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mathematics
calculus with applications

Calculus With Applications 12th Edition Margaret L. Lial - Solutions

Suppose the labor cost (in dollars) for manufacturing a precision camera can be approximated bywhere x is the number of hours required by a skilled craftsperson and y is the number of hours required by a semiskilled person. Find values of x and y that minimize the labor cost. Find the minimum labor
For the functions defined as follows, find all values of x and y such that both fx(x, y) = 0 and fy(x, y) = 0.ƒ(x, y) = 50 + 4x - 5y + x2 + y2 + xy
Find the volume under the given surface z = f(x, y) and above the rectangle with the given boundaries. 2 || ху (x² + y²)²³ 1 < x < 2, 1 sys4
Find the level curve at a production of 500 for the production functions in Exercises. Graph each level curve in the xy-plane.A study of the connection between immigration and the fiscal problems associated with the aging of the baby boom generation considered a production function of the form z =
An ice cream cone has a radius of approximately 1 in. and a height of approximately 4 in. By what factor does a change in the radius affect the volume compared with a change in the height?
Find fxx(x, y) and fxy(x, y). f(x, y) = 3x + y x - 1
A cylindrical can is to be made that will hold 250p in3 of candy. Find the dimensions of the can with minimum surface area.
For the function in Exercise 38, what is the effect on z of doubling x? Of doubling y? Of doubling both?Exercise 38Find the level curve at a production of 500 for the production functions in Exercises. Graph each level curve in the xy-plane.In their original paper, Cobb and Douglas estimated the
An ordinary 12-oz beer or soda can holds about 25 in3. Find the dimensions of a can with minimum surface area. Measure a can and see how close its dimensions are to the results you found.
The total cost (in dollars) to produce x units of electrical tape and y units of packing tape is given by C(x, y) = 2x2 + 2y2 - 3xy + 4x - 94y + 4200. Find the number of units of each kind of tape that should be produced so that the total cost is a minimum. Find the minimum total cost.
Although it is often true that a double integral can be evaluated by using either dx or dy first, sometimes one choice over the other makes the work easier. Evaluate the double integrals in Exercises in the easiest way possible. If xety xexy dx dy; 0≤x≤ 2,0 ≤ y ≤ 1 R
Find the volume under the given surface z = f(x, y) and above the rectangle with the given boundaries.z = ex+y; 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
The multiplier functioncompares the growth of an Individual Retirement Account (IRA) with the growth of the same deposit in a regular savings account. The function M depends on the three variables n, i, and t, where n represents the number of years an amount is left at interest, i represents the
If labor (x) costs $250 per unit, materials (y) cost $150 per unit, and capital (z) costs $75 per unit, write a function for total cost.
Find fxx(x, y) and fxy(x, y).ƒ(x, y) = 4x2e2y
A rectangular box with no top is to be built from 500 m2 of material. Find the dimensions of such a box that will enclose the maximum volume.
The total revenue (in hundreds of dollars) from the sale of x spas and y solar heaters is approximated by R(x, y) = 15 + 169x + 182y - 5x2 - 7y2 - 7xy. Find the number of each that should be sold to produce maximum revenue. Find the maximum revenue.
Although it is often true that a double integral can be evaluated by using either dx or dy first, sometimes one choice over the other makes the work easier. Evaluate the double integrals in Exercises in the easiest way possible. | 20tery 2x³e³ dx dy; 0≤x≤ 1,0 ≤ y ≤ 1 R
Find fx(x, y, z), fy(x, y, z), fz(x, y, z), and fyz(x, y, z) for the following.ƒ(x, y, z) = x4 + 2yz2 + z4
Find fxx(x, y) and fxy(x, y).ƒ(x, y) = yex2
A 1-lb cracker box has a volume of 185 in3. The end of the box is square. Find the dimensions of such a box that has minimum surface area.
The rate of heat loss (in watts) in harbor seal pups has been approximated bywhere m is the body mass of the pup (in kg), and T and A are the body core temperature and ambient water temperature, respectively (in °C). Find the heat loss for the following data.(a) Body mass = 21 kg; body core
The profit (in thousands of dollars) that Aunt Mildred’s Metalworks earns from producing x tons of steel and y tons of aluminum can be approximated by P(x, y) = 36xy - x3 - 8y3. Find the amounts of steel and aluminum that maximize the profit, and find the value of the maximum profit.
Find fx(x, y, z), fy(x, y, z), fz(x, y, z), and fyz(x, y, z) for the following.ƒ(x, y, z) = 6x3 - x2y2 + y5
Find fx(x, y, z), fy(x, y, z), fz(x, y, z), and fyz(x, y, z) for the following. f(x, y, z) = = 6x — 5у 4z + 5
What is the multiplier when money is invested for 40 years at 6% interest and the income tax rate is 28%? Which account grows faster?
Find fxx(x, y) and fxy(x, y).ƒ(x, y) = ln |2 - x2y|
A rectangular closed box is to be built at minimum cost to hold 125 m3. Since the cost will depend on the surface area, find the dimensions that will minimize the surface area of the box.
The probability that a three-person jury will make a correct decision is given bywhere 0 < a < 1 is the probability that the person is guilty of the crime, r is the probability that a given jury member will vote “guilty” when the defendant is indeed guilty of the crime, and s is the
The time (in hours) that a branch of Amalgamated Entities needs to spend to meet the quota set by the main office can be approximated by T(x, y) = x4 + 16y4 - 32xy + 40, where x represents how many thousands of dollars the factory spends on quality control and y represents how many thousands of
Find fx(x, y, z), fy(x, y, z), fz(x, y, z), and fyz(x, y, z) for the following. f(x, y, z) = 2x² + xy yz - 2
Find fxx(x, y) and fxy(x, y). ƒ(x, y) = ln |1 + 3xy2|
RD Enterprises is designing an open top box to transport 400 yd3 of material from one place to another in several trips. Two of the sides and bottom of the box can be made of a free material that the company has on hand, but only 8 yd2 of the material is available. Because of the hauling
The table below illustrates the dramatic increase in the number of transistors per square mm on computer chips since 1974.(a) To fit the data to a function of the form y = abt, where t is the number of years since 1970 and y is the number of transistors per mm2, we could take natural logarithms of
Find the dimensions that will minimize the surface area (and hence the cost) of a rectangular fish aquarium, open on top, with a volume of 32 ft3.
The surface area of a human (in square meters) has been approximated by A = 0.024265h0.3964 m0.5378, where h is the height (in cm) and m is the mass (in kg). Find A for the following data.(a) Height, 178 cm; mass, 72 kg(b) Height, 140 cm; mass, 65 kg(c) Height, 160 cm; mass, 70 kg
Find fx(x, y, z), fy(x, y, z), fz(x, y, z), and fyz(x, y, z) for the following. f(x, y, z) = ln |x² - 5xz² + y4|
An article titled “How Dinosaurs Ran” explains that the locomotion of different-sized animals can be compared when they have the same Froude number, defined aswhere v is the velocity, g is the acceleration of gravity (9.81 m per sec2), and l is the leg length (in meters).(a) One result
Find fx(x, y, z), fy(x, y, z), fz(x, y, z), and fyz(x, y, z) for the following. f(x, y, z)= In 8xy + 5yz - x³
In Exercises, find all points where the functions defined below have any relative extrema. Find any saddle points.z = x2 + y2 + 9x - 8y + 1
The probability that the majority of a three-person jury will convict a guilty person is given by the formula: P(r, s, t) = rs(1 - t) + (1 - r)st + r(1 - s)t + rst subject to the constraint that r + s + t = α, where r, s, and t represent each of the three jury members’ probability of reaching a
The process of frying food changes its quality, texture, and color. According to research done at the University of Saskatchewan, the total change in color E (which is measured in the form of energy as kJ/mol) of blanched potato strips can be estimated by the function E(t, T) = 436.16 - 10.57t -
According to research at the Great Swamp in New York, the percentage of fish that are intolerant to pollution can be estimated by the function P(W, R, A) = 48 - 2.43W - 1.81R - 1.22A, where W is the percentage of wetland, R is the percentage of residential area, and A is the percentage of
In Exercises, find all points where the functions defined below have any relative extrema. Find any saddle points. Z || 1 -x² + 2 1 2 + 2xy = 5x - 7y + 10 -
In Exercises, approximate the indicated derivative for each function by using the definition of the derivative with small values of h.ƒ(x, y) = (x + y/2)x+y/2(a) ƒx(1, 2)(b) ƒy(1, 2)
In tropical regions, dengue fever is a significant health problem that affects nearly 100 million people each year. Using data collected from the 2002 dengue epidemic in Colima, Mexico, researchers have estimated that the incidence I (number of new cases in a given year) of dengue can be predicted
In Exercises, approximate the indicated derivative for each function by using the definition of the derivative with small values of h.ƒ(x, y) = (x + y2)2x+y(a) ƒx(2, 1) (b) ƒy(2, 1)
Using data collected by the U.S. Forest Service, the annual number of deer-vehicle accidents for any given county in Ohio can be estimated by the function A(L, T, U, C) = 53.02 + 0.383L + 0.0015T + 0.0028 - 0.0003C, where A is the estimated number of accidents, L is the road length (in kilometers),
Suppose that the manufacturing cost of a laptop is approximated by M(x, y) = 45x2 + 40y2 - 20xy + 50, where x is the cost of processors and y is the cost of labor. Find the following.(a) My(4, 2)(b) Mx(3, 6)(c) (∂M/∂x)(2, 5)(d) (∂M/∂y)(6, 7)(e) Mxx(x, y) (f) Myy(x, y)(g) Mxy(x, y)
Use the region R with the indicated boundaries to evaluate each double integral. [Sc (5x + 8y) dy dx; 1 ≤ x ≤ 3,0 ≤ y ≤ x - 1 R
Researchers found that the transmission of the COVID-19 virus is affected by the air temperature and relative humidity according to the formula R = 3.968 - 0.0383T - 0.0224H, where R is the reproductive number (a quantity measuring the severity of infectiousness), T is the temperature in degrees
Suppose the production function of a company is given bywhere x represents units of labor and y represents units of capital. Find the following when 6 units of labor and 8 units of capital are used.(a) The marginal productivity of labor(b) The marginal productivity of capital P(x, y) = 250 Vx² +
A manufacturer estimates that production (in hundreds of units) is a function of the amounts x and y of labor and capital used, as follows.(a) Find the number of units produced when 16 units of labor and 81 units of capital are utilized.(b) Find and interpret ƒx(16, 81) and ƒy(16, 81).(c) What
In Exercises, find all points where the functions defined below have any relative extrema. Find any saddle points.ƒ(x, y) = 2x2 + 4xy + 4y2 - 3x + 5y - 15
The revenue from the sale of x units of a sedative and y units of an antibiotic is given by R(x, y) = 5x2 + 9y2 - 4xy. Suppose 9 units of sedative and 5 units of antibiotic are sold.(a) What is the approximate effect on revenue if 10 units of sedative and 5 units of antibiotic are sold?(b) What is
Pregnant sows tethered in stalls often show high levels of repetitive behavior, such as bar biting and chain chewing, indicating chronic stress. Researchers from Great Britain have developed a function that estimates the relationship between repetitive behavior, the behavior of sows in adjacent
Use the region R with the indicated boundaries to evaluate each double integral. || (2x (2x + 6y) dy dx; 2 ≤ x ≤ 4, 2 ≤ y ≤ 3x R
In Exercises, find all points where the functions defined below have any relative extrema. Find any saddle points.z = x3 + y2 + 2xy - 4x - 3y - 2
Refer to the figure for Exercise 53. Assume L, W, and H are in feet. Write a function in terms of L, W, and H that gives the total area of the material required to build the box.Exercise 53Extra postage is charged for parcels sent by U.S. mail that are more than 84 in. in length and girth combined.
Extra postage is charged for parcels sent by U.S. mail that are more than 84 in. in length and girth combined. (Girth is the distance around the parcel perpendicular to its length. See the figure.) Express the combined length and girth as a function of L, W, and H. H L W Length FIRST CLASS Girth
Researchers have estimated the maximum life span (in years) for various species of mammals according to the formula L(E, P) = 23E0.6P-0.267, where E is the average brain mass and P is the average body mass (both in g). Find L for the following species. Do these values seem reasonable?(a) Black rat,
Use the region R with the indicated boundaries to evaluate each double integral. R 1 -dy dx; 1 ≤ x ≤ 2,0 ≤ y ≤ x - 1 X
Use the region R with the indicated boundaries to evaluate each double integral. R (4-4x²) dy dx; 0≤x≤ 1,0 ≤ y ≤ 2 - 2x
In Exercises, find all points where the functions defined below have any relative extrema. Find any saddle points.ƒ(x, y) = 7x2 + y2 - 3x + 6y - 5xy
Use the region R with the indicated boundaries to evaluate each double integral. ffer R ex/² dx dy; 1 ≤ y ≤ 2,0 ≤ x ≤ y²
Describe the different types of points that might occur when ƒx(x, y) = ƒy(x, y) = 0.
The holes cut in a roof for vent pipes require elliptical templates. A formula for determining the length of the major axis of the ellipse is given bywhere D is the (outside) diameter of the pipe and H is the “rise” of the roof per D units of “run”; that is, the slope of the roof is H/D.
The production function z for the United States was once estimated as z = Cx0.75y0.25, where x stands for the amount of labor and y the amount of capital and C is a constant. Find the marginal productivity of labor and of capital.
Use the region R with the indicated boundaries to evaluate each double integral. [[ (x² − y) dy dx; −1 ≤ x ≤ 1, −x² ≤ y ≤r² - R
The total cost of producing M units of a product, the optimal number of units in each batch is given bywhere ƒ is the fixed setup cost to manufacture the product and k is the cost of storing one unit of the product for one year. (a) InkWe’R produces 40,000 gallons a year of laser toner. It
Use Lagrange multipliers to find the extrema of the functions defined in Exercises.ƒ(x, y) = x2y; x + y = 4
A similar production function for New South Wales in Australia is z = Cx0.65y0.35, with x, y, z, and C as in Exercise 53. Find the marginal productivity of labor and of capital.Exercise 53The production function z for the United States was once estimated as z = Cx0.75y0.25, where x stands for the
Use Lagrange multipliers to find the extrema of the functions defined in Exercises.ƒ(x, y) = x2 + y2; x = y - 6
Use the region R with the indicated boundaries to evaluate each double integral. [x³y dy dx; R bounded by y = x², y = 2x R
Use the region R with the indicated boundaries to evaluate each double integral. If ; dy y R -dy dx; R bounded by y = x, y x, y 1 = , x = X 2
A manufacturer of automobile batteries estimates that his total production (in thousands of units) is given by ƒ(x, y) = 3x1/3y2/3, where x is the number of units of labor and y is the number of units of capital utilized.(a) Find and interpret ƒx(64, 125) and ƒy(64, 125) if the current
Find positive numbers x and y, whose sum is 80, such that x2y is maximized.
A study defined administrative intensity as the number of administrative personnel A per production personnel X, or I = A/X. To seek the optimum administrative intensity for an industry, the following production function was defined: Q = CSσ AαKβXγ, where S is the number of distinct
The rate of heat loss (in watts) in harbor seal pups has been approximated bywhere m is the body mass of the pup (in kg), and T and A are the body core temperature and ambient water temperature, respectively (in°C). Find the approximate change in heat loss under the following conditions.(a) The
Find positive numbers x and y, whose sum is 75, such that xy2 is maximized.
Use the total differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places. V5.12 + 12.05²
Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration, as in Exercises - In 2-2 1 - dx dy In x
Evaluate dz using the given information. Z || x + 5y X -2y x = 1, y = -2, dx = -0.04, dy = 0.02
Use the region R with the indicated boundaries to evaluate each double integral. R 2y/x dy dx; R bounded by y = x², y = 0, x = 2
Notice in the previous two exercises that we specified that x and y must be positive numbers. Does a maximum exist without this requirement? Explain why or why not.
Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration, as in Exercises -2.1 en 0 Jy/2 dx dy
Evaluate dz using the given information.z = 6x2 - 7y2 + 4xy; x = 3, y = -1, dx = 0.03, dy = 0.01
According to the Fick Principle, the quantity of blood pumped through the lungs depends on the following variables (in milliliters):b = quantity of oxygen used by the body in one minutea = quantity of oxygen per liter of blood that has just gone through the lungsv = quantity of oxygen per liter of
A weight-loss counselor has prepared a program of diet and exercise for a client. If the client sticks to the program, the weight loss that can be expected (in pounds per week) is given bywhere c is the average daily calorie intake for the week and n is the number of 40-minute aerobic workouts per
Use the total differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places. V4.06 0.04
The idea of the average value of a function, discussed earlier for functions of the form y = f (x), can be extended to functions of more than one independent variable. For a function z = f (x, y), he average value of f over a region R is defined aswhere A is the area of the region R. Find the
The surface area of a human (in square meters) has been approximated by A = 0.024265h0.3964 m0.5378, where h is the height (in cm) and m is the mass (in kg).(a) Find the approximate change in surface area when the mass changes from 72 kg to 73 kg, while the height remains at 180 cm.(b) Find
Give an example of a region that cannot be expressed by either of the forms shown in Figure 6. 0 a y=h(x) y = g(x) g(x) = h(x) for all x in [a, b] (a) d 0 Figure 6 x = g(y) x=h(y) g(y) h(y) for all y in [c, d] (b) X
An alternative to the body mass index (BMI) for estimating the health risks of obesity is A Body Shape Index (ABSI), given bywhere w and h are the waist and height, respectively, in meters, and b is the BMI. (See the definition of BMI in the previous exercise.) The National Heart, Lung and Blood
Evaluate each iterated integral. -35 [[²2² (2x + 6y + y²) dy dx
The idea of the average value of a function, discussed earlier for functions of the form y = f (x), can be extended to functions of more than one independent variable. For a function z = f (x, y), he average value of f over a region R is defined aswhere A is the area of the region R. Find the
Recall from the Volume and Average Value section in the previous chapter that volume could be found with a single integral. In this section volume is found using a double integral. Explain when volume can be found with a single integral and when a double integral is needed.
Evaluate each iterated integral. 24 10 Jo (x²y² + 5x) dx dy
In 1908, J. Haldane constructed diving tables that provide a relationship between the water pressure on body tissues for various water depths and dive times. The tables were successfully used by divers to virtually eliminate decompression sickness. The pressure in atmospheres for a no-stop dive is
Evaluate each iterated integral. 4 5 ST. 13 J2 V6x + 3y dx dy

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