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mathematics
calculus

Questions and Answers of Calculus

Suppose the Lorenz curve for the distribution of income of a certain country is given by Y=xe x-1 Find the Gini coefficient of income.
Suppose the income from an Internet access business is a continuous income stream with annual rate of flow given by f (t) = 100te -0.1t in thousands of dollars per year. Find the total income over
The number of millions of White non-Hispanic individuals in the U.S. civilian non-institutional population age 16 and older for selected years from 1980 and projected to 2050 can be modeled by P(x) =
The percent of natural gas in the United States extracted from shale rock from 2012 and projected to 2040 is given by the function y=-0.843+13.8 ln x, where x is the is the number of years after 2010
Evaluate the improper integrals that converge.1.2.3.4.5.
For what value of c does f∞0 c/e 0.5t dt = 1?
For what value of c does f∞10 c/x3 dx = 1?
Find the area, if it exists, of the region under the graph of y f (x) and to the right of x 1.1.2.3.4.
Show that the function is a probability density function. f (x) = { 200/x3 of x ≥ 10 0 otherwise
For what value of c is the function f(x) = { c/x2 if x ≥ 1 0 otherwise a probability density function?
For what value of c is the function f(x) = { c/x3 if x ≥ 1 0 otherwise a probability density function?
Find the value of c so that f(x) = { ce -x/4 x ≥ 0 0 x ˂ 0 is a probability density function.
Find the value of c (in terms of k) so that f(x) = { ce -kx x ≥ 0 0 x ˂ 0 is a probability density function.
Find the mean of the probability distribution if the probability density function is f(x) = { 200/x3x ≥ 10 0 otherwise
Find the mean of the probability distribution if the probability density function is f(x) = { c/x4 if x ≥ 10 0 otherwise
Find the area below the graph of y = f(x) and above the x-axis for f (x) = 24xe-3x. Use the graph of y = f (x) to find the interval for which f (x) ≥ 0 and the graph of the integral of f (x) over
Find the area below the graph of y = f (x) and above the x-axis for f (x) = x2e - x and x ≥ 0. Use the graph of the integral of f (x) over this interval to find the area.
Suppose that a continuous income stream has an annual rate of flow at time t given by f (t) =A, where A is a constant. If the interest rate is r (as a decimal, r ˃ 0), compounded continuously, show
Suppose that a donor wishes to provide a cash gift to a hospital that will generate a continuous income stream with an annual rate of flow at time t given by f (t)= $20,000 per year. If the annual
Suppose that a business provides a continuous income stream with an annual rate of flow at time t given by f (t) =120e0.04t in thousands of dollars per year. If the interest rate is 9% compounded
Suppose that the output of the machinery in a factory can be considered as a continuous income stream with annual rate of flow at time t given by f (t)=450e -0.09t (in thousands of dollars per year).
A business has a continuous income stream with an annual rate of flow at time t given by f (t)=56,000e0.02t (dollars per year). If the interest rate is 10% compounded continuously, find the capital
Suppose that a business provides a continuous income stream with an annual rate of flow at time t given by f (t) = 10,800e0.06t (dollars per year). If money is worth 12% compounded continuously, find
In a manufacturing process involving several machines, the average down time t (in hours) for a machine that needs repair has the probability density function f (t) = 0.5e -0.5t t ≥ 0 Find the
The duration t (in minutes) of customer service calls received by a certain company is given by the probability density function f (t) = 0.4e -0.4t t ≥ 0 Find the probability that a call selected
The probability density function for the life span of an electronics part is f (t)=0.08e -0.08t, where t is the number of months in service. Find the probability that any given part of this type
A transmission repair firm that wants to offer a lifetime warranty on its repairs has determined that the probability density function for transmission failure after repair is f (t)=0.3e -0.3t, where
Suppose that the rate at which a nuclear power plant produces radioactive waste is proportional to the number of years it has been operating, according to f (t)500t (in pounds per year). suppose
For each interval [a, b] and value of n given in Problems 1-5, find h and the values of x0, x1,........ xn.1.2. 3. 4. 5.
In Problems below approximate each integral by(a) the Trapezoidal Rule.(b) Simpson's Rule.Use n 4 and round answers to 3 decimal places.1.2.3.4.
Use the table of values given in each of Problems below to approximate f ba f(x) dx. Use Simpson's Rule whenever n is even; otherwise, use the Trapezoidal Rule. Round answers to 1 decimal
Suppose that the production from an assembly line can be considered as a continuous income stream with annual rate of flow given by f(t) = 100/e0.lt/ t+1 (in thousands of dollars per year) Use
Suppose that the rate of flow of a continuous income stream is given by f (t) =500t (in thousands of dollars per year). If money is worth 7% compounded continuously, then the present value of this
Suppose that a company's total cost (in dollars) of producing x items is given by C(x) = (x2+1)3/2 1000.Use the Trapezoidal Rule with n=3 to approximate the average total cost for the production of
Suppose that the demand for q units of a certain product at $p per unit is given by p = 850+ 100/q2+1Use Simpson's Rule with n6 to approximate the average price as demand ranges from 3 to 9 items.
Use the supply and demand schedules in Problems 27 and 28, with p in dollars and x as the number of units.Use Simpson's Rule to approximate the producer's surplus at market equilibrium. Note that
Use the supply and demand schedules in Problems 27 and 28, with p in dollars and x as the number of units.Use Simpson's Rule to approximate the consumer's surplus at market equilibrium. Round all
Suppose that the rate of production of a product (in units per week) is measured at the end of each of the first 5 weeks after start-up, and the data in the table are obtained.Approximate the total
The manufacturer of a medicine wants to test how a new 300-milligram capsule is released into the bloodstream. After a volunteer is given a capsule, blood samples are drawn every half hour, and the
If the Lorenz curves for years a and b are given by La(x) and Lb(x), respectively, then from year a to year b, the change in the Gini coefficient (Gbâ”€ Ga) is given by 2 f10 [la(x) - Lb (x)] dx
If the Lorenz curves for years a and b are given by La(x) and Lb(x), respectively, then from year a to year b, the change in the Gini coefficient (Gbâ”€ Ga) is given by2 f10 [la(x) - Lb (x)] dx
Suppose that the presence of phosphates in certain waste products dumped into a lake promotes the growth of algae. Rampant growth of algae affects the oxygen supply in the water, so an environmental
A land developer is planning to dig a small lake and build a group of homes around it. To estimate the cost of the project, the area of the lake 848 CHAPTER 13 Definite Integrals: Techniques of
For each integral in Problems below do the following.(a) Approximate its value by using the Trapezoidal Rule.(b) Approximate its value by using Simpson's Rule.(c) Find its exact value by
Find the area between the curves in below. 1. y=x2-3x+2 and y=x2+4 from x=0 to x=5 2. Y=x2 and y=4x+5 3. Y=x3 and y=x from x=-1 to x=0 4. y=x3-1 and y=x-1
Evaluate the integrals in problems below, using the formulas in Table 13.2.1.2. 3. 4.
In below, use integration by parts to evaluate.1.2. 3. 4.
Use 6 subintervals of the same size to approximate the area under the graph of y=3x2 from x=0 to x=1. Use the right-hand endpoints of the subintervals to find the heights of the rectangles.
Evaluate the improper integrals in Problems below.1.2. 3. 4.
Evaluate f 31 2/x3 dx (a) Exactly. (b) by using the Trapezoidal Rule with n=4 (to 3 decimal places). (c)s by using Simpson's Rule with n=4 (to 3 decimal places).
Use the Trapezoidal Rule with n=5 to approximate f 10 4 dx / x2+1 Round your answer to 3 decimal places
Use the table that follows to approximate f 221 f (x) dx by using Simpson's Rule. Round your answer to 1 decimal place
Maintenance costs for buildings increase as the buildings age. If the rate of increase in maintenance costs for a building is M ' (t) = 14,000 / √ t + 16 Where M is in dollars and t is time in
Use rectangles to find the exact area under the graph of y=3x2 from x=0 to x=1. Use n equal subintervals.
Suppose the total income in dollars from a machine is given by I=50e0.2t, 0 ≤ t ≤ 4, t in hours Find the average income over this 4-hour period.
In 1969, after the "Great Society" initiatives of the Johnson administration, the Lorenz curve for the U.S. income distribution was L(x)=x2.1936. In 2000, after the stock market's historic 10-year
The demand function for a product under pure competition is p = √ 64-4x, and the supply function is p=x-1, where x is the number of units and p is in dollars. (a) Find the market equilibrium. (b)
Find the producer's surplus at market equilibrium for Problem 44. The demand function for a product under pure competition is p = √ 64 - 4x, and the supply function is p=x-1, where x is the number
Suppose that a machine's production is considered a continuous income stream with an annual rate of flow at time t given by f (t)=150e-0.2t in thousands of dollars per year. Money is worth 8%,
Suppose the cost function for x units of a product is given by C(x) = √ 40,000 + x2 dollars. Find the average cost over the first 150 units.
Suppose the supply function for x units of a certain lamp is given by p = 0.02x + 50.01 - 10 / √ x2 + 1 Where p is in dollars find the producer's surplus if the equilibrium price is $70 and the
Suppose the present value of a continuous income stream over the next 5 years is given by P = 9000 f50 te -0.08t -dt, P in dollars, t in years
If the marginal cost for x units of a product is MC=3+60(x+1) ln (x+1) dollars per unit and if the fixed cost is $2000, find the total cost function.
Find the probability that a phone lasts more than 1 year if the probability density function for its life expectancy is given by f(x) = { 1.42 -1.4x x ≥ 0 0 x ˂ 0
Find the capital value of a business if its income is considered a continuous income stream with annual rate of flow given by f (t)=120e0.03t in thousands of dollars per year, and the current
Suppose that a continuous income stream has an annual rate of flow f (t) =100e-0.01t 2(in thousands of dollars per year). Use Simpson's Rule with n=4 to approximate the total income from this stream
A company has the data shown in the table from the sale of its product.If x represents hundreds of units and revenue R is in hundreds of dollars, approximate the total revenue from the sale of 1000
Find the area between the graph of y=x3-4x+5 and the x-axis from x=1 to x=3.
Evaluate the integrals in below.1.2. 3. 4. 5.
Give the domain of each function in Problems 1-8. 1. z = x2 + y2 2. z = 4x - 3y 3. z = 4x - 3 / y 4. z = x + y2 / √x
In Problems 1-3, evaluate each function as indicated.1.Find q1(40,35). 2. Find q1(50,10). 3. z(x, y) = xex+y; find z(3, - 3).
The future value S of an investment earning 6% compounded continuously is a function of the principal P and the length of time t that the principal has been invested. It is given by S =f (P, t) =
If $100,000 is borrowed to purchase a home, then the monthly payment R is a function of the interest rate i (expressed as a percent) and the number of years n before the mortgage is paid. It is given
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 = f (K, M, h) = √12KM / h where K is the
Suppose that a gas satisfies the universal gas law, V = nRT/P, with n equal to 10 moles of the gas and R, the universal gas constant, equal to 0.082054. What is V if T = 10 K (kelvins, the units in
At the Dallas-Fort Worth Airport, the average daily temperatures and humidities for July are Maximum: 97.8°F with 44% humidity Minimum: 74.7°F with 80% humidity** Calculate the Summer Simmer Index
In Orlando, Florida, the following represent the average daily temperatures and humidities for August. Maximum: 91.6°F with 60% humidity Minimum: 73.4°F with 92% humidity** Calculate the Summer
The tables show that a monthly mortgage payment, R, is a function of the amount financed, A, in thousands of dollars; the duration of the loan, n, in years; and the annual interest rate, r, as a
Wind and cold temperatures combine to make the air temperature feel colder than it actually is. This combination is reported as wind chill. The table shows the latest wind chill calculations from the
Suppose that the utility function for two goods X and Y is given by U = xy2, and a consumer purchases 9 units of X and 6 units of Y. (a) If the consumer purchases 9 units of Y, how many units of X
Suppose that an indifference curve for two goods, X and Y, has the equation xy =400. (a) If 25 units of X are purchased, how many units of Y must be purchased to remain on this indifference
Suppose that a company's production for Q units of its product is given by the Cobb-Douglas production function Q = 30K1/4L3/4 where K is dollars of capital investment and L is labor hours. (a) Find
Suppose that a company's production for Q units of its product is given by the Cobb-Douglas production function Q = 70K2/3L1/3 where K is dollars of capital investment and L is labor hours. (a) Find
Suppose that the number of units of a good produced, z, is given by z = 20xy, where x is the number of machines working properly and y is the average number of work-hours per machine. Find the
The Kirk Kelly Kandy Company makes two kinds of candy, Kisses and Kreams. The profit, in dollars, for the company is given by P(x, y) = 10x + 6.4y -0.001x2 - 0.025y2 where x is the number of pounds
1. The cost per day to society of an epidemic is C(x, y) = 20x +200y where C is in dollars, x is the number of people infected on a given day, and y is the number of people who die on a given day. If
In Problems 1-3, evaluate the functions at the given values of the independent variables. 1. z = x3 + 4xy + y2; x = 1, y = - 1 2. z = 4x2 -3xy3; x = 2, y = 2 3. z = x -y / x + y; x = 4, y = - 1
If z = x4 - 5x2 + 6x + 3y3 - 5y + 7, find ∂z / ∂x and ∂z / ∂y.
If C(x, y) = 1000 - 4x + xy2, find ∂C/∂x and ∂C/∂y.
If Q(s, t) = 2s - 3t / s2 + t2, find ∂Q/∂s and ∂Q/∂t.
If q = 5p1 + 4p2 / p1 + p2, find ∂q/∂p1 and ∂q/∂p2.
If z = e2x + y ln x, find zx and zy.
If z = ln (1 + x2y) - ye-x, find zx and zy.
If f (x, y) = ln (xy + 1), find ∂f/∂x and ∂f/∂y.
If f (x, y) = 100exy, find ∂f/∂x and ∂f/∂y.
Find the partial derivative of f (x, y) = 4x3 - 5xy + y2 with respect to x at the point (1, 2, -2).
Find the partial derivative of f (x, y) = 3x2 +4x + 6xy with respect to y at x = 2, y = - 1.
1. Find the slope of the tangent in the positive x-direction to the surface z = 5x3 - 4xy at the point (1, 2, -3). 2. Find the slope of the tangent in the positive y-direction to the surface z = x3 -
If z = x5 - 6x + 4y4 - y2, find ∂z / ∂x and ∂z/∂y.
Find the slope of the tangent in the positive x-direction to the surface z = 2xy + ln (4x + 3y) at (1, - 1, - 2).

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