppose that the utility function for two products is given by U=x2y, and the budget constraint is 2x+3y=120. Find the values of x and y that maximize utility. Check by graphing the budget constraint with the indifference curve for maximum utility and with two other indifference curves.
Suppose that the utility function for two commodities is given by U=x2y3, and the budget constraint is 10x+15y=250. Find the values of x and y that maximize utility. Check by graphing the budget constraint with the indifference curve for maximum utility and with two other indifference curves.
A company has the Cobb-Douglas production function z = 400x 0.6 y 0.4 where x is the number of units of labor, y is the number of units of capital, and z is the units of production. Suppose labor costs $150 per unit, capital costs $100 per unit, and the total cost of labor and capital is limited to
Find the minimum value of z=4x2+y2 subject to the constraint x + y=5.
Suppose a company has the Cobb-Douglas production function z = 100 0.75 y0.25 where x is the number of units of labor, y is the number of units of capital, and z is the units of production. Suppose further that labor costs $90 per unit, capital costs $150 per unit, and the total costs of labor and
A firm has two plants, X and Y. suppose that the cost of producing x units at plant X is x2+1200 Dollars and the cost of producing y units of the same product at plant Y is given by 3y2+800 dollars. If the firm has an order for 1200 units, how many should it produce at each plant to fill this order
Suppose that the cost of producing x units at plant X is (3x+4)x dollars and that the cost of producing y units of the same product at plant Y is (2y+8)y dollars. If the firm that owns the plants has an order for 149 units, how many should it produce at each plant to fill this order and minimize
On the basis of past experience a company has determined that its monthly sales revenue (in dollars) is related to its advertising according to the formula s=20x+y2+4xy, where x is the amount spent on print advertising and y is the amount spent on cable advertising. If the company plans to spend
Find the dimensions x, y, and z (in inches) of the rectangular box with the largest volume that satisfies 3x+ 4y + 12z = 12
Find the dimensions (in centimeters) of the box with square base, open top, and volume 500,000 cubic centimeters that requires the least materials.
Show that a box with a square base, an open top, and a fixed volume requires the least material to build if it has a height equal to one-half the length of one side of the base.
Find the minimum value of z =3x2+5y2-2xy subject to the constraint x + y=5.

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