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calculus with applications

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

Use substitution to find each indefinite integral. 1² + 2 2 t³ + 6t+ 3 19 dt
Find the area between the curves in Exercises.y = x4/3, y = 2x1/3
Use substitution to find each indefinite integral. 2x + 1 |(₁² (x² + x)³ dx
In Exercises, estimate the area under each curve by summing the area of rectangles. Use the left endpoints, then the right endpoints, and then give the average of those answers.The following graph shows the rate of use of electrical energy (in millions of kilowatts) in a certain city on a very hot
In this exercise, we investigate the value of ∫01 x2 dx using larger and larger values of n in the definition of the definite integral.(a) First let n = 10, so Δx = 0.1. Fill a list on your calculator with values of x2 as x goes from 0.1 to 1. (On a TI-84 calculator, use the command
Find the area between the curves in Exercises.y = √x, y = x√x
Repeat Exercise 25 for ∫01 x3 dx.Exercise 25In this exercise, we investigate the value of ∫01 x2 dx using larger and larger values of n in the definition of the definite integral.(a) First let n = 10, so Δx = 0.1. Fill a list on your calculator with values of x2 as x goes from 0.1 to 1.
Use substitution to find each indefinite integral. y² + y (2y³ + 3y² + 1)²/3 dy J
Find the area between the curves in Exercises.x = 0, x = 3, y = 2e3x, y = e3x + e6
Use substitution to find each indefinite integral. [4rV 4rV8 - r dr
Use substitution to find each indefinite integral. [P(P p(p + 1)5 dp
The following graph shows the U.S. wind energy consumption (trillion BTUs) for various years. Estimate the total consumption for the 18-year period from 2001 to 2019 using rectangles of width 3 years. Consumption (trillion BTUS) 3000 2500 2000 1500- 1000 500 0 2001 70 142 341 2004
Find the area between the curves in Exercises.x = 0, x = 3, y = ex, y = e4-x
Suppose a company wants to introduce a new machine that will produce a rate of annual savings (in dollars) given by S′(t) = 150 - t2, where t is the number of years of operation of the machine, while producing a rate of annual costs (in dollars) of(a) For how many years will it be profitable to
Repeat Exercise 30 using the demand curve for a fruit juice (in dollars per liter) given byfor 0 ≤ q ≤ 25, where q is the demand in liters. Estimate the total revenue using rectangles of 5 liters' width. D (q) = 2 59 125 4 59 9 + 15
The total revenue for a product can be calculated as the area under the demand curve. Suppose that the demand curve for a certain wine (in dollars per liter) is given byfor 0 ≤ q ≤ 50, where q is the demand in liters. Estimate the total revenue using rectangles of width 10 liters. D(q) =
In Exercises, use a graphing calculator to find the values of x where the curves intersect and then to find the area between the two curves.y = ex, y = -x2 - 2x
Use substitution to find each indefinite integral. 2.x (x + 5)6 dx
Use substitution to find each indefinite integral. n Vu - 1 du
In Exercises, use a graphing calculator to find the values of x where the curves intersect and then to find the area between the two curves.y = ln x, y = x3 - 5x2 + 6x - 1
A new smog-control device will reduce the output of sulfur oxides from automobile exhausts. It is estimated that the rate of savings (in millions of dollars per year) to the community from the use of this device will be approximated by S′(t)= -t2 + 4t + 8, after t years of use of the device. The
On March 13, 2020, President Trump signed a proclamation declaring that the COVID-19 outbreak in the U. S. constituted a national emergency. Sweeping changes were made throughout the nation, including social distancing, in an effort to control the spread of the virus. The graph on the next page
The graph below shows the rate of inhalation of oxygen (in liters per minute) by a person riding a bicycle very rapidly for 10 minutes. Estimate the total volume of oxygen inhaled in the first 20 minutes after the beginning of the ride. Use rectangles with widths of 1 minute. Liters per
Use substitution to find each indefinite integral. [(VP² √(√x² - 6x - 6x)(x − 3) dx
Use substitution to find each indefinite integral. [(√x (√x² + 12x) (x + 6) dx
Use substitution to find each indefinite integral. dx X z(xu[ε + I)
The graph below shows the number of fatal automobile accidents in California for various years. Estimate the total number of accidents in the 8-year period from 2009 to 2017 using rectangles of width 2 years.The next two graphs are from the Road & Track website. The curves show the velocity at
Perdaris Enterprises had an expenditure rate of E'(x) = e0.¹x dollars per day and an income rate of I'(x) = 98.80.1 - e0.¹x dollars per day on a particular job, where x was the number of days from the start of the job. The company's profit on that job will equal total income less
Use substitution to find each indefinite integral. √2 + ln x X dx
A factory of Starr Smith Industries has installed a new process that will produce an increased rate of revenue (in thousands of dollars per year) of R′(t) = 104 - 0.4et/2, where t is time measured in years. The new process produces additional costs (in thousands of dollars per year) at the rate
Estimate the distance traveled by the Lamborghini Gallardo LP560-4 using the graph below. Use rectangles with widths of 3 seconds, except for the last rectangle, which should have a width of 2 seconds. The circle marks the point where the car has gone a quarter mile. Does this seem correct?
Estimate the distance traveled by the Alfa Romeo 8C Competizione using the graph on the next page. Use rectangles with widths of 4 seconds, except for the last rectangle, which should have a width of 3.5 seconds. The circle marks the point where the car has gone a quarter mile. Does this seem
Find the producers’ surplus if the supply function for pork bellies is given (in dollars) by S(q) = q5/2 + 2q3/2 + 50. Assume supply and demand are in equilibrium at q = 16.
In Exercises, use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval. f(x) = 2x - 14; [6, 10]
Find the consumers’ surplus if the demand function for grass seed is given (in dollars) byassuming supply and demand are in equilibrium at q = 3. D(q) = = 200 (3q + 1)²¹
Use substitution to find each indefinite integral. 2x e²x e²x + 5 е 2x dx
Suppose the supply function for concrete is given (in dollars) by S(q) = 100 + 3q3/2 + q5/2, and that supply and demand are in equilibrium at q = 9. Find the producers’ surplus.
Use substitution to find each indefinite integral. 1 x(In x) dx
Using the table below, estimate the distance traveled by the Mercedes-Benz C63 AMG Edition 507. Acceleration Zero to 30 mph 40 mph 50 mph 60 mph 70 mph 80 mph 90 mph 100 mph 110 mph 120 mph 130 mph 140 mph 150 mph 160 mph Seconds 1.7 2.3 3.1 3.9 5.1 6.3 7.5 9.2 10.9 12.9 15.1 18.2 21.7 27.0
In Exercises, use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval.f(x) = 4x − 32; [5, 10] -
Find the consumers’ surplus if the demand function for extra virgin olive oil is given (in dollars) byand if supply and demand are in equilibrium at q = 6. D(q) = 32,000 (2q + 8)³¹
Use substitution to find each indefinite integral. log x X dx
Using the table below, estimate the distance traveled by the BMW M3.The following graphs show the typical heat gain, in BTUs per hour per square foot, for windows (one with plain glass and one that is triple glazed) in Pittsburgh in June, one facing east and one facing south. The horizontal axis
In Exercises, use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval.f(x) = 2 − 2x²; [0,5] -
When data are given in tabular form, you may need to vary the size of the interval to calculate the area under the curve. The next two exercises include data from Car and Driver magazine. To estimate the total distance traveled by the car (in feet) during the time it took to reach its maximum
In Exercises, use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval.f(x) = 9 - x²; [0,6]
Suppose the supply function for a certain item is given (in dollars) by S(q) = (q + 1)2,and the demand function is given by(a) Graph the supply and demand curves.(b) Find the point at which supply and demand are in equilibrium.(c) Find the consumers’ surplus.(d) Find the producers’ surplus.
Use substitution to find each indefinite integral. (log₂ (5x + 1))² 5r + 1 - dx
In Exercises, use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval.f(x) = x³; [-1, 3]
In Exercises, use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval.f(x) = x³ − 2x; [-2,4] -
(a) Estimate the total heat gain per square foot for a plain glass window facing south.(b) Estimate the total heat gain per square foot for a triple glazed window facing south.
Suppose that with the supply and demand for oil as in Exercise 39, the government sets the price at $264 per unit.(a) Use the supply function to calculate the quantity that will be produced at the new price.(b) Find the consumers’ surplus for the new price, using the quantity found in part (a) in
The Lorenz curve for Hispanic households in the U.S. in 2018 can be estimated byfor 0 ≤ x ≤ 1.(a) Find and interpret I(0.6).(b) Find the Gini index of inequality. I(x) = 331 240 कर 91 240 X
Use the Fundamental Theorem to show that the following are true. b [F(x) dx = [1(x) dx + [ f(x) dx a
Use substitution to find each indefinite integral. [1831²+ x83x²+1 dx
Musk the friendly pit bull has escaped again! Here is her velocity during the first 4 seconds of her romp.Give two estimates for the total distance Musk traveled during her 4-second trip, one using the left endpoint of each interval and one using the right endpoint. t (sec) v (ft/sec) 0 1 0 2 8
Use substitution to find each indefinite integral. 105√x+2 Vx - dx
The speed of a particle in a test laboratory was noted every second for 3 seconds. The results are shown in the following table. Use the left endpoints and then the right endpoints to estimate the total distance the particle moved in the first three seconds. t (sec) v (ft/sec) 0 1 10 6.5 2 6 3 5.5
In Exercises, use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval.f(x)= ex 1; [-1,2]
For a family with an income in the lowest fifth of the U.S. population, the mean household income for the years 2000 to 2018 can be approximated (in thousands of dollars) by ƒ(t) = 0.00354t3 - 0.0762t2 + 0.608t + 13.71, where t is the number of years since 2000. The mean household for a family in
Stan and Ollie work on the integralStan lets u = x3 and proceeds to getOllie tries u = ex3 and proceeds to getDiscuss which procedure you prefer, and why. 3x² et³ dx.
In Exercises, use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval. √(x) = 1 - 1 ; [e ¹, e] X
In Exercises, use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval. 1 f(x) = — - ! ; X e [1, ²]
In 2009, Jamaica’s Usain Bolt set a world record in the 100-m sprint. His speed at various times in the race is given in the following table.(a) Use the information in the table and left endpoints to estimate the distance that Bolt ran in miles. You will first need to calculate Δt for each
In Exercises, use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval.f(x) = 1 - ex; [-1,2]
Stan and Ollie work on the integralStan lets u = x2 + 2 and proceeds to getOllie multiplies out the function under the integral and getsHow can they both be right? 2x(x² + 2) dx.
Pollution begins to enter a lake at time t = 0 at a rate (in gallons per hour) given by the formula ƒ(t) = 10(1 - e-0.5t), where t is the time (in hours). At the same time, a pollution filter begins to remove the pollution at a rate g(t) = 0.4t as long as pollution remains in the lake.(a) How much
A company incurs debt at a rate ofdollars per year, where t is the amount of time (in years) since the company began. By the fourth year, the company had accumulated $16,260 in debt.(a) Find the total debt function.(b) How many years must pass before the total debt exceeds $40,000? D'(t) = 90(t +
Find the area of each shaded region. 0 3 1 f 2 f(x)=x²-2x 4 X
Find the area of each shaded region. y 4 2 0 -2 * y=4-x? 2 4 X
Repeat the steps of Exercise 43, using the functions ƒ(t) = 15(1 - e-0.05t) and g(t) = 0.3t. Exercise 43Pollution begins to enter a lake at time t = 0 at a rate (in gallons per hour) given by the formula ƒ(t) = 10(1 - e-0.5t), where t is the time (in hours). At the same time, a pollution
Suppose that the Lorenz curve for a population is given by I(x) = 0.9x2 + 0.1x for 0 ≤ x ≤ 1.(a) Find and interpret I(0.1).(b) Find and interpret I(0.4).(c) Find the Gini index of inequality.
The marginal revenue (in thousands of dollars) from the sale of x handheld gaming devices is given by R′(x) = 4x(x2 + 27,000)-2/3.(a) Find the total revenue function if the revenue from 125 devices is $29,591.(b) How many devices must be sold for a revenue of at least $40,000?
Find the area of each shaded region. 4 2 0 -2 1 2 y=e* -e 3 X
A worker sketches the curves y = √x and y = x/2 on a sheet of metal and cuts out the region between the curves to form a metal plate. Find the area of the plate.
Find an equation of the curve whose tangent line has a slope of ƒ′(x) = x2/3, given that the point (1, 3/5) is on the curve.
The slope of the tangent line to a curve is given by ƒ′(x) = 6x2 - 4x + 3. If the point (0, 1) is on the curve, find an equation of the curve.
Harby Communications, a small company of science writers, found that its rate of profit (in thousands of dollars) after t years of operation is given by P′(t) = (3t + 3)(t2 + 2t + 2)1/3.(a) Find the total profit in the first three years.(b) Find the profit in the fourth year of operation.(c) What
Exercises refer to Examples 9 and 10 in this section.Suppose v(t) = 9t2 - 3√t and s(1) = 8. Find s(t).Examples 9 Suppose the velocity of an object at time t is v(t) = 6t2 - 8t and that the object is at 5 when time is 0. Find s(t), the function that gives the position of the object at time t.
In the 1999 movie October Sky, Homer Hickam was accused of launching a rocket that started a forest fire. Homer proved his innocence by showing that his rocket could not have flown far enough to reach where the fire started. He used the following reasoning.(a) Using the fact that a(t) = -32 (see
Show that an object thrown from an initial height h0 with an initial velocity v0 has a height at time t given by the functionwhere g is the acceleration due to gravity, a constant with value -32 ft /sec2. h(t) = jgt + vot + ho,
Suppose that the rate of consumption of a natural resource is c′(t), where c′(t) = kert. Here t is time in years, r is a constant, and k is the consumption in the year when t = 0. In 2020, an oil company sold 1.2 billion barrels of oil. Assume that r = 0.04.(a) Write c′(t) for the
A small rocket was launched straight up from a platform.After 5 seconds, the rocket reached a maximum height of 412 ft. Find the initial velocity and height of the rocket.
The rate of consumption of oil (in billions of barrels) by the company in Exercise 75 was given as 1.2e0.04t, where t = 0 corresponds to 2020. Find the total amount of oil used by the company from 2020 to year T. At this rate, how much will be used in 5 years?
The density of sediment (in grams per cubic centimeter) at the bottom of Lake Coeur d’Alene, Idaho, is given by p(x) = p0e0.0133x, where x is the depth (in centimeters) and p0 is the density at the surface. The total mass of a square-centimeter column of sediment above a depth of h cm is given
Exercises refer to Examples 9 and 10 in this section.Suppose a(t) = (15/2)√t + 3e-t, v(0) = -3, and s(0) = 4. Find s(t).Examples 9 Suppose the velocity of an object at time t is v(t) = 6t2 - 8t and that the object is at 5 when time is 0. Find s(t), the function that gives the position of the
Based on data from the U.S. Census Bureau, an approximate family income distribution for the U.S. is given by the function ƒ(x) = 0.00462x3 - 0.160x2 + 1.30x + 2.85, where ƒ(x) is the number of families (in millions) and x is the annual income in units of $10,000, with 0 ≤ x ≤ 20. For
Exercises refer to Examples 9 and 10 in this section.Suppose a(t) = 18t + 8, v(1) = 15, and s(1) = 19. Find s(t).Examples 9 Suppose the velocity of an object at time t is v(t) = 6t2 - 8t and that the object is at 5 when time is 0. Find s(t), the function that gives the position of the object
The U.S. Census Bureau gives an age distribution that is approximately modeled (in millions) by the function ƒ(x) = 38.5 + 5.66x - 1.10x2 where x varies from 0 to 9 decades. The population of a given age group can be found by integrating this function over the interval for that age group.(a) Find
Exercises refer to Examples 9 and 10 in this section.An object is dropped from a small plane flying at 6400 ft. Assume that a(t) = -32 ft per second and v(0) = 0. Find s(t). How long will it take the object to hit the ground?Examples 9 Suppose the velocity of an object at time t is v(t) = 6t2
In an example from an earlier chapter, the velocity v of the blood in a blood vessel was given as v = k(R2 - r2), where R is the (constant) radius of the blood vessel, r is the distance of the flowing blood from the center of the blood vessel, and k is a constant. Total blood flow (in millimeters
The daily energy requirements of female beagles who are at least 1 year old change with respect to time according to the function E(t) = 753t-0.1321, where E(t) is the daily energy requirement (in kJ/W0.67), where W is the dog’s weight (in kilograms) for a beagle that is t years old.(a) Assuming
Exercises refer to Examples 9 and 10 in this section.For a particular object, a(t) = 5t2 + 4 and v(0) = 6. Find v(t).Examples 9 Suppose the velocity of an object at time t is v(t) = 6t2 - 8t and that the object is at 5 when time is 0. Find s(t), the function that gives the position of the
The average annual increment in the horn length (in centimeters) of bighorn rams born in recent years can be approximated by y = 0.1762x2 - 3.986x + 22.68, where x is the ram’s age (in years) for x between 3 and 9. Integrate to find the total increase in the length of a ram’s horn during this
The number of degrees in dentistry (D.D.S. or D.M.D.) conferred to females in the United States has been increasing steadily in recent decades. Based on data from the National Center for Education Statistics, the rate of change of the number of dentistry degrees can be approximated by the function
According to Fick’s law, the diffusion of a solute across a cell membrane is given bywhere A is the area of the cell membrane, V is the volume of the cell, c(t) is the concentration inside the cell at time t, C is the concentration outside the cell, and k is a constant. If c0 represents the
The number of bachelor’s degrees conferred in the United States has been increasing steadily in recent decades. Based on data from the National Center for Education Statistics, the rate of change of the number of bachelor’s degrees (in thousands) can be approximated by the function B′(t) =
If ƒ(x) is the instantaneous death rate for members of a population at time x, then the number of individuals who survive to age T is given byIn 1825, the biologist Benjamin Gompertz proposed that ƒ(x) = kbx. Find a formula for F(T). T ·(²² f(x) dx. 0 F(T) =
A population of E. coli bacteria will grow at a rate given by w′(t) = (3t + 2)1/3, where w is the weight (in milligrams) after t hours. Find the change in weight of the population from t = 0 to t = 3.
The rate of change of the volume V(t) of blood in the aorta at time t is given by V′(t) = -kP(t), where P(t) is the pressure in the aorta at time t and k is a constant that depends upon properties of the aorta. The pressure in the aorta is given by P(t) = P0 e-mt, where P0 is the pressure at time

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