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mathematics
calculus

Questions and Answers of Calculus

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,
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
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
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
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
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
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.
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
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
(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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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,
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 /
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
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
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
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,
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
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
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
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
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
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.
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
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
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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