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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
Find the value of each integral that converges. xp z_x J-
Find the area between the graph of the given function and the x-axis over the given interval, if possible. f(x) = 1 (x - 1)²³ for (-∞, 0]
Use integration by parts to derive the following formula from the table of integrals. [.r. x". In x dx = xn+1 In |x| n + 1 1 1²] + (n + 1)². + C, n-1
Find the area between the graph of the given function and the x-axis over the given interval, if possible. 00 -00 xex dx.
A stock analyst plots the price per share of a certain common stock as a function of time and finds that it can be approximated by the function S(t) = 37 + 6e-0.03t, where t is the time (in years) since the stock was purchased. Find the average price of the stock over the first six years.
Find the value of each integral that converges. I беx dx
Find the area between the graph of the given function and the x-axis over the given interval, if possible. ,z (z²x + 1) dx. X .00
Find the value of each integral that converges. -0 -00 X 2 x² + 3 dx
Show thatconverges if p > 1 and diverges if p ≤ 1. 8 ,00 P x dx
Find the value of each integral that converges. 4 In (5x) dx
(a) One way to integrate ∫x √x + 1 dx is to use integration by parts. Do so to find the antiderivative.(b) Another way to evaluate the integral in part (a) is by using the substitution u = x + 1. Do so to find the antiderivative.(c) Compare the results from the two methods. If they do not look
The Federal debt held by the public as a percent of the gross domestic product (GDP) was modeled by the function y = -0.000224x4 + 0.0206x3 - 0.569x2 + 5.62x + 23.7, where t was the time (in years) since 1980.(a) Find the average public debt, as a percent of GDP, from 1990 to 2000.(b) Find the
The U.S. corn production (in billions of bushels) was modeled by the function p(t) = 1.757(1.0248)t, where t was the time (in years) since 1930.(a) Find the average corn production from 1930 to 1950.(b) Find the average corn production from 2000 to 2018.
Recall from the previous chapter that the consumers’ surplus is defined bywhere D(q) is the demand function, q0 is the equilibrium quantity, and p0 = D(q0) is the equilibrium price. Find the consumers’ surplus for each of the following demand functions and equilibrium quantities.D(q) = (q +
Recall from the previous chapter that the producers’ surplus is defined bywhere S(q) is the supply function, q0 is the equilibrium quantity, and p0 S(q0) is the equilibrium price. Find the producers’ surplus for each of the following supply functions and equilibrium quantities.S(q) = qe0.2q +
The intensity of the reaction to a certain drug, in appropriate units, is given by R(t) = te-0.1t, where t is time (in hours) after the drug is administered. Find the average intensity during the following hours.(a) Second hour(b) Twelfth hour(c) Twenty-fourth hour
Consider the functions ƒ(x) = 1/√1 + x2 and g(x) = 1/√1 + x4.(a) Use your calculator to approximate ∫1b f(x) for b = 20, 50, 100, 1000, and 10,000.(b) Based on your answers from part (a), would you guess that ∫1∞ f(x) dx is convergent or divergent?(c) Use your calculator to
Find the area between the graph of each function and the x-axis over the given interval, if possible.ƒ(x) = 3e-x, for [0,∞)
(a) Use your calculator to approximate ∫0b e-x2 dx for b = 1, 5, 10, and 20.(b) Based on your answers to part (a), does ∫0∞e-x2 dx appear to be convergent or divergent? If convergent, what seems to be its approximate value?(c) Explain why this integral should be convergent by
How is the present value of money flow found? The accumulated amount of money flow?
(a) Use your calculator to approximate ∫0b e-0.00001x dx for b = 10, 50, 100, and 1000.(b) Based on your answers to part (a), does ∫0∞ e-0.00001x dx appear to be convergent or divergent?(c) To what value does the integral actually converge?
The rate of change of revenue from the sale of x toaster ovens is R′(x) = x(x - 50)1/2. Find the total revenue from the sale of the 50th to the 75th ovens.
Find the capital values of the properties in Exercises.A castle for which annual rent of $225,000 will be paid in perpetuity; the interest rate is 6% compounded continuously.
The rate of reaction to a drug is given by r′(t) = 2t2e-t, where t is the number of hours since the drug was administered. Find the total reaction to the drug from t = 1 to t = 6.
Find the capital values of the properties in Exercises.A fort on a strategic peninsula in the North Sea; the annual rent is $1,000,000, paid in perpetuity; the interest rate is 5% compounded continuously.
The rate of growth of a microbe population is given by m′(t) = 27te3t, where t is time in days. What is the total accumulated growth during the first 2 days?
Find the capital value of an asset that generates $7200 yearly income if the interest rate is as follows.(a) 5% compounded continuously(b) 10% compounded continuously
A study of the sales lost when a store has insufficient inventory to meet the demand and impatient customers are unwilling to wait for the store to reorder has found that when the demand is a decreasing exponential function, the expected sales lost per period is given bywhere B is the base stock
The area covered by a patch of moss is growing at a rate of A′(t) = √t ln t cm2 per day, for t ≥ 1. Find the additional amount of area covered by the moss between 4 and 9 days.
Suppose income from an investment starts (at time 0) at $6000 a year and increases linearly and continuously at a rate of $200 a year. Find the capital value at an interest rate of 5% compounded continuously.
Assume that each function gives the rate of flow of money in dollars per year over the given period, with continuous compounding at the given annual interest rate. Find the accumulated amount of money flow at the end of the time period.ƒ(t) = 1000, 5 years, 6%
When harvesting a population, such as fish, the present value of the resource is given bywhere r is a discount factor, n(t) is the net revenue at time t, and y(t) is the harvesting effort. Suppose y(t) = K and n(t) = at + b. Find the present value. P = 0 en(t)y(t)dt,
Suppose income from an endowment is generated at the annual rate of $42,000 per year. Find the capital value of this endowment at an interest rate of 7.5% compounded continuously.
Assume that each function gives the rate of flow of money in dollars per year over the given period, with continuous compounding at the given annual interest rate. Find the accumulated amount of money flow at the end of the time period.ƒ(t) = 500e-0.04t, 8 years, 10%
Assume that each function gives the rate of flow of money in dollars per year over the given period, with continuous compounding at the given annual interest rate. Find the accumulated amount of money flow at the end of the time period.ƒ(t) = 20t, 6 years, 4%
In an epidemiological model used to study the spread of drug use, a single drug user is introduced into a population of N non-users. Under certain assumptions, the number of people expected to use drugs as a result of direct influence from each drug user is given bywhere a, b, and k are constants.
Assume that each function gives the rate of flow of money in dollars per year over the given period, with continuous compounding at the given annual interest rate. Find the accumulated amount of money flow at the end of the time period.ƒ(t) = 1000 + 200t, 10 years, 9%
An investment scheme is expected to produce a continuous flow of money, starting at $1000 and increasing exponentially at 5% a year for 7 years. Find the present value at an interest rate of 11% compounded continuously.
The rate of reaction to a drug is given by r′(t) = 2t2e-t, where t is the number of hours since the drug was administered. Find the total reaction to the drug over all the time since it was administered, assuming this is an infinite time interval.
The proceeds from the sale of a building will yield a uniform continuous flow of $10,000 a year for 10 years. Find the final amount at an interest rate of 10.5% compounded continuously.
Find the capital value of an office building for which annual rent of $50,000 will be paid in perpetuity, if the interest rate is 9%.
The percentage of men aged 65 and older in the labor force between 1950 and 2018 could be approximated by the function ƒ(t) = 0.0137t2 - 2.63t + 143, where t is the number of years since 1900. Find the average percentage of men aged 65 and older in the labor force between 1950 and 2018.
The number of U.S. deaths by firearms has declined dramatically in recent decades. The number of deaths each year can be approximated by d(t) = 2632e-0.03725t, where t is the number of years since 1970. Assuming this trend continues, estimate the total number of U.S. deaths from firearms from the
The rate at which radioactive waste is entering the atmosphere at time t is decreasing and is given by Pe-kt, where P is the initial rate. Use the improper integralwith P =50 to find the total amount of the waste that will enter the atmosphere for each value of k.k = 0.06 00 0 Pe-kt dt
The amount of coal (in thousands of short tons) consumed by U.S. utilities to produce electricity between 2001 and 2018 could be approximated by the function ƒ(t) = -2.67t2 + 32.1t + 926, where t is the number of years since 2000. Find the average amount of coal consumed by U.S. utilities to
The rate at which radioactive waste is entering the atmosphere at time t is decreasing and is given by Pe-kt, where P is the initial rate. Use the improper integralwith P = 50 to find the total amount of the waste that will enter the atmosphere for each value of k.k = 0.04 00 0 Pe-kt dt
The reaction rate to a new drug t hours after the drug is administered is r′(t) = 0.5te-t. Find the total reaction over the first 5 hours.
The decline in carbon monoxide emissions in the U.S. in recent decades can be approximated by c(t) = 237e-0.0277t,where t is the number of years since 1970 and c(t) gives the annual carbon monoxide emissions in millions of tons. Assuming this trend continues, estimate the total amount of carbon
An important function in many areas of mathematics, such as statistics, is the gamma function, defined aswhere t is not 0 or a negative integer. Find each of the following.(a) Г(1) (b) Г(2) (c) Г(3) (d) Г(4) Γ(t) =√√x+16 0 || x-¹ex dx,
Determine whether each improper integral converges or diverges, and find the value of each that converges. ["Cett, Ce-kt dt, k>0
A frustum is what remains of a cone when the top is cut off by a plane parallel to the base. Suppose a right circular frustum (that is, one formed from a right circular cone) has a base with radius r, a top with radius r/2, and a height h. (See the figure below.) Find the volume of this frustum by
An oil leak from an uncapped well is polluting a bay at a rate of ƒ(t) = 125e-0.025t gallons per year. Use an improper integral to find the total amount of oil that will enter the bay, assuming the well is never capped.
Suppose that u and v are differentiable functions of x with ∫01 du = 4 and the following functional values.Use this information to determine ∫01 du. X 0 1 u(x) 2 3 v(x) 1 -4
The decline in sulfur dioxide emissions in the U.S. in recent decades can be approximated by s(t) = 55,545e-0.05056t, where t is the number of years since 1970 and s(t) gives the annual sulfur dioxide emissions in millions of tons. Assuming this trend continues, estimate the total amount of sulfur
Suppose the temperature (degrees F) in a river at a point x meters downstream from a factory that is discharging hot water into the river is given by T(x0 = 160 - 0.05x2. Find the average temperature over each interval.(a) [0, 10] (b) [10, 40] (c) [0, 40]
Determine whether each improper integral converges or diverges, and find the value of each that converges. 00 a C dt, tk k > 1
Find the average value of each function on the given interval.ƒ(x) = x ln x; [1, e]
Use the integration feature on a graphing calculator to find the volume of the solid of revolution by rotating about the x-axis each region bounded by the given curves. f(x) = = 1 4+x²² y = 0, x = -2, x = 2
Find the area between the graph of the given function and the x-axis over the given interval, if possible. f(x) 1 X- - 1' for (-∞, 0]
How is the average value of a function found?
Suppose that u and v are differentiable functions of x with ∫020 v du = -1 and the following functional values.Use this information to determine ∫020 v du. X 1 20 u(x) 5 15 -2 6
Find the average value of each function on the given interval.ƒ(x) = x2e2x; [0, 2]
Find the average value of ƒ(x) = 2x + 1 over the interval [0, 8].
Find the value of each integral that converges. fox 10 x dx
Find the area between the graph of the given function and the x-axis over the given interval, if possible. f(x) = e, for (-∞, e]
Use the integration feature on a graphing calculator to find the volume of the solid of revolution by rotating about the x-axis each region bounded by the given curves.ƒ(x) = e-x2, y = 0, x = -1, x = 1
Find the average value of ƒ(x) = 7x2(x3 + 1)6 over the interval [0, 2].
Suppose we know that the functions r and s are everywhere differentiable and that r(0) = 0. Suppose we also know that for 0 ≤ x ≤ 2, the area between the x-axis and the nonnegative function h(x) = s(x) dr/dx is 5, and that on the same interval, the area between the x-axis and the nonnegative
Arman plots the price per share of a stock that he owns as a function of time and finds that it can be approximated by the function S(t) = t(25 - 5t) + 18, where t is the time (in years) since the stock was purchased. Find the average price of the stock over the first five years.
Suppose we know that the functions u and v are everywhere differentiable and that u(3) = 0. Suppose we also know that for 1 ≤ x ≤ 3, the area between the x-axis and the nonnegative function h(x) = u(x) dv/dx is 15, and that on the same interval, the area between the x-axis and the nonnegative
The difference between accumulated amount of money flow and total money flow is that the first includes interest received on the money after it comes in.Determine whether each statement is true or false, and explain why.
Integration by parts should be used to determine ∫ln(4x) dx.Determine whether each of the following statements is true or false, and explain why.
We would need to apply the method of integration by parts twice to determineDetermine whether each of the following statements is true or false, and explain why. [x³e³dx.
The capital value of an asset is often defined as the present value of all future net earnings of the asset.Determine whether each statement is true or false, and explain why.
Integration by parts cannot be used on a definite integral.Determine whether each statement is true or false, and explain why.
The average value of a function on an interval is the integral of the function over the interval.Determine whether each statement is true or false, and explain why.
The present value of money flow for T years is the integral from 0 to T of the function giving the rate of money flow.Determine whether each statement is true or false, and explain why.
An integral from 0 to ∞ is not necessarily an improper integral.Determine whether each statement is true or false, and explain why.
Integration by parts can be used n times to integrate ∫xnex dx for any positive integer n.Determine whether each statement is true or false, and explain why.
A right circular cone is an example of a volume of revolution.Determine whether each statement is true or false, and explain why.
The total money flow is found by taking the derivative of the function giving the rate of money flow.Determine whether each statement is true or false, and explain why.
Integration by parts should be used to evaluateDetermine whether each of the following statements is true or false, and explain why. Jo 10x xel dx.
An integral from 0 to ∞ always has an infinite value.Determine whether each statement is true or false, and explain why.
Integration by parts can be used only when the integrand can be written as the product of two factors, u and dv.Determine whether each statement is true or false, and explain why.
A solid of revolution has circular cross sections.Determine whether each statement is true or false, and explain why.
In Exercises, find each indefinite integral. 3 X e ³x² 3x dx
The most important function in probability and statistics is the density function for the standard normal distribution, which is the familiar bell-shaped curve. The function isUse Simpson’s rule with n = 20 for the following.(a) The area under this curve between x = -1 and x = 1 represents the
In Exercises, find each indefinite integral. 3√t dt
In Exercise , we estimated the total U.S. wind energy consumption (in trillion BTUs) for the 12-year period from 2001 to 2019 using rectangles and the following data.(a) Approximate the total wind energy consumption using the trapezoidal rule.(b) Approximate the total wind energy consumption using
In Exercises, find each indefinite integral. (2x + 3) dx
In Exercises, find each indefinite integral. (5x - 1) dx
In Exercises, find each indefinite integral. √(x² (x² - 3x + 2) dx
In Exercises, find each indefinite integral. [16. (6 - x²) dx
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
Explain why the limits of integration are changed when u is substituted for an expression in x in a definite integral.Determine whether each of the following statements is true or false, and explain why.
Suppose that ƒ(x) > 0 and ƒ′(x) > 0 for all x between a and b, where a 6 b. Which of the following cases is true of a trapezoidal approximation T for the integral ∫ab ƒ(x)dx? Explain.(a)(b)(c) Can’t say which is larger T< rb a f(x) dx
Describe the type of integral for which numerical integration is useful.Determine whether each of the following statements is true or false, and explain why.
In Exercises, use n = 4 to approximate the value of the given integrals by the following methods: (a) The trapezoidal rule, and (b) Simpson’s rule. (c) Find the exact value by integration. So 4xe* dx
In Exercises, use n = 4 to approximate the value of the given integrals by the following methods: (a) The trapezoidal rule, and (b) Simpson’s rule. (c) Find the exact value by integration. 4 [xV₂ S xV2x² + 1 dx
Explain under what circumstances substitution is useful in integration.Determine whether each of the following statements is true or false, and explain why.
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