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mathematics
calculus with applications
Calculus For Business, Economics And The Social And Life Sciences 11th Brief Edition Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price - Solutions
The population of a small town is currently 85,000. A study commissioned by the mayor’s office finds that people are settling in the town at the rate of R(t) = 1,200e0.01t per year and that the fraction of the population who continue to live in the town t years after arriving is given by S(t) =
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. 0 (2x + 6)* dx J-3
A pharmaceutical firm has been granted permission by the FDA to test the effectiveness of a new drug in combating a virus. The firm administers the drug to a test group of uninfected but susceptible individuals, and using statistical methods, determines that t months after the test begins, people
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. X ਦਿਲ 4 - 3x dx
In Exercises 29 through 34, find the Gini index for the given Lorenz curve. L(x) ex et - 1 e - 1
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. So 1 V 6t + 1 dt
In Exercises 31 through 38, sketch the indicated region R and find its area by integration.R is the region under the curve y = 1/x + x2 over the interval 1 ≤ x ≤ 2.
In Exercises 31 through 34, solve the given initial value problem for y = f(x). dy dx x + 1 Vx where y = 5 when x = 4
Oil is being pumped from an oil field t years after its opening at the rate of P'(t) = 1.3e0.04t billion barrels per year. The field has a reserve of 20 billion barrels, and the price of oil holds steady at $112 per barrel.a. Find P(t), the amount of oil pumped from the field at time t. How much
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. 1² (x³ + 1)² fa dx
In Exercises 31 through 38, sketch the indicated region R and find its area by integration.R is the region bounded by the curve y = 4/x and the line x + y = 5.
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. 1 √x(√x + 1) dx
In Exercises 35 through 42, the slope f'(x) at each point (x, y) on a curve y = f(x) is given along with a particular point (a, b) on the curve. Use this information to find f(x).f'(x) = 4x + 1; (1, 2)
After t months on the job, a postal clerk can sort Q(t) = 700 − 400e−0.5t letters per hour. What is the average rate at which the clerk sorts mail during the first 3 months on the job?
Answer the questions in Exercise 35 for another oil field with a pumping rate of P'(t) = 1.5e0.03t and with a reserve of 16 billion barrels. You may assume that the price of oil is still $112 per barrel and that the prevailing annual interest rate is 5%.Data from Exercises 35Oil is being
In Exercises 31 through 38, sketch the indicated region R and find its area by integration.R is the region bounded by the curves y = 8/x and y = √x and the line x = 8.
In a certain undeveloped country, the life expectancy of a person t years old is L(t) years, where L(t) = 41.6(1 + 1.07t)0.13a. What is the life expectancy of a person in this country at birth? At age 50?b. What is the average life expectancy of all people in this country between the ages of 10 and
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. 2/3 (²-1) ²³. dx
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. La= (x³ + x)√x² + 2x² + 1 dx
In Exercises 35 through 42, the slope f'(x) at each point (x, y) on a curve y = f(x) is given along with a particular point (a, b) on the curve. Use this information to find f(x).f'(x) = 3 − 2x; (0, −1)
Answer the questions in Exercise 36 for a country whose life expectancy function isData from Exercises 36In a certain undeveloped country, the life expectancy of a person t years old is L(t) years, where L(t) = 41.6(1 + 1.07t)0.13a. What is the life expectancy of a person in this country at birth?
An inventory of 60,000 kilograms of a certain commodity is used at a constant rate and is exhausted after 1 year. What is the average inventory for the year?
In Exercises 31 through 38, sketch the indicated region R and find its area by integration.R is the region bounded by the curve y = 2 + x − x2 and the x axis.
A ruptured pipe in an offshore oil rig produces a circular oil slick that is T feet thick at a distance r feet from the rupture, whereAt the time the spill is contained, the radius of the slick is 7 feet. We wish to find the volume of oil that has been spilled.a. Sketch the graph of T(r). Notice
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. fe+1 X x-1 dx
Repeat Exercise 39 with the inspiration rate function R(t) = −1.2t3 + 5.72t liters/sec and sketch the graph of R(t).Data from Exercises 39A pneumotachograph is a device used by physicians to graph the rate of air flow into and out of the lungs as a patient breathes. The graph in the accompanying
A pneumotachograph is a device used by physicians to graph the rate of air flow into and out of the lungs as a patient breathes. The graph in the accompanying figure shows the rate of inspiration (breathing in) for a particular patient. The area under the graph measures the total volume of air
Rework Exercise 41 for a situation with spill thickness(T and r in feet) and radius of containment 9 feet.Data from Exercises 41A ruptured pipe in an offshore oil rig produces a circular oil slick that is T feet thick at a distance r feet from the rupture, whereAt the time the spill is contained,
In Exercises 35 through 42, the slope f(x) at each point (x, y) on a curve y = f(x) is given along with a particular point (a, b) on the curve. Use this information to find f(x). 2 f(x) = x²³ - 1²/1/² + 2; (1, 3)
In Exercises 37 through 42, solve the given initial value problem for y = f(x).where y = 1 when x = 0 dy dx 1 x + 1
A marketing survey indicates that t months after a new type of computerized air purifier is introduced to the market, sales will be generating profit at the rate of P'(t) thousand dollars per month, wherea. When is the rate of profitability positive and when is it negative? When is the rate
In Exercises 35 through 42, the slope f'(x) at each point (x, y) on a curve y = f(x) is given along with a particular point (a, b) on the curve. Use this information to find f(x).f'(x) = 3x2 + 6x − 2; (0, 6)
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. 2 La 1 (t + 1)(t - 2)° dt
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. (In x)² X xp. dx
In Exercises 39 through 42, find the average value of the given function over the indicated interval. f(t) = t√/8 - 7t²; over 0 ≤ t ≤ 1
Suppose that t years from now, one investment plan will be generating profit at the rate of P1'(t) = 100 + t2 hundred dollars per year, while a second investment will be generating profit at the rate of P2'(t) = 220 + 2t hundred dollars per year.a. For how many years does the rate of profitability
Marya invests $10,000 for 5 years in a bank that pays 5% annual interest.What is the average value of her account over this time period if interest is compounded continuously?
In Exercises 39 through 42, find the average value of the given function over the indicated interval.f(x) = x3 − 3x + 12x; over 1 ≤ x ≤ 8
In Exercises 37 through 42, solve the given initial value problem for y = f(x).where y = 2 when x = 1 dy dx In Vx X
The winner of a state lottery is offered a choice of either receiving $10 million now as a lump sum or of receiving A dollars a year for the next 6 years as a continuous income stream. If the prevailing annual interest rate is 5% compounded continuously and the two payouts are worth the same, what
An investment produces a continuous income stream at the rate of A(t) thousand dollars per year at time t, where A(t) = 10e1−0.05t The prevailing rate of interest is 5% per year compounded continuously.a. What is the future value of the investment over a term of 5 years (0 ≤ t ≤ 5)?b. What is
A star baseball free agent is the object of a bidding war between two rival teams. The first team offers a 3 million dollar signing bonus and a 5-year contract guaranteeing him 8 million dollars this year and an increase of 3% per year for the remainder of the contract. The second team offers $9
In Exercises 37 through 42, solve the given initial value problem for y = f(x).where y = 3 when x = −1 dy dx x+2 ² + 4x + 5
In Exercises 39 through 42, find the average value of the given function over the indicated interval. h(x) et 1 + 2et 2eti over 0≤x≤ 1
In Exercises 37 through 42, solve the given initial value problem for y = f(x).dy/dx = e2−x where y = 0 when x = 2
Particulate matter emitted from a smokestack is distributed in such a way that r miles from the stack, the pollution density is p(r) units per square mile, wherea. What is the total amount of pollution within a 3-mile radius of the smokestack?b. Suppose a health agency determines that it is unsafe
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. 1 x ln x -dx
Answer the questions in Exercise 40 for two investments with respective rates of profitability P1'(t) = 60e0.12t thousand dollars per year and P2'(t) = 160e0.08t thousand dollars per year.Data from Exercises 40Suppose that t years from now, one investment plan will be generating profit at the rate
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. (1/2 el/x dx 1₁3 2² a
In Exercises 35 through 42, the slope f(x) at each point (x, y) on a curve y = f(x) is given along with a particular point (a, b) on the curve. Use this information to find f(x).f'(x) = x−1/2 + x; (1, 2)
In Exercises 35 through 42, the slope f(x) at each point (x, y) on a curve y = f(x) is given along with a particular point (a, b) on the curve. Use this information to find f(x). f'(x) = 3 X 4; (1, 0)
In Exercises 35 through 42, the slope f(x) at each point (x, y) on a curve y = f(x) is given along with a particular point (a, b) on the curve. Use this information to find f(x).f'(x) = e−x + x2; (0, 4)
In Exercises 39 through 42, find the average value of the given function over the indicated interval.g(v) = ve−v2; over 0 ≤ v ≤ 2
In Exercises 43 through 46, solve the given separable initial value problem.dy/dx = −2y; y = 3 when x = 0
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. 1 ¹(√x - 1)³/2 Vx dx
Answer the questions in Exercise 40 for two investments with respective rates of profitabilityP1'(t) = 90e0.1t thousand dollars per year and P2'(t) = 140e0.07t thousand dollars per yearData from Exercises 40Suppose that t years from now, one investment plan will be generating profit at the rate of
Everett Dunn’s advertising firm has been hired to promote a new television series for 3 weeks before its debut and 2 weeks afterward. After t weeks of the advertising campaign, Everett estimates that P(t) percent of the viewing public is aware of the series, wherea. What is the average percentage
In Exercises 43 through 46, the slope f'(x) at each point (x, y) on a curve y = f(x) is given, along with a particular point (a, b) on the curve. Use this information to find f(x).f'(x) = (1− 2x)3/2; (0, 0)
In Exercises 43 through 46, solve the given separable initial value problem.dy/dx = xy; y = 1 when x = 0
In Exercises 43 through 46, p = D(q) is the demand curve for a particular commodity; that is, q units of the commodity will be demanded when the price is p = D(q) dollars per unit. In each case, for the given level of production q0, find p0 = D(q0) and compute the corresponding consumers’
Consider the following problem: A certain oil well that yields 300 barrels of crude oil a month will run dry in 3 years. It is estimated that t months from now the price of crude oil will be P(t) = 118 + 0.3√t dollars per barrel. If the oil is sold as soon as it is extracted from the
Records indicate that t months after the beginning of the year, the price of ground beef in local supermarkets wasdollars per pound. What was the average price of ground beef during the first 3 months of the year? P(t) = 0.09² 0.2t + 4
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. 6t 1² + 1 2 dt
In Exercises 31 through 38, sketch the indicated region R and find its area by integration.R is the triangular region with vertices (0, 0), (2, 4), and (0, 6).
Answer the questions in Exercise 35 for an oil field with a pumping rate of P'(t) = 1.2e0.02t and with a reserve of 12 billion barrels. Assume that the prevailing interest rate is 5% as before, but that the price of oil after t years is given by A(t) = 112e0.015t.Data from Exercises 35Oil is being
In Exercises 35 through 42, the slope f'(x) at each point (x, y) on a curve y = f(x) is given along with a particular point (a, b) on the curve. Use this information to find f(x).f'(x) = −x(x + 1); (−1, 5)
In Exercises 37 through 42, solve the given initial value problem for y = f(x).dy/dx = (3 − 2x)2 where y = 0 when x = 0
In an investigation by V. A. Tucker and K. Schmidt-Koenig,* it was determined that the energy E expended by a bird in flight varies with the speed v (km/hr) of the bird. For a particular kind of parakeet, the energy expenditure changes at a rate given bywhere E is given in joules per gram mass per
Luisa, a $2 million state lottery winner, is given a $250,000 check now and a continuous income flow at the rate of $200,000 per year for 10 years. If the prevailing rate of interest is 5% per year compounded continuously, is this a good deal for Luisa or not? Explain.
In Exercises 37 through 42, solve the given initial value problem for y = f(x).where y = 3 when x = 1 dy dx √4x + 5
In Exercises 65 through 68, the demand function q = D(p) for a particular commodity is given in terms of a price p per unit at which all q units can be sold. In each case:(a) Find the elasticity of demand and determine the values of p for which the demand is elastic, inelastic, and of unit
A hot drink is taken outside on a cold winter day when the air temperature is −5°C. According to a principle of physics called Newton’s law of cooling, the temperature T (in degrees Celsius) of the drink t minutes after being taken outside is given by a function of the formwhere A and k are
The concentration of a certain drug in an organ t minutes after an injection is given bygrams per cubic centimeter (g/cm3).a. What is the initial concentration of the drug (when t = 0)?b. What is the concentration 20 minutes after an injection? After 1 hour?c. What is the average rate of change of
The general probability density function has the formwhere μ and σ are constants, with σ > 0.a. Show that f (x) has an absolute maximum at x =μ and inflection points at x = μ + σ and x =μ − σ.b. Show that f(μ + c) = f(μ − c) for every numberc. What does this tell you about
The concentration of a certain drug in an organ t minutes after an injection is given bygrams per cubic centimeter (g/cm3).a. What is the initial concentration of the drug (when t = 0)?b. What is the concentration 10 minutes after an injection? After 1 hour?c. What is the average rate of change of
Beth owns an asset whose value t years from now will beIf the prevailing interest rate remains constant at 5% per year compounded continuously, when will it be most advantageous to sell the collection and invest the proceeds? V(t) = 2,000e 2t dollars.
In an experiment designed to test short-term memory,* L. R. Peterson and M. J. Peterson found that the probability p(t) of a subject recalling a pattern of numbers and letters t seconds after being given the pattern isa. What is the probability that the subject can recall the pattern immediately (t
In Exercises 65 through 68, the demand function q = D(p) for a particular commodity is given in terms of a price p per unit at which all q units can be sold. In each case:(a) Find the elasticity of demand and determine the values of p for which the demand is elastic, inelastic, and of unit
In Exercises 69 through 72, the cost C(x) of producing x units of a particular commodity is given. In each case:(a) Find the marginal cost C'(x).(b) Determine the level of production x for which the average costis minimized.C(x) = e0.2x A(x) = C(x) X
In Exercises 69 through 72, the cost C(x) of producing x units of a particular commodity is given. In each case:(a) Find the marginal cost C'(x).(b) Determine the level of production x for which the average costis minimized.C(x) = 100e0.01x A(x) = C(x) X
A naturalist at an animal sanctuary has determined that the functionprovides a good measure of the number of animals in the sanctuary that are x years old. Sketch the graph of f(x) for x > 0, and find the most likely age among the animals, that is, the age for which f(x) is largest. f(x) = 4e
It is determined that the volume of the yolk of a house fly egg shrinks according to the formula V(t) = 5e−1.3t mm3 (cubic millimeters), where t is the number of days from the time the egg is produced. The egg hatches after 4 days.a. What is the volume of the yolk when the egg hatches?b. Sketch
In Exercises 47 through 52, find an equation for the tangent line to y = f (x) at the specified point.f(x) = xe−x; where x = 0
Investors are often interested in knowing how long it takes for a particular investment to double. A simple means for making this determination is the “rule of 70,” which says: The doubling time of an investment with an annual interest rate r% compounded continuously is given by d = 70/r.For
In Exercises 69 through 72, the cost C(x) of producing x units of a particular commodity is given. In each case:(a) Find the marginal cost C'(x).(b) Determine the level of production x for which the average costis minimized.C(x) = 12√x ex/10 A(x) = C(x) X
A radioactive substance decays exponentially with half-life l. Suppose the amount of the substance initially present (when t = 0) is Q0.a. Show that the amount of the substance that remains after t years will be Q(t) = Q0e−(ln 2/λ)t.b. Find a number k so that the amount in part (a) can be
Use the graphing utility of your calculator to sketch the graph of f(x) = x(e−x + e−2x). Use ZOOM and TRACE to find the largest value of f(x). What happens to f(x) as x → + ∞?
“Ötzi the Iceman” is the name given a neolithic corpse found frozen in an Alpine glacier in 1991. He was originally thought to be from the Bronze Age because of the hatchet he was carrying. However, the hatchet proved to be made of copper rather than bronze. Read an article on the Bronze Age,
The half-life of radium is 1,690 years. How long will it take for a 50-gram sample of radium to be reduced to 5 grams?
The amount of a certain radioactive substance remaining after t years is given by a function of the form Q(t) = Q0e−0.003t. Find the half-life of the substance.
According to Fick’s law* f(t) = C(1 − e−kt), where f(t) is the concentration of solute inside a cell at time t, C is the (constant) concentration of solute surrounding the cell, and k is a positive constant. Suppose that for a particular cell, the concentration on the inside of the cell after
Plant life exists only in the top 10 meters of a lake or sea, primarily because the intensity of sunlight decreases exponentially with depth. Specifically, the Bouguer-Lambert law says that a beam of light that strikes the surface of a body of water with intensity I0 will have intensity I at a
Glottochronology is the methodology used by linguists to determine how many years have passed since two modern languages “branched” from a common ancestor. Experiments suggest that if N words are in common use at a base time t = 0, then the number N(t) of them still in use with essentially the
In Exercises 69 through 72, the cost C(x) of producing x units of a particular commodity is given. In each case:(a) Find the marginal cost C'(x).(b) Determine the level of production x for which the average costis minimized.C(x) = x2 + 10xe−x A(x) = C(x) X
The air pressure f (s) at a height of s meters above sea level is given bya. The atmospheric pressure outside an airplane is 0.25 atmosphere. How high is the plane?b. A mountain climber decides she will wear an oxygen mask once she has reached an altitude of 7,000 meters. What is the atmospheric
Jayla falls into a lake where the water temperature is −3°C. Her body temperature after t minutes in the water is T(t) = 35e−0.32t. She will lose consciousness when her body temperature reaches 27°C. How long do rescuers have to save her? How fast is Jayla’s body temperature dropping at the
The amount of a sample of a radioactive substance remaining after t years is given by a function of the form Q(t) = Q0e−0.0001t. At the end of 5,000 years, 200 grams of the substance remain. How many grams were present initially?
Instant coffee is made by adding boiling water (212ºF) to coffee mix. If the air temperature is 70ºF, Newton’s law of cooling says that after t minutes, the temperature of the coffee will be given by a function of the form f(t) = 70 + Ae−kt. After cooling for 2 minutes, the coffee is still
Two graphs y = f(x) and y = g(x) are reflections of one another in the y axis if whenever (a, b) is a point on one of the graphs, then (−a, b) is a point on the other, as indicated in the accompanying figure. Use this criterion to show that the graphs ofare reflections of one another in the y
The temperature T (in degrees Celsius) of the body of a murder victim found in a room where the air temperature is 20°C is given bywhere t is the number of hours after the victim’s death.a. Graph the body temperature T(t) for t ≥ 0. What is the horizontal asymptote of this graph, and what does
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