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mathematics
calculus with applications
Calculus For Business, Economics And The Social And Life Sciences 11th Brief Edition Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price - Solutions
In Exercises 65 through 72, find the average value of the function f(x, y) over the given region R. y f(x, y) = = + = X y R: 1 ≤ x ≤ 4, 1 ≤ y ≤ 3
The smaller the resistance to flow in a blood vessel, the less energy is expended by the pumping heart. One of Poiseuille’s laws* says that the resistance to the flow of blood in a blood vessel satisfieswhere L is the length of the vessel, r is its radius, and k is a constant that depends on the
In Exercises 65 through 72, find the average value of the function f(x, y) over the given region R. In x f(x, y) ху R: 1 ≤ x ≤ 2, 2 ≤ y ≤ 3
Each of Exercises 61 through 68 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.An editor estimates that if x thousand dollars are spent on development and y thousand dollars are spent on promotion, approximately Q(x, y)
Each of Exercises 61 through 68 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.The output at a certain plant is Q(x, y) = 0.08x2 + 0.12xy + 0.03y2 units per day, where x is the number of hours of skilled labor used and
In Exercises 65 through 72, find the average value of the function f(x, y) over the given region R.f(x, y) = xy(x − 2y); R: −2 ≤ x ≤ 3, −1 ≤ y ≤ 2
There are two sources of air pollution that affect the health of a certain community. Health officials have determined that at a point located r miles from source A and s miles from source B, there will beunits of pollution. A housing project lies in a region R for which 2 ≤ r ≤ 3 and 1 ≤ s
In Exercises 65 through 72, find the average value of the function f(x, y) over the given region R.f(x, y) = xyex2y; R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2
In Exercises 73 through 76, evaluate the double integral over the specified region R. Choose the order of integration carefully. Super R xey dA; R: 0≤x≤ 1,0 ≤ y ≤1
In Exercises 73 through 76, evaluate the double integral over the specified region R. Choose the order of integration carefully. In(xy) RY dA; R: 1 ≤ x ≤ 3, 2 ≤ y ≤ 5
The flow of blood from an artery into a small capillary is given by the formulawhere c is a positive constant, x is the diameter of the capillary, y is the pressure in the artery, and z is the pressure in the capillary. What function gives the rate of change of blood flow with respect to capillary
For the production function given by Q = xayb, where a > 0 and b > 0, show thatIn particular, if b = 1 − a with 0 ag + ax aQ ау = (a + b)Q
At a certain factory, the amount of air pollution generated each day is measured by the function , Q(E, T ) = 125E2/3T1/2 where E is the number of employees and T (°C) is the average temperature during the workday. Currently, there are E = 151 employees and the average temperature is T = 10°C. If
A demographer sets up a grid to describe location within a suburb of a major metropolitan area. In relation to this grid, the population density at each point (x, y) is given by f(x, y) = 1 + 3y2 hundred people per square mile, where x and y are in miles. A housing project occupies the region R
In Exercises 73 through 76, evaluate the double integral over the specified region R. Choose the order of integration carefully. [[ye'y R yey dA; R: -1 ≤ x ≤ 1,1 ≤ y ≤ 2
In Exercises 65 through 72, find the average value of the function f(x, y) over the given region R.f(x, y) = 6xy; R is the triangle with vertices (0, 0), (0, 1), (3, 1).
To estimate the amount of blood that flows through a patient’s lung, cardiologists use the empirical formulawhere P is a percentage of the total blood flow, x is the carbon dioxide output of the lung, y is the arteriovenous carbon dioxide difference in the lung, u is the carbon dioxide output of
In Exercises 65 through 72, find the average value of the function f(x, y) over the given region R.f(x, y) = ex2; R is the triangle with vertices (0, 0), (1, 0), (1, 1).
In Exercises 65 through 72, find the average value of the function f(x, y) over the given region R.f(x, y) = exy−1/2; R is the region bounded by x = √y, y = 0, and x = 1.
The function z = f (x, y) is said to satisfy Laplace’s equation if zxx + zyy = 0. Functions that satisfy such an equation play an important role in a variety of applications in the physical sciences, especially in the theory of electricity and magnetism. In Exercises 73 through 76, determine
In Exercises 65 through 72, find the average value of the function f(x, y) over the given region R.f(x, y) = x; R is the region bounded by y = 4 − x2 and y = 0.
The function z = f (x, y) is said to satisfy Laplace’s equation if zxx + zyy = 0. Functions that satisfy such an equation play an important role in a variety of applications in the physical sciences, especially in the theory of electricity and magnetism. In Exercises 73 through 76, determine
The function z = f (x, y) is said to satisfy Laplace’s equation if zxx + zyy = 0. Functions that satisfy such an equation play an important role in a variety of applications in the physical sciences, especially in the theory of electricity and magnetism. In Exercises 73 through 76, determine
The function z = f (x, y) is said to satisfy Laplace’s equation if zxx + zyy = 0. Functions that satisfy such an equation play an important role in a variety of applications in the physical sciences, especially in the theory of electricity and magnetism. In Exercises 73 through 76, determine
A constant elasticity of substitution (CES) production function is one with the general formwhere K is capital expenditure; L is the level of labor; and A, α, and β are constants that satisfy A > 0, 0 −1. Exercises 31 through 33 involve such production functions. Suppose you wish to
The accompanying table relates the weight C of the large claw of a fiddler crab to the weight W of the rest of the crab’s body, both measured in milligrams (mg).a. For each data point (W, C) in the table, plot the point (ln W, ln C) on a graph. Note that the scatter diagram suggests that y = ln C
Let P(K, L) be a production function, where K and L represent the capital and labor required for a certain manufacturing procedure. Suppose we wish to maximize P(K, L) subject to a cost constraint, C(K, L) = A, for constant A. Use the method of Lagrange multipliers to show that optimal production
A constant elasticity of substitution (CES) production function is one with the general formwhere K is capital expenditure; L is the level of labor; and A, α, and β are constants that satisfy A > 0, 0 −1. Exercises 31 through 33 involve such production functions. Use the method of Lagrange
In Exercises 25 through 36, evaluate the given double integral for the specified region R.where R is the region bounded by y = √x , y = 1, and x = 0. New R dA,
In Exercises 29 through 34, find the second partials (including the mixed partials). f(s, t) = √² + f²
In Exercises 29 through 35, assume that the required extreme value is a relative extremum.A manufacturer with exclusive rights to a sophisticated new industrial machine is planning to sell a limited number of the machines to both foreign and domestic firms. The price the manufacturer can expect to
In Exercises 25 through 36, evaluate the given double integral for the specified region R.where R is the triangle bounded by the lines y = 1/2x, y = −x, and y = 2. R 1 y² + 1 dA,
In Exercises 29 through 35, assume that the required extreme value is a relative extremum.A manufacturer with exclusive rights to a new industrial machine is planning to sell a limited number of them and estimates that if x machines are supplied to the domestic market and y to the foreign market,
A constant elasticity of substitution (CES) production function is one with the general formwhere K is capital expenditure; L is the level of labor; and A, α, and β are constants that satisfy A > 0, 0 −1. Exercises 31 through 33 involve such production functions.Use the method of Lagrange
The Easy-Gro agricultural company estimates that when 100x worker-hours of labor are employed on y acres of land, the number of bushels of wheat produced is f(x, y) = Axayb, where A, a, and b are positive constants. Suppose the company decides to double the production factors x and y. Determine how
In Exercises 31 through 34, use the method of Lagrange multipliers to find the maximum and minimum values of the given function f(x, y) subject to the indicated constraint.f(x, y) = x2 + y3; x2 + 3y = 4
The determination of relationships between measurements of various parts of a particular organism is a topic of interest in the branch of biology called allometry.* Suppose a biologist observes that the shoulder height h and antler size w of an elk, both in centimeters (cm), are related as
In Exercises 29 through 34, find the second partials (including the mixed partials).f(x, y) = x2yex
The output at a certain factory is Q(K, L) = 120K2/3L1/3 units, where K is the capital investment measured in units of $1,000 and L the size of the labor force measured in worker-hours.a. Compute the output if the capital investment is $125,000 and the size of the labor force is 1,331
In Exercises 25 through 36, evaluate the given double integral for the specified region R.where R is the region bounded by y = 1/x2, y = x and x = 2. √√2 R 2x dA,
In Exercises 31 through 34, use the method of Lagrange multipliers to find the maximum and minimum values of the given function f(x, y) subject to the indicated constraint. 1 1 f(x, y) = 4x + y; = + = = 1 - X y
In Exercises 29 through 35, assume that the required extreme value is a relative extremum.Repeat Exercise 31 for the case where p = 20 − 5x, q = 4 − 2y, and C = 2xy + 4.Data from Exercises 31A company produces x units of commodity A and y units of commodity B. All the units can be sold for p =
In Exercises 29 through 34, find the second partials (including the mixed partials).f(u, v) = ln(u2 + v2)
A paint store carries two brands of latex paint. Sales figures indicate that if the first brand is sold for x1 dollars per gallon and the second for x2 dollars per gallon, the demand for the first brand will be D1(x1, x2) = 200 − 10x1 + 20x2 gallons per month and the demand for the second brand
In Exercises 28 through 30, let Q(x, y) be a production function, where x and y represent units of labor and capital, respectively. If unit costs of labor and capital are given by p and q, respectively, then px + qy represents the total cost of production.Show that with the fixed production level
The number of reported cases of AIDS in the United States by year of reporting at 4-year intervals since 1980 is given in this table:a. Plot these data on a graph with time t (years after 1980) on the x axis.b. Find the equation of the least-squares line for the given data.c. How many cases of AIDS
In Exercises 31 through 34, use the method of Lagrange multipliers to find the maximum and minimum values of the given function f(x, y) subject to the indicated constraint.f(x, y) = x + 2y; 4x2 + y2 = 68
In Exercises 25 through 36, evaluate the given double integral for the specified region R.where R is the triangle with vertices (−1, 0), (1, 0), and (0, 1). SS₁2²x R (2x + 1) dA,
Victor Murray’s firm manufactures scientific graphing calculators that cost $40 each to produce and business calculators that cost $20 each.a. If x graphing calculators and y business calculators are produced each month, express the total monthly cost of production as a function of x and y.b.
In Exercises 29 through 35, assume that the required extreme value is a relative extremum.A company produces x units of commodity A and y units of commodity B. All the units can be sold for p = 100 − x dollars per unit of A and q = 100 − y dollars per unit of B. The cost (in dollars) of
In Exercises 31 through 34, use the method of Lagrange multipliers to find the maximum and minimum values of the given function f(x, y) subject to the indicated constraint.f(x, y) = x2 + 2y2 + 2x + 3; x2 + y2 = 4
In Exercises 29 through 34, find the second partials (including the mixed partials).f(x, y) = ex2y
In Exercises 29 through 35, assume that the required extreme value is a relative extremum.The telephone company is planning to introduce two new types of executive communications systems that it hopes to sell to its largest commercial customers. It is estimated that if the first type of system is
In Exercises 29 through 34, find the second partials (including the mixed partials). f(x, y) = x+1 y - 1
Using x skilled workers and y unskilled workers, a manufacturer can produce Q(x, y) = 10x2y units per day. Currently there are 20 skilled workers and 40 unskilled workers on the job.a. How many units are currently being produced each day?b. By how much will the daily production level change if 1
In Exercises 23 through 30, sketch the indicated level curve f(x, y) = C for each choice of constant C.f(x, y) = ln(x2 + y2); C = 4, C = ln 4
In Exercises 25 through 36, evaluate the given double integral for the specified region R.where R is the region bounded by y = x2 and y = 6 − x. SS₁ R 12x dA,
A biologist studying a bacterial colony measures its population each hour and records these data:a. Plot these data on a graph. Does the scatter diagram suggest that the population growth is linear or exponential?b. If you think the scatter diagram in part (a) suggests linear growth, find a
On election day, the polls in a certain state open at 8:00 A.M. Every 2 hours after that, an election official determines what percentage of the registered voters have already cast their ballots. The data through 6:00 P.M. are shown here:a. Plot these data on a graph.b. Find the equation of the
In Exercises 25 through 36, evaluate the given double integral for the specified region R.where R is the region bounded by y = x2 and y = 2x. Sf₂ R (2y - x) dA,
In Exercises 25 through 30, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = (y − 1)ex − y2 on the square region R with vertices (0, 0), (1, 0), (1, 1), and (0, 1).
In Exercises 29 through 35, assume that the required extreme value is a relative extremum.A T-shirt shop carries two competing shirts, one endorsed by Tim Duncan and the other by LeBron James. The owner of the store can obtain both types at a cost of $2 per shirt and estimates that if Duncan shirts
In Exercises 28 through 30, let Q(x, y) be a production function, where x and y represent units of labor and capital, respectively. If unit costs of labor and capital are given by p and q, respectively, then px + qy represents the total cost of production.Show that the inputs x and y that maximize
In Exercises 23 through 28, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = xy2 on the quarter circular region R bounded by x2 + y2 = 12 with x ≥ 0, y ≥ 0.
In Exercises 21 through 28, evaluate the partial derivatives fx(x, y) and fy(x, y) at the given point (x0, y0). f(x, y) = xy In + In(2x - 3y)²; at (1, 1) X
In Exercises 25 through 30, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = ex2+4x+y2 bounded by x2 + 4x + y2 = 0. on the circular region R
In Exercises 28 through 30, let Q(x, y) be a production function, where x and y represent units of labor and capital, respectively. If unit costs of labor and capital are given by p and q, respectively, then px + qy represents the total cost of production.Use Lagrange multipliers to show that
In Exercises 29 through 34, find the second partials (including the mixed partials).f(x, y) = 5x4y3 + 2xy
In Exercises 23 through 30, sketch the indicated level curve f(x, y) = C for each choice of constant C.f(x, y) = xey; C = 1, C = e
In Exercises 23 through 30, sketch the indicated level curve f(x, y) = C for each choice of constant C.f(x, y) = yex; C = 0, C = 1
In Exercises 25 through 36, evaluate the given double integral for the specified region R.where R is the region bounded by y = x3 and y = √x. JS4 R 48xy dA,
In a study of five industrial areas, a researcher obtained these data relating the average number of units of a certain pollutant in the air and the incidence (per 100,000 people) of a certain disease:a. Plot these data on a graph.b. Find the equation of the least-squares line.c. Use the
Modify the least-squares procedure, as illustrated in Example 7.4.4, to find a function of the form P(t) = Aert whose graph best fits the population data in Exercise 26, where P(t) is the U.S. population t decades after 1950.a. Roughly at what percentage rate is the U.S. population growing?b.
In Exercises 23 through 28, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = x4 + 2y3 on the circular region R bounded by x2 + y2 = 1.
In Exercise 26, how does the maximum utility change if k is increased by 1 dollar?Data from Exercises 26A consumer has k dollars to spend on two commodities, the first of which costs a dollars per unit and the second b dollars per unit. Suppose that the utility derived by the consumer from x units
In Exercises 21 through 28, evaluate the partial derivatives fx(x, y) and fy(x, y) at the given point (x0, y0).f(x, y) = xe−2y + ye−x + xy2; at (0, 0)
In Exercises 25 through 30, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = x3 − 4xy + 4x + y2 on the square region R with vertices (1, 2), (4, 2), (1, 5), and (4, 5).
In Exercises 23 through 28, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = x2 + 3y2 − 4x + 6y − 3 on the square region R with vertices (0, 0), (3, 0), (3, −3), (0, −3).
In Exercises 25 through 36, evaluate the given double integral for the specified region R.where R is the triangle with vertices (0, 0), (1, 0), and (1, 1). R xe" dA,
In Exercises 23 through 28, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = 2x2 + y2 + xy2 − 2 on the square region R with vertices (5, 5), (−5, 5), (5, −5), (−5, −5).
In Exercises 23 through 30, sketch the indicated level curve f(x, y) = C for each choice of constant C.f(x, y) = xy; C = 1, C = −1, C = 2, C = −2
A consumer has k dollars to spend on two commodities, the first of which costs a dollars per unit and the second b dollars per unit. Suppose that the utility derived by the consumer from x units of the first commodity and y units of the second commodity is given by the Cobb-Douglas utility function
Over the past 4 years, a college admissions officer has compiled the following data (measured in units of 1,000) relating the number of college catalogs requested by high school students by December 1 to the number of completed applications received by March 1:a. Plot these data on a graph.b. Find
In Exercises 25 through 30, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = x2 + 2x + y2 − 4y +12 on the triangular region R with vertices (4, 0), (1, 0), and (0, 4).
The accompanying table gives the U.S. decennial census figures (in millions) for the period 1950–2000:a. Find the least-squares line y = mt + b for these data, where y is the U.S.population t decades after 1950.b. Use the least-squares line found in part (a) to predict the U.S. population for the
In Exercises 25 through 30, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = x2 − 2x + 4y2 − 6y + 15 on the triangular region R with vertices (0, 0), (5, 5), and (−5, 5).
In Exercises 23 through 28, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = 4xy − 8x − 4y + 5 on the triangular region R with vertices (0, 0), (2, 0), (0, 3).
For each of five different years, the accompanying table gives the percentage of high school students who had used cocaine at least once in their lives up to that year:a. Plot these data on a graph, with the number of years after 1991 on the x axis and the percentage of cocaine users on the y
A consumer has $280 to spend on two commodities, the first of which costs $2 per unit and the second $5 per unit. Suppose that the utility derived by the consumer from x units of the first commodity and y units of the second is given by U(x, y) = 100x0.25y0.75.a. How many units of each commodity
In Exercises 25 through 36, evaluate the given double integral for the specified region R.where R is the rectangle bounded by the lines x = −1, x = 2, y = −1, and y = 0. R 3xy² dA,
In Exercises 23 through 30, sketch the indicated level curve f(x, y) = C for each choice of constant C.f(x, y) = x/y; C = −2, C = 2
In Exercises 23 through 28, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = xy − x − 3y on the triangular region R with vertices (0, 0), (5, 0), (5, 5).
a. If unlimited funds are available, how much should the manufacturer in Exercise 22 spend on development and how much on promotion to generate the largest possible profit?b. Suppose the allocation problem in part (a) is solved by the method of Lagrange multipliers. What is the value of l that
In Exercises 1 through 22, find the critical points of the given functions and classify each as a relative maximum, a relative minimum, or a saddle point. f(x, y) = = X x² + y² + 4
This table lists the gross domestic product (GDP) figures for China (billions of yuan) for the period 2004–2009:a. Find the least-squares line y = mt + b for these data, where y is the GDP of China t years after 2004.b. Use the least-squares line found in part (a) to predict the GDP of China for
In Exercises 21 through 28, evaluate the partial derivatives fx(x, y) and fy(x, y) at the given point (x0, y0). f(x, y) = y 2x + y at (0, -1)
In Exercises 21 through 28, evaluate the partial derivatives fx(x, y) and fy(x, y) at the given point (x0, y0).f(x, y) = (x − 2y)2 + (y − 3x)2 + 5; at (0, −1)
In Exercises 17 through 24, find all critical points of the given function and use the second partials test to classify each as a relative maximum, a relative minimum, or a saddle point.f(x, y) = xe2x2+5xy+2y2
When x thousand dollars are spent on labor and y thousand on equipment, the output of a certain factory is Q units, whereSuppose $120,000 is available for labor and equipment.a. How should the money be allocated between labor and equipment to generate the largest possible output?b. Use the Lagrange
In Exercises 21 through 28, evaluate the partial derivatives fx(x, y) and fy(x, y) at the given point (x0, y0).f (x, y) = 3x2 − 7xy + 5y3 − 3(x + y) − 1; at (−2, 1)
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