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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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; = +
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
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
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,
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
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
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
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
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 +
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
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
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
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
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
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
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,
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
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) =
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,
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
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.
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 +
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
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
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 +
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
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
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
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 +
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
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
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
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,
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
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
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
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
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) =
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
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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