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mathematics
calculus

Questions and Answers of Calculus

The rate of growth of world population can be modeled by dN/dt =N0rert, r < 1 where t is the time in years from the present and N0 and r are constants. What function describes world population if the
Compound interest If $P is invested for n years at 10% compounded continuously, the rate at which the future value is growing is dS / dn = 0.1Pe0.1n (a) What function describes the future value at
When an object is moved from one environment to another, its temperature T changes at a rate given by dT / dt = kCekt where t is the time in the new environment (in hours), C is the temperature
The rate at which blood pressure decreases in the aorta of a normal adult after a heartbeat iswhere t is time in seconds. (a) What function describes the blood pressure in the aorta if p = 95 when t
A store finds that its sales decline after the end of an advertising campaign, with its daily sales for the period declining at the rate S'(t)= - 1477.8e-0.2t, 0 ≤ t ≤ 35, where t is the number
Suppose the rate of change of the expected life span l at birth of people born in the United States can be modeled bywhere t is the number of years past 1920. (a) Use integration and the data point
Suppose the rate of change of the percent P of U.S. households with cable/satellite TV can be modeled bywhere t is the number of years past 1975. (a) Use integration and the data point for 2010 to
The Social Security Administration makes projections about the consumer price index (CPI) in order to understand the effects of inflation on Social Security benefits and to plan for cost-of-living
The following table shows the U.S. average annual wage in thousands of dollars for selected years from 2012 and projected to 2050. Suppose the rate of change of the U.S. average annual wage can be
If the daily marginal cost for a product is MC = 2x + 100, with fixed costs amounting to $200, find the total cost function for each day. Cost, revenue, and profit are in dollars and x is the number
Suppose that the marginal cost for a product is MC = 60√x + 1 and its fixed cost is $340.00. If the marginal revenue for the product is MR = 80x, find the profit or loss from production and sale
The average cost of a product changes at the rate C̅(x) = - 6x-2 + 1 / 6 and the average cost of 6 units is $10.00. (a) Find the average cost function. (b) Find the average cost of 12 units. Cost,
The average cost of a product changes at the rateand the average cost of 10 units is $20.00. (a) Find the average cost function. (b) Find the average cost of 20 units. Cost, revenue, and profit are
Suppose that marginal cost for a certain product is given by MC = 1.05(x + 180)0.05 and marginal revenue is given by MR = (1 / √0.5x + 4) + 2.8, where x is in thousands of units and both revenue
Suppose that the marginal cost for a certain product is given by MC = 1.02(x + 200)0.02 and marginal revenue is given by MR = (2 / √4x + 1) + 1.75, where x is in thousands of units and revenue and
If consumption is $7 trillion when disposable income is $0 and if the marginal propensity to consume is 0.80, find the national consumption function (in trillions of dollars).
If national consumption is $9 trillion when income is $0 and if the marginal propensity to consume is 0.30, what is consumption when disposable income is $20 trillion?
If consumption is $8 trillion when income is $0 and if the marginal propensity to consume is dC / dy = 0.3 + 0.2 / √y find the national consumption function.
If consumption is $5 trillion when disposable income is $0 and if the marginal propensity to consume is dC/dy = 0.4 + 0.3/√y find the national consumption function.
If consumption is $6 trillion when disposable income is $0 and if the marginal propensity to consume is dC/dy = 1/√y + 1 + 0.4 find the national consumption function.
If the monthly marginal cost for a product is MC = x + 30 and the related fixed costs are $5000, find the total cost function for the month. Cost, revenue, and profit are in dollars and x is the
If consumption is $5.8 trillion when disposable income is $0 and if the marginal propensity to consume is dC/dy = 1/√2y + 9 + 0.8 find the national consumption function.
Suppose that the marginal propensity to consume is dC / dy = 0.7 - e-2y and that consumption is $5.65 trillion when disposable income is $0. Find the national consumption function.
Suppose that the marginal propensity to consume isand that consumption is $6.04 trillion when disposable income is $0. Find the national consumption function.
Suppose that the marginal propensity to save is dS/dy = 0.15 and that consumption is $5.15 trillion when disposable income is $0. Find the national consumption function.
Suppose that the marginal propensity to save is dS / dy = 0.22 and that consumption is $8.6 trillion when disposable income is $0. Find the national consumption function.
Suppose that the marginal propensity to save isand that consumption is $6 trillion when disposable income is $0. Find the national consumption function.
If consumption is $3 trillion when disposable income is $0 and if the marginal propensity to save is dS / dy = 0.2 + e-1.5y find the national consumption function.
If the marginal cost for a product is MC = 4x + 2 and the production of 10 units results in a total cost of $300, find the total cost function. Cost, revenue, and profit are in dollars and x is the
If the marginal cost for a product is MC = 3x + 50 and the total cost of producing 20 units is $2000, find the total cost function. Cost, revenue, and profit are in dollars and x is the number of
If the marginal cost for a product is MC = 4x + 40 and the total cost of producing 25 units is $3000, find the cost of producing 30 units. Cost, revenue, and profit are in dollars and x is the number
If the marginal cost for producing a product is MC = 5x + 10, with a fixed cost of $800, find the cost of producing 20 units. Cost, revenue, and profit are in dollars and x is the number of units.
A firm knows that its marginal cost for a product is MC = 3x + 20, that its marginal revenue is MR = 44 - 5x, and that the cost of production of 80 units is $11,400. (a) Find the optimal level of
A certain firm's marginal cost for a product is MC = 6x + 60, its marginal revenue is MR = 180 + 2x, and its total cost of production of 10 items is $1000. (a) Find the optimal level of
Suppose that the marginal revenue for a product is MR = 900 and the marginal cost is MC = 30 √x + 4, with a fixed cost of $1000. (a) Find the profit or loss from the production and sale of 5
In Problems 1-4, show that the given function is a solution to the differential equation. 1. y = x2; 4y - 2xy' = 0 2. y = x3; 3y - xy' = 0 3. y = 3x2 + 1; 2y dx - x dy = 2 dx 4. y = 4x3 + 2; 3y dx -
In Problems 1-4, find the particular solution. 1. y' = ex-3; y(0) = 2 2. y' = e2x+1; y(0) = e 3. dy = (1/x -x) dx; y(1) = 0 4. dy = (x2 - 1/x+1)dx; y(0) = 1/3
In Problems 1-3, find the general solution to the given differential equation. 1. dy/dx = x2/y 2. y3 dx = dy/x3 3. dx = x3y dy
In Problems 1-3, find the particular solution to each differential equation. 1. dy/dx = x2/y3 when x = 1, y =1 2. dy/dx = x + 1/xy when x = 1, y =3 3. 2y2 dx = 3x2 dy when x = 2, y = - 1
If x and y are measurements of certain parts of an organism, then the rate of change of y with respect to x is proportional to the ratio of y to x. That is, if k is a constant, then these
A bimolecular chemical reaction is one in which two chemicals react to form another substance. Suppose that one molecule of each of the two chemicals reacts to form two molecules of a new substance.
(a) If $10,000 is invested at 6%, compounded continuously, find an equation for the future value of the investment as a function of time t in years. (b) What is the future value of the investment
(a) If $2000 is invested at 8%, compounded continuously, find an equation for the future value of the investment as a function of time t, in years. (b) How long will it take for the investment to
When the interest on an investment is compounded continuously, the investment grows at a rate that is proportional to the amount in the account, so that if the amount present is P, then dP/dt = kP
When the interest on an investment is compounded continuously, the investment grows at a rate that is proportional to the amount in the account. If $20,000 is invested (when t0) and the amount in
Suppose that the growth of a certain population of bacteria satisfies dy/dt = ky where y is the number of organisms, t is the number of hours, and k is a constant. If initially there are 10,000
Suppose that, for a certain population of bacteria, growth occurs according to dy/dt = ky (t in hours, k constant) If the doubling rate depends on temperature, find how long it takes for the number
Suppose that in a certain company, the relationship between the price per unit p of its product and the weekly sales volume y, in thousands of dollars, is given bySolve this differential equation if
Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that the relationship between its price p for an oil change and its monthly sales volume y, in thousands of dollars, isSolve
A breeder reactor converts uranium-238 into an isotope of plutonium-239 at a rate proportional to the amount present at any time. After 10 years, 0.03% of the radioactivity has dissipated (that is,
A certain radioactive substance has a half-life of 50 hours. Find how long it will take for 90% of the radioactivity to be dissipated if the amount of material x satisfies dx/dt = kx (t in hours, k
Suppose that a liquid carries a drug into a 100-cc organ at a rate of 5 cc s and leaves the organ at the same rate. Suppose that the concentration of the drug entering is 0.06 g cc. If initially
In Problems 1-5, use integration to find the general solution to each differential equation. 1. dy = xex2+1 dx 2. dy = x2ex3-1 dx 3. 2y dy = 4x dx 4. 4y dy = 4x3 dx 5. 3y2 dy = (2x - 1) dx
Drug in an organ Suppose that a liquid carries a drug into a 250-cc organ at a rate of 10 cc s and leaves the organ at the same rate. Suppose that the concentration of the drug entering is 0.15 g cc.
Drug in an organ Suppose that a liquid carries a drug with concentration 0.1 g cc into a 200-cc organ at a rate of 5 cc s and leaves the organ at the same rate. If initially there is 10 g of the drug
Suppose that a liquid carries a drug with concentration 0.05 g cc into a 150-cc organ at a rate of 6 cc s and leaves at the same rate. If initially there is 1.5 g of drug in the organ, find the
Let V denote the volume of a tumor, and suppose that the growth rate of the tumor satisfies dV/dt = 0.2Ve-0.1t If the initial volume of the tumor is 1.86 units, find an equation for V as a function
The differential equation dx/dt = x(a - b ln x) where x represents the number of objects at time t, and a and b are constants, is the model for Gompertz curves. Recall from Section 5.3, "Solutions of
If V is the volume of a spherical cell, then in certain cell growth and for some fetal growth models, the rate of change of V is given by dV/dt = kV 2/3 where k is a constant depending on the
The rate of change of atmospheric pressure P with respect to the altitude above sea level h is proportional to the pressure. That is, dP/dh = kP (k constant) Suppose that the pressure at sea level is
Newton's law of cooling (and warming) states that the rate of change of temperature u = u(t) of an object is proportional to the temperature difference between the object and its surroundings, where
Newton's law of cooling can be used to estimate time of death. (Actually the estimate may be quite rough because cooling does not begin until metabolic processes have ceased.) Suppose a corpse is
Carbon in the atmosphere is due to carbon dioxide (CO2) emissions from fossil-fuel burning and is considered to be a primary contributor to climate change. Using data from the Organization for
With Social Security Administration data from 1995 and projected to 2070, the billions of dollars G of gross domestic product (GDP) can be modeled by dG/dt = 0.05317G, G(15) = 12,145 where t is the
The impact of a 5% inflation rate on an $80,000-per-year pension can be severe. If P represents the purchasing power (in dollars) of an $80,000 pension, then the effect of a 5% inflation rate can be
The rate of change of the number of Americans over age 65 with Alzheimer's disease (in millions per year) for the years 2000 through 2050 can be modeled by the differential equation dA/dt = 0.0228A
Evaluate the integrals in Problems 1-4. 1. ∫ x6 dx 2. ∫ x1/2 dx 3. ∫ (12x3 - 3x2 + 4x + 5) dx 4. ∫ 7(x2 - 1)2 dx
In Problems 1-3, find the general solution to each differential equation. 1. dy/dt = 4.6e-0.05t 2. dy = (64 + 76x - 36x2) dx 3. dy/dx = 4x/y - 3
In Problems 1 and 2, find the particular solution to each differential equation. 1. y' = x2/y + 1, y (0) = 4 2. y' = 2x/1 + 2y' y (2) = 0
If the marginal revenue for a month for a product is M̅̅̅̅R̅ = 0.06x + 12 dollars per unit, find the total revenue from the sale of x = 800 units of the product.
Suppose that the rate of change of production of the average worker at a factory is given by dp/dt = 27 + 24t - 3t2, 0 ≤ t ≤ 8 where p is the number of units the worker produces in t hours. How
The rate of change of the oxygen level (in mmol/l) per month in a body of water after an oil spill is given bywhere t is the number of months after the spill. What function gives the oxygen level P
A population of bacteria grows at the rate dp/dt = 100,000/(t = 1002 where p is the population and t is time. If the population is 1000 when t = 1, write the equation that gives the size of the
The rate of change of the market share (as a percent) a firm expects for a new product is dy/dt = 2.4e-0.04t where t is the number of months after the product is introduced. (a) Write the equation
If the marginal revenue for a product is MR = 800/x + 2, find the total revenue function.
The marginal cost for a product is MC = 6x + 4 dollars per unit, and the cost of producing 100 items is $31,400. (a) Find the fixed costs. (b) Find the total cost function.
Suppose a product has a daily marginal revenue MR = 46 and a daily marginal cost MC = 30 + 1/5x, both in dollars per unit. If the daily fixed cost is $200, how many units will give maximum profit and
If consumption is $8.5 trillion when disposable income is $0, and if the marginal propensity to consume isfind the national consumption function.
Suppose that the marginal propensity to save is dS/dy = 0.2 - 0.1e-2y and consumption is $7.8 trillion when disposable income is $0. Find the national consumption function.
For many species of fish, the length L and weight W of a fish are related by dW/dL = 3W/L The general solution to this differential equation expresses the allometric relationship between the length
When the interest on an investment is compounded continuously, the investment grows at a rate that is proportional to the amount in the account, so that if the amount present is P, then dP/dt = kP (k
Radioactive beryllium is sometimes used to date fossils found in deep-sea sediment. The amount of radioactive material x satisfies dx/dt = kx Suppose that 10 units of beryllium are present in a
Suppose that a liquid carries a drug into a 120-cc organ at a rate of 4 cc/s and leaves the organ at the same rate. If initially there is no drug in the organ, and if the concentration of drug in the
A 300-gal tank initially contains a solution with 100 lb of a chemical. A mixture containing 2 lb/gal of the chemical enters the tank at 3 gal min, and the well-stirred mixture leaves at the same
over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the right-hand endpoints of the subintervals. (See Example 1.) 1. f (x)=4x-x2
Compare the right-hand and left-hand values by finding SR - SL for n=10, for n=100, and as nâ†’ âˆž. When the area under f (x) = x2 x from x = 0 to x = 2 is approximated, the formulas for the
Because f (x) = x2+x is increasing over the interval from x=0 to x=2, function values at the right-hand endpoints are maximum values for each subinterval, and function values at the left-hand
Find the value of each sum.1.2. 3. 4. 5.
Use the sum formulas I-V to express each of the following without the summation symbol. In Problems 20-23, find the numerical value.1.2.3.4.5.
Use the function y 2x from x 0 to x 1 and n equal subintervals with the function evaluated at the left-hand endpoint of each subinterval.What is the area of the(a) First rectangle?(b) Second
Use the function y 2x from x 0 to x 1 and n equal subintervals with the function evaluated at the left-hand endpoint of each subinterval.(a) Find a formula for the sum of the areas of the n
How do your answers to Problems 27(a)-(e) compare with the corresponding calculations in the discussion (after Example 1) of the area under y=2x using right hand endpoints?
For parts (a)-(e), use the function y=x2 from x=0 to x=1 with n equal subintervals and the function evaluated at the right-hand endpoints.(a) Find a formula for the sum of the areas of the n
How do your answers to Problems 29(a)-(e) compare with the corresponding calculations in Example 2?
Use rectangles to find the area between y=x2 -6x+8 and the x-axis from x=0 to x=2. Divide the interval [0, 2] into n equal subintervals so that each subinterval has length 2/ n.
Use rectangles to find the area between y=4x-x2 and the x-axis from x=0 to x=4. Divide the interval [0, 4] into n equal subintervals so that each subinterval has length 4/ n.
The annual per capita out-of-pocket expenses (to the nearest dollar) for U.S. health care for selected years from 2013 and projected to 2021 are shown in the table and figure.(a) Use n=4 equal
Crude oil and petroleum products are imported continuously by the United States. The following table and figure show the net expenditures for U.S. oil imports for selected years (in billions of
The figure gives the times that it takes a Porsche 911 to reach speeds from 0 mph to 100 mph, in increments of 10 mph, with a curve connecting them. The area under this curve from t=0 seconds to t=14
The figure gives the times that it takes a Mitsubishi Eclipse GSX to reach speeds from 0 mph to 100 mph, in increments of 10 mph, with a curve connecting them. The area under this curve from t=0

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