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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

In Exercises 1 through 14, approximate the given integral using (a) The trapezoidal rule (b) Simpson’s rule with the specified number of subintervals. So Vx dx; n = n = 10
In Exercises 1 through 10, determine whether the given function is a probability density function. f(x) = for 0≤x≤ I e 0 otherwise
In Exercises 1 through 10, use integration by parts to find the given integral. [(5 (5 + 3x)e-x/2 dx
In Exercises 1 through 14, approximate the given integral using (a) The trapezoidal rule (b) Simpson’s rule with the specified number of subintervals. 2 x² dx; n = 4 J1
In Exercises 1 through 26, use integration by parts to find the given integral. xex/2 dx
Use integration by parts to find each of these indefinite and definite integrals. a. √2x In x² dx C. c. Lxv -4 xV1 - 2x dx b. d. foxeo xe0.2x dx X 1 f ===¹ dx et
In Exercises 1 through 30, either evaluate the given improper integral or show that it diverges. 1 +00 x-3/2 dx
Repeat Exercise 94 for the curvesData from Exercises 94Use the graphing utility of your calculator to draw the graphs of the curves y = −x3 − 2x2 + 5x − 2 and y = x ln x on the same screen. Use
In Exercises 1 through 10, use integration by parts to find the given integral. tel-t dt
In Exercises 1 through 30, either evaluate the given improper integral or show that it diverges. I oot. 1 1 X dx
In Exercises 1 through 10, determine whether the given function is a probability density function. f(x) = 10 0 for 30 ≤ x ≤ 40 otherwise
In Exercises 1 through 26, use integration by parts to find the given integral. xe -x dx
A car is driven so that after t hours its speed is S(t) miles per hour.a. Write down a definite integral that gives the average speed of the car during the first N hours.b. Write down a definite
Use the graphing utility of your calculator to draw the graphs of the curves y = −x3 − 2x2 + 5x − 2 and y = x ln x on the same screen. Use ZOOM and TRACE or some other feature of your
You have seen that the definite integral can be used to compute the area under a curve, but the “area as an integral” formula works both ways.a. Computeb. Compute SVT 1 - x² dx.
A sprinkler system sprays water onto a garden in such a way that 11e−r2/10 inches of water per hour are delivered at a distance of r feet from the sprinkler. What is the total amount of water laid
A lake has roughly the same shape as the bottom half of the solid formed by rotating the curve 2x2 + 3y2 = 6 about the x axis, for x and y measured in miles. Conservationists want the lake to contain
The population (in thousands) of a certain city t years after January 1, 2005, is given by the functionWhat is the average population of the city during the decade 2005–2015? P(t) = 150e0.03t 1 +
A study conducted by a certain state determines that the Lorenz curves for high school teachers and real estate brokers are given by the functionsL1(x) = 0.67x4 + 0.33x3L2(x) = 0.72x2 +
A study suggests that the distribution of incomes for social workers and physical therapists may be represented by the Lorenz curves y = L1(x) and y = L2(x), respectively, where L1(x) = x1.6 and
A toxin is introduced to a bacterial colony, and t hours later, the population P(t) of the colony is changing at the rateIf there were 1 million bacteria in the colony when the toxin was introduced,
Verify the sum rule for definite integrals; that is, if f(x) and g(x) are continuous on the interval a ≤ x ≤ b, then [₁ - fro [f(x) + g(x)] dx = f(x) dx + g(x) dx + f²86x
The surface area S of the body of an average person 4 feet tall who weighs w lb changes at the rateThe body of a particular child who is 4 feet tall and weighs 50 lb has surface area 1,365 in2. If
After t minutes, an object moving along a line has velocity v(t) = 1 + 4t + 3t2 meters per minute. How far does the object travel during the third minute?
Jan opens a stock account with $5,000 at the beginning of January and subsequently deposits $200 a month. Unfortunately, the market is depressed, and she finds that t months after depositing a
Find dx 1 + et
At t hours past midnight, the temperature T(°C) in a certain northern city is found to be changing at a rate given by T'(t) = −0.02(t − 7)(t − 14) °C/hour. By how much does the temperature
Records indicate that t months after the beginning of the year, the price of bacon in local supermarkets was P(t) = 0.06t2 − 0.2t + 6.2 dollars per pound. What was the average price of bacon during
Suppose that the consumers’ demand function for a certain commodity is D(q) = 50 − 3q − q2 dollars per unit.a. Find the number of units that will be bought if the market price is $32 per
The concentration of a drug in a patient’s bloodstream t hours after an injection is decreasing at the rateBy how much does the concentration change over the first 4 hours after the injection? C'
Find 1 1 + e X dx.
It is estimated that x months from now, the population of a certain town will be changing at the rate of P'(x) = 2 + 1.5√x people per month. The current population is 5,000.a. Find a function P(x)
A tree has been transplanted and after x years is growing at the rate ofmeters per year. By how much does the tree grow during the second year? h'(x) = = 0.5 + 1 (x + 1)²
Find x¹/³ (x²/3 + 1)³/2 dx.
Find fe=³(1 + e¹² dx.
A ball is thrown upward from the top of a building, and t seconds later has velocity v(t) = −32t + 80 ft/sec. What is the change in the ball’s position over the first 3 seconds?
A study conducted by an environmental group in the year 2005 determined that t years later, the population of a certain endangered bird species will be decreasing at the rate ofindividuals per year.
Paloma, while driving at a constant speed of 45 mph, begins to speed up in such a way that her velocity t hours later is v(t) = 45 + 12t mph. How far does she travel in the first 2 hours?
A certain oil well that yields 900 barrels of crude oil per month will run dry in 3 years. The price of crude oil is currently $92 per barrel and is expected to rise at the constant rate of 80 cents
A nuclear power plant produces radioactive waste at a constant rate of 300 pounds per year. The waste decays exponentially with a half-life of 35 years. How much of the radioactive waste from the
Find 2x 1 + et dx.
In a learning experiment, subjects are given a series of facts to memorize, and it is determined that t minutes after the experiment begins, the average subject is learning at the ratewhere L(t) is
a. Prove the constant rule:b. Prove the exponential rule: fkd k dx = kx + C.
In Exercises 72 through 75, the velocity v(t) = x'(t) at time t of an object moving along the x axis is given, along with the initial position x(0) of the object. In each case, find:(a) The position
In Exercises 72 through 75, the velocity v(t) = x'(t) at time t of an object moving along the x axis is given, along with the initial position x(0) of the object. In each case, find:(a) The position
Answer the question in Exercise 75 if the rate of disintegration is given byData from Exercises 75A protein with mass m (grams) disintegrates into amino acids at a rate given byHow much more protein
A protein with mass m (grams) disintegrates into amino acids at a rate given byHow much more protein is there after 2 hours than after 5 hours? dm dt -2 t + 1 g/hr
Our spy, intent on avenging the death of Siggy Leiter (Exercise 75 in Section 4.2), is driving a sports car toward the lair of the fiend who killed his friend. To remain as inconspicuous as possible,
A bicycle manufacturer expects that x months from now consumers will be buying 5,000 bicycles per month at the price of P(x) = 200 + 3√x dollars per bicycle. What is the average revenue the
In Exercises 72 through 75, the velocity v(t) = x'(t) at time t of an object moving along the x axis is given, along with the initial position x(0) of the object. In each case, find:(a) The position
In a certain community the fraction of the homes placed on the market that remain unsold for at least t weeks is approximately f(t) = e−0.2t. If 200 homes are currently on the market and if
What is the present value of an investment that will generate income continuously at a constant rate of $1,000 per year for 10 years if the prevailing annual interest rate remains fixed at 7%
The city manager of Playa Linda estimates that water is being consumed by his community at the rate of C'(t) = 10 + 0.3e0.03t billion gallons per year, where C(t) is the water consumption t years
Money is transferred continuously into an account at the constant rate of $1,200 per year. The account earns interest at the annual rate of 8% compounded continuously. How much will be in the account
If H'(x) = 0 for all real numbers x, what must be true about the graph of H(x)? Explain how your observation can be used to show that if G'(x) = F'(x) for all x, then G(x) = F(x) + C for constant C.
A study indicates that t months from now the population of a certain town will be growing at the rate of P'(t) = 5 + 3t2/3 people per month. By how much will the population of the town increase over
In Exercises 1 through 30, either evaluate the given improper integral or show that it diverges. +00 Vx dx
In a certain section of the country, the price of large Grade A eggs is currently $2.50 per dozen. Studies indicate that x weeks from now, the price p(x) will be changing at the rate of p'(x) = 0.2 +
In a certain suburb of Los Angeles, the level of ozone L(t) at 7:00 A.M. is 0.25 parts per million (ppm). A 12-hour weather forecast predicts that the ozone level t hours later will be changing at
A roast is removed from the freezer of a refrigerator and left on the counter to defrost. The temperature of the roast was −4°C when it was removed from the freezer and t hours later, was
In Exercises 72 through 75, the velocity v(t) = x'(t) at time t of an object moving along the x axis is given, along with the initial position x(0) of the object. In each case, find:(a) The position
The concentration C(t) in milligrams per cubic centimeter (mg/cm3) of a drug in a patient’s bloodstream is 0.5 mg/cm3 immediately after an injection and t minutes later is decreasing at the rateA
Ashok’s investment portfolio changes value at the ratewhere V is in thousands of dollars and t is the number of years after 2006. By how much does the value of Ashok’s portfolio change between
It is estimated that t years from the beginning of the year 2012, the demand for oil in a certain country will be changing at the rate of D'(t) = (1 + 2t)−1 billion barrels per year. Will more oil
The Mitscherlich model for agricultural production specifies that the size Q(t) of a crop changes at a rate proportional to the difference between the maximum possible crop size B and Q; that is,a.
The different members or organs of an individual often grow at different rates, and an important part of allometry involves the study of relationships among these growth rates (recall the Think About
An advertising agency begins a campaign to promote a new product and determines that t days later, the number of people N(t) who have heard about the product is changing at a rate given byHow many
A protein with mass m (grams) disintegrates into amino acids at a rate given byWhat is the net change in mass of the protein during the first 4 hours? dm dt || - 15t t + 5 -
An oil spill in the ocean is roughly circular in shape, with radius R(t) feet t minutes after the spill begins. The radius is increasing at the ratea. Find an expression for the radius R(t), assuming
Repeat Exercise 64 for the curvesData from Exercises 64.Use the graphing utility of your calculator to draw the graphs of the curves y = x2e−x and y = 1/x on the same screen. Use ZOOM and TRACE or
Money is transferred continuously into an account at the rate of 5,000e0.015t dollars per year at time t (years). The account earns interest at the annual rate of 5% compounded continuously. How much
An object is moving so that its velocity after t minutes is v(t) = 3 + 2t + 6t2 meters per minute. How far does the object travel during the second minute?
The Weber-Fechner law in experimental psychology specifies that the rate of change of a response R with respect to the level of stimulus S is inversely proportional to the stimulus; that is,Let S0 be
One of Poiseuille’s laws for the flow of blood in an artery says that if v(r) is the velocity of flow r cm from the central axis of the artery, then the velocity decreases at a rate proportional to
Bejax Corporation has set up a production line to manufacture a new type of cellular telephone. The rate of production of the telephones isHow many telephones are produced during the third month?
An environmental study of a certain community suggests that t years from now the level L(t) of carbon monoxide in the air will be changing at the rate of L'(t) = 0.1t + 0.1 parts per million (ppm)
The output of a factory is changing at the ratewhere t is the number of hours after the morning shift begins at 8 A.M. How many units are produced between 10 A.M. and noon? Q'(t) = 2t³ - 3t² + 10t
A tree has been transplanted and after x years is growing at the rate ofmeters per year. After 2 years, it has reached a height of 5 meters. How tall was it when it was transplanted? 1 + 1 (x + 1)²
In Exercises 63 through 66, the supply S(t) and demand D(t) functions for a commodity are given in terms of the unit price p(t) at time t. Assume that price changes at a rate proportional to the
It is estimated that x weeks from now, the number of commuters using a new subway line will be increasing at the rate of 18x2 + 500 per week. Currently, 8,000 commuters use the subway. How many will
A new medical procedure is applied to a cancerous tumor with volume 30 cm3, and t days later the volume is found to be changing at the rate V'(t) = 0.15 − 0.09e0.006t cm3/daya. Find a formula for
Consider an object moving along a straight line. Explain why the object’s average velocity over any time interval equals the average value of its velocity over that interval.
The promoters of a county fair estimate that t hours after the gates open at 9:00 A.M. visitors will be entering the fair at the rate of −4(t + 2)3 + 54(t + 2)2 people per hour. How many people
The resale value of a certain industrial machine decreases at a rate that changes with time. When the machine is t years old, the rate at which its value is changing is 200(t − 6) dollars per year.
The marginal cost of producing a certain commodity is C'(q) = 6q + 1 dollars per unit when q units are being produced.a. What is the total cost of producing the first 10 units?b. What is the cost of
Let f(x) represent the total number of items a subject has memorized x minutes after being presented with a long list of items to learn. Psychologists refer to the graph of y = f(x) as a learning
Rob is taking a learning test in which the time he takes to memorize items from a given list is recorded. Let M(t) be the number of items he can memorize in t minutes. His learning rate is found to
A researcher models the temperature T (in C) during the time period from 6 A.M. to 6 P.M. in a certain northern city by the functionwhere t is the number of hours after 6 A.M.a. What is the average
It is estimated that x years from now, the value V(x) of an acre of Carter Hightower’s farmland will be increasing at the rate ofdollars per year. The land is currently worth $500 per acre.a. Find
In Exercises 63 through 66, the supply S(t) and demand D(t) functions for a commodity are given in terms of the unit price p(t) at time t. Assume that price changes at a rate proportional to the
It is estimated that t days from now a farmer’s crop will be increasing at the rate of 0.5t2 + 4(t + 1)−1 bushels per day. By how much will the value of the crop increase during the next 6 days
A conservationist finds that the population P(t) of a certain endangered species is growing at a rate given by P'(t) = 0.51e−0.03t, where t is the number of years after records began to be kept. If
Statistics compiled by the local department of corrections indicate that x years from now the number of inmates in county prisons will be increasing at the rate of 280e0.2x per year. Currently, 2,000
Show that the average value V of a continuous function f(x) over the interval a ≤ x ≤ b may be computed as the slope of the line joining the points (a, F(a)) and (b, F(b)) on the curve y = F(x),
In Exercises 63 through 66, the supply S(t) and demand D(t) functions for a commodity are given in terms of the unit price p(t) at time t. Assume that price changes at a rate proportional to the
Use the graphing utility of your calculator to draw the graphs of the curves y = x2e−x and y = 1/x on the same screen. Use ZOOM and TRACE or some other feature of your calculator to find where the
The resale value of a certain industrial machine decreases over a 10-year period at a rate that changes with time. When the machine is x years old, the rate at which its value is changing is 220(x
In Exercises 63 through 66, the supply S(t) and demand D(t) functions for a commodity are given in terms of the unit price p(t) at time t. Assume that price changes at a rate proportional to the
It is estimated that the demand for a manufacturer’s product is increasing exponentially at the rate of 2% per year. If the current demand is 5,000 units per year and if the price remains fixed at
Find the function whose tangent line has slope xe−2x2 for each x and whose graph passes through the point (0, −3).

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