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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
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
In 1941, explorers Paul Siple and Charles Passel discovered that the amount of heat lost when an object is exposed to cold air depends on both the temperature of the air and the velocity of the wind. They developed the Wind Chill Index as a way to measure the danger of frostbite while doing outdoor
Evaluate each iterated integral. 25 1 J3 e2x-7y dx dy
As we saw in the previous section, 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 the approximate change in life span
The manufacturer of a fruit juice drink has decided to try innovative packaging in order to revitalize sagging sales. The fruit juice drink is to be packaged in containers in the shape of tetrahedra in which three edges are perpendicular, as shown in the figure below. Two of the perpendicular edges
The Heat Index chart in the next column shows the heat index, which combines the effects of temperature with humidity to give a measure of the apparent temperature, or how hot it feels to the body. For example, when the outside temperature is 90°F and the relative humidity is 40%, then the
Evaluate each iterated integral. 44 - dx dy J2 J2 y
Find each double integral over the region R with boundaries as indicated. [[ (x² + 2y²) dx dy; 0≤x≤ 5,0 ≤ y ≤ 2 R
Evaluate each iterated integral. 2-2 1 - dx dy X
A company’s total cost for operating its two warehouses is C(x, y) = 1/9x2 + 2x + y2 + 5y + 100 dollars, where x represents the number of units stored at the first warehouse and y represents the number of units stored at the second. Find the average cost to store a unit if the first warehouse has
A production function is given by P(x, y) = 500x0.2y0.8, where x is the number of units of labor and y is the number of units of capital. Find the average production level if x varies from 10 to 50 and y from 20 to 40.
The gravitational attraction F on a body a distance r from the center of Earth, where r is greater than the radius of Earth, is a function of its mass m and the distance r as follows:where R is the radius of Earth and g is the force of gravity— about 32 feet per second per second (ft per
The table at the bottom of this page above accompanies the Voldyne® 5000 Volumetric Exerciser. The table gives the typical lung capacity (in milliliters) for women of various ages and heights. Based on the chart, it is possible to conclude that the partial derivative of the lung capacity with
Find each double integral over the region R with boundaries as indicated. R V2x + y dx dy; 1 ≤ x ≤ 3, 2 ≤ y ≤ 5
Fitts’s law is used to estimate the amount of time it takes for a person, using his or her arm, to pick up a light object, move it, and then place it in a designated target area. Mathematically, Fitts’s law for a particular individual is given bywhere s is the distance (in feet) the object is
The profit (in dollars) from selling x units of one product and y units of a second product is P = -(x - 100)2 - (y - 50)2 + 2000. The weekly sales for the first product vary from 100 units to 150 units, and the weekly sales for the second product vary from 40 units to 80 units. Estimate average
In 1931, Albert Einstein developed the following formula for the sum of two velocities, x and y:where x and y are in miles per second and c represents the speed of light, 186,282 miles per second.(a) Suppose that, relative to a stationary observer, a new super space shuttle is capable of traveling
Find each double integral over the region R with boundaries as indicated. R yey² + dx dy; 0≤x≤ 1,0 ≤ y ≤ 1
A developmental mathematics instructor at a large university has determined that a student’s probability of success in the university’s pass/fail remedial algebra course is a function of s, n, and a, where s is the student’s score on the departmental placement exam, n is the number of
Find each double integral over the region R with boundaries as indicated. R Vy +xdx dy; 0
A company sells two products. The demand functions of the products are given by q1 = 300 - 2p1 and q2 = 500 - 1.2p2, where q1 units of the first product are demanded at price p1 and q2 units of the second product are demanded at price p2. The total revenue will be given by R = q1 p1 + q2 p2. Find
Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration. -2x JJ 0 xy dy dx
We saw that 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
Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration. ·x [C 1x² x³y dy dx
Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration. -22x² 1 J2 y dy dx
We saw that 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. If the amount of steel produced varies from 0 to 8 tons, and the amount of aluminum produced varies from 0
Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration. Vy 0 Jy x dx dy
Find the volume under the given surface z = f(x, y) and above the given rectangle.z = x + 8y + 4; 0 ≤ x ≤ 3, 1 ≤ y ≤ 2
Find the volume under the given surface z = f(x, y) and above the given rectangle.z = x2 + y2; 3 ≤ x ≤ 5, 2 ≤ y ≤ 4
Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration. 1 Ch Jo Jx/2 y² + 1 dy dx
Use the region R, with boundaries as indicated, to evaluate the given double integral. || ₁² (2-12 (2 - x² - y²) dy dx; 0 ≤ x ≤1, x² ≤ y ≤ x R
The charge (in dollars) for painting a sports car is given by C(x, y) = 4x2 + 5y2 - 4xy + √x, where x is the number of hours of labor needed and y is the number of gallons of paint and sealant used. Find the following.(a) The charge for 10 hours and 5 gal of paint and sealant(b) The charge for 15
The manufacturing cost (in dollars) for a certain computer is given by c(x, y) = 2x + y2 + 4xy + 25, where x is the memory capacity of the computer in gigabytes (GB) and y is the number of hours of labor required. For 640 GB and 6 hours of labor, find the following.(a) The approximate change in
The production function z for one country is z = x0.7y0.3, where x represents the amount of labor and y the amount of capital. Find the marginal productivity of the following.(a) Labor (b) Capital
The cost (in dollars) to manufacture x solar cells and y solar collectors is c(x, y) = x2 + 5y2 + 4xy - 70x - 164y + 1800.(a) Find values of x and y that produce minimum total cost.(b) Find the minimum total cost.
Maximize each of the following utility functions, with the cost of each commodity and total amount available to spend given.ƒ(x, y) = xy3, cost of a unit of x is $2, cost of a unit of y is $4, and $80 is available.
Maximize each of the following utility functions, with the cost of each commodity and total amount available to spend given.ƒ(x, y) = x5y2, cost of a unit of x is $10, cost of a unit of y is $6, and $42 is available.
The cost (in dollars) to produce x satellite receiving dishes and y transmitters is given by C(1x, y) = 100 ln(x2 + y) + exy/20. Production schedules now call for 15 receiving dishes and 9 transmitters. Use differentials to approximate the change in costs if 1 more dish and 1 fewer transmitter are
Approximate the volume of material needed to manufacture a cone of radius 2 cm, height 8 cm, and wall thickness 0.21 cm.
Exercises require both the trapezoidal rule and Simpson’s rule. They can be worked without calculator programs if such programs are not available, although they require more calculation than the other problems in this exercise set.The difference between the true value of an integral and the value
In the study of bioavailability in pharmacy, a drug is given to a patient. The level of concentration of the drug is then measured periodically, producing blood level curves such as the ones shown in the figure.The areas under the curves give the total amount of the drug available to the patient
In Exercises, find each indefinite integral. | x2 dx (x³ + 5)4
The shown below amount of property damage (in dollars) due to automobile accidents in California in recent years. In this exercise we are interested in the total amount of property damage due to automobile accidents over the 8-year period from mid-2009 to mid-2017 using the data for the 9 years.(a)
The average value of a function on an interval is an indefinite integral.Determine whether each statement is true or false, and explain why.
The average value of the function ƒ(x) = 2x2 + 3 on [1, 4] is given byDetermine whether each of the following statements is true or false, and explain why. 7 3 -4 1 π(2x² + 3)² dx.
Column integration is another way of doing integration by parts.Determine whether each statement is true or false, and explain why.
If an integral from -∞ to ∞ is written as the sum of two integrals, one from -∞ to c and the other from c to ∞, the integral is convergent so long as one of the two integrals in the sum is convergent.Determine whether each statement is true or false, and explain why.
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 1000
Use integration by parts to find the integrals in Exercises. xex dx
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = x, y = 0, x = 0, x = 3
Determine whether each improper integral converges or diverges, and find the value of each that converges. J3 1 2 X dx
The volume of the solid formed by revolving the function ƒ(x) = x + 4 about the x-axis on the interval [-4, 5] is given byDetermine whether each of the following statements is true or false, and explain why. -5 -4 T(x + 4)² dx.
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 2000
The volume of the solid formed by revolving the function ƒ(x) = √x2 + 1 about the x-axis on the interval [1, 2] is given byDetermine whether each of the following statements is true or false, and explain why. 2 TV x² + 1 dx.
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 300
Use integration by parts to find the integrals in Exercises f (x (x + 6)e* dx
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = 3x, y = 0, x = 0, x = 2
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 500
If ƒ(t) = 1000e0.05t represents the rate of flow of money for a vending machine over the first five years of income, then the total money flow for that time period is given byDetermine whether each of the following statements is true or false, and explain why. -5 Jo 1000e0.05t dt.
Use integration by parts to find the integrals in Exercises. [ (4x (4x − 12)e-8x dx
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = 2x + 1, y = 0, x = 0, x = 4
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = x - 4, y = 0, x = 4, x = 10
Use integration by parts to find the integrals in Exercises. [ (6x (6x + 3)e-2x dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. J27 SW/N dx
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 400e0.03t
If a company expects an annual flow of money during the next five years to be ƒ(t) = 1000e0.05t, the present value of this income, assuming an annual interest rate of 4.5% compounded continuously is given byDetermine whether each of the following statements is true or false, and explain why.
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves. f(x) = 1 x + 2, y = 0, x = 1, x = 3 3
Use integration by parts to find the integrals in Exercises. /.x x ln x dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. -1 -00 ~/m²/₂ X dx
Determine whether each of the following statements is true or false, and explain why. 00 xe 2x dx = lim C-0 -00 C xe-2x dx C
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves. f(x) : = -x + 4, y = 0,x = 0,x = 5 2
Use integration by parts to find the integrals in Exercises. J₁ x³ ln x dx
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 800e0.05t
Use integration by parts to find the integrals in Exercises. 2x + 1 ox dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. -4 3 -00 x 4 dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. 1 .1.0001 X dx
Describe the type of integral for which integration by parts is useful.
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 5000e-0.01t
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = √x , y = 0, x = 1, x = 4
Use integration by parts to find the integrals in Exercises. -3 3 - x 3ex - dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. 00 1 .0.999 Xx dx
Use integration by parts to find the integrals in Exercises. -9 In 3x dx
Compare finding the average value of a function with finding the average of n numbers.
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 1000e-0.01t
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = √x + 5, y = 0, x = 1, x = 3
Use integration by parts to find the integrals in Exercises. -2 In 5x dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. Lax -00 -3 (x - 2)-³ dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. -10 -00 -2 x² dx
Find each integral, using techniques from this or the previous chapter. 3x Vx - 2 dx
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = √2x + 1, y = 0, x = 1, x = 4
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 25t
What is an improper integral? Explain why improper integrals must be treated in a special way.
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 50t
Find each integral, using techniques from this or the previous chapter. xe do
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = √4x + 2, y = 0, x = 0, x = 2
Determine whether each improper integral converges or diverges, and find the value of each that converges. 00- -8/3 dx E/8_X I-
Determine whether each improper integral converges or diverges, and find the value of each that converges. .-27 x-5/3 dx
Each of the functions in Exercises represents the rate of flow of money in dollars per year. Assume a 10-year period at 8% compounded continuously and find the following: (a) The present value; (b) The accumulated amount of money flow at T = 10.ƒ(t) = 0.01t + 100
Find the volume of the solid of revolution formed by rotating about the x-axis each region bounded by the given curves.ƒ(x) = ex, y = 0, x = 0, x = 2
Find the area between y = (x - 2)ex and the x-axis from x = 2 to x = 4.
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