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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

Find the function whose tangent line has slope x(x2 + 1)−1 for each x and whose graph passes through the point (1, 5).
In Exercises 59 through 62, solve the given initial value problem. dy dt t + 1 t where y 3 when t = 1
An environmentalist finds that a certain type of tree grows in such a way that its height h(t) after t years is changing at the rate ofIf the tree was 2 feet tall when it was planted, how tall will
It is estimated that t days from now a farmer’s crop will be increasing at the rate of 0.3t2 + 0.6t + 1 bushels per day. By how much will the value of the crop increase during the next 5 days if
Anita Bellman is a retailer who specializes in grains. She receives a shipment of 10,000 kilograms of rice that will be used up over a 5-month period at the constant rate of 2,000 kilograms per
After t hours of observation, the population P(t) of a bacterial colony is found to be changing at the rateIf the population was 200,000 bacteria when observation began, what will it be 12 hours
In certain biological models, the human body’s reaction to a particular kind of medication is measured by a function of the formwhere k is a positive constant and M is the amount of medication
A retiree deposits S dollars into an account that earns interest at an annual rate r compounded continuously, and annually withdraws W dollars.a. Explain why the account changes at the ratewhere V(t)
A model* of the cardiovascular system relates the stroke volume V(t) of blood in the aorta at time t during systole (the contraction phase) to the pressure P(t) in the aorta at the same time by the
In Exercises 59 through 62, solve the given initial value problem.dy/dx = x(x − 1), where y = 1 when x = 1
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 N'(t) people per hour. Find an expression for the number of
An investor makes regular deposits totaling D dollars each year into an account that earns interest at the annual rate r compounded continuously.a. Explain why the account grows at the ratewhere V(t)
A biomass is growing at the rate of M'(t) = 0.5e0.2t g/hr. By how much does the mass change during the second hour?
The owner of a fast-food chain determines that if x thousand units of a new meal item are supplied, then the marginal price at that level of supply is given bywhere p(x) is the price (in dollars) per
A patient is injected with a drug, and t hours later, the concentration of the drug remaining in the patient’s bloodstream is given byWhat is the average concentration of drug during the first 8
The price p (dollars per unit) of a particular commodity is increasing at the ratewhen x hundred units of the commodity are supplied to the market. The manufacturer supplies 200 units (x = 2) when
Normally, the metabolism of an organism functions at an essentially constant rate, called the basal metabolic rate of the organism. However, the metabolic rate may increase or decrease depending on
The manager of a shoe store determines that the price p (dollars) for each pair of a popular brand of sports sneakers is changing at the rate ofwhen x (hundred) pairs are demanded by consumers. When
In Exercises 51 through 58, find the area of the region R that lies under the given curve y = f(x) over the indicated interval a ≤ x ≤ b. Under y 3 X over 1 ≤x≤ e²
In Exercises 55 through 58, find the volume of the solid of revolution formed by rotating the specified region R about the x axis.R is the region under the curvefrom x = 1 to x = 4. y = x + 1 √x
It is estimated that t months from now the population of a certain town will be increasing at the rate of 4 + 5t2/3 people per month. If the current population is 10,000, what will the population be
In Exercises 51 through 58, find the area of the region R that lies under the given curve y = f(x) over the indicated interval a ≤ x ≤ b. Under y 3 5- 2x over -2 ≤ x ≤ 1
Repeat Exercise 55 for marginal revenueand for the marginal cost C'(x) = 2 + x + x2.Data from Exercises 55A company determines that the marginal revenue from the production of x units of its product
In Exercises 59 through 62, solve the given initial value problem.dy/dx = 2, where y = 4 when x = −3
Suppose the owner of the oil well in Exercise 57 decides to step up production so that 600 barrels per month are extracted but everything else remains the same.a. How many months pass before the well
For several weeks, the highway department has been recording the speed of freeway traffic flowing past a certain downtown exit. The data suggest that between the hours of 1:00 and 6:00 P.M. on a
It is estimated that t years from now the value of a certain parcel of land will be increasing at the rate of V'(t) dollars per year. Find an expression for the amount by which the value of the land
In Exercises 55 through 58, find the volume of the solid of revolution formed by rotating the specified region R about the x axis.R is the region under the curvefrom x = 1 to x = 3. y = 1 Vx
The aerobic rating of a person x years old is given byWhat is a person’s average aerobic rating from age 15 to age 25? From age 60 to age 70? A(x) = 110(In x - 2) X for x ≥ 10
A second study of the country in Exercise 54 indicates that natural restrictive forces are at work that make the actual rate of growthinstead of the exponential rate P'1(t) = 10e0.02t. If this rate
A manufacturer estimates marginal revenue to be 200q−1/2 dollars per unit when the level of production is q units. The corresponding marginal cost has been found to be 0.4q dollars per unit. If the
Suppose the consumption function for a particular country is c(x), where x is national disposable income. Then the marginal propensity to consume is c'(x). Suppose x and c are both measured in
A certain oil well that yields 400 barrels of crude oil per month will run dry in 2 years. The price of crude oil is currently $98 per barrel and is expected to rise at the constant rate of 40 cents
At a certain factory, when K thousand dollars is invested in the plant, the production Q is changing at a rate given byunits per thousand dollars invested. When $8,000 is invested, the level of
In Exercises 55 through 58, find the volume of the solid of revolution formed by rotating the specified region R about the x axis.R is the region under the curve y = e−x/10 from x = 0 to x = 10.
In Exercises 51 through 58, find the area of the region R that lies under the given curve y = f(x) over the indicated interval a ≤ x ≤ b. Under y 3 V9 - 2x over -8 ≤ x ≤0
In Exercises 51 through 58, find the area of the region R that lies under the given curve y = f(x) over the indicated interval a ≤ x ≤ b.Under y = xe−x2 over 0 ≤ x ≤ 3
In a certain section of the country, it is estimated that t weeks from now, the price of chicken will be increasing at the rate ofcents per kilogram per week. If chicken currently costs $2.30 per
In Exercises 55 through 58, find the volume of the solid of revolution formed by rotating the specified region R about the x axis.R is the region under the curve y = x2 + 1 from x = −1 to x = 2.
The marginal revenue from the sale of x units of a particular commodity is estimated to bewhere R(x) is revenue in dollars.a. Find R(x), assuming that R(0) = 0.b. What revenue should be expected from
In Exercises 51 through 58, find the area of the region R that lies under the given curve y = f(x) over the indicated interval a ≤ x ≤ b.Under y = e2x, over 0 ≤ x ≤ ln 3
A company determines that the marginal revenue from the production of x units of its product is R'(x) = x(5 − x)3 hundred dollars per unit, while the corresponding marginal cost is C'(x) = 5 + 2x
The population of a third-world country grows exponentially at the unsustainable rate of P'1(t) = 10e0.02t thousand people per year where t is the number of years after 2005. A study indicates that
The population of a certain community t years after the year 2000 is given byWhat was the average population of the community during the decade from 2000 to 2010? P(t) e0.21 4 + e0.2t million people
In Exercises 51 through 54, an initial population P0 is given along with a renewal rate R(t) and a survival function S(t). In each case, use the given information to find the population at the end of
In Exercises 51 through 54, an initial population P0 is given along with a renewal rate R(t) and a survival function S(t). In each case, use the given information to find the population at the end of
The marginal profit of a certain commodity is P'(q) = 100 − 2q when q units are produced. When 10 units are produced, the profit is $700.a. Find the profit function P(q).b. What production level q
The number of bacteria present in a certain culture after t minutes of an experiment was Q(t) = 2,000e0.05t. What was the average number of bacteria present during the first 5 minutes of the
In Exercises 51 through 58, find the area of the region R that lies under the given curve y = f(x) over the indicated interval a ≤ x ≤ b.Under y = (3x + 4)1/2, over 0 ≤ x ≤ 4
In Exercises 51 through 54, an initial population P0 is given along with a renewal rate R(t) and a survival function S(t). In each case, use the given information to find the population at the end of
The resale value of a certain industrial machine decreases at a rate that depends on its age. When the machine is t years old, the rate at which its value is changing is −960e−t/5 dollars per
Suppose a study indicates that the distribution of income for professional baseball players is given by the Lorenz curve y = L1(x), wherewhile those of professional football players and basketball
In a certain state, it is found that the distribution of income for lawyers is given by the Lorenz curve y = L1(x), wherewhile that of surgeons is given by y = L2(x), whereCompute the Gini index for
The monthly sales at an import store are currently $10,000 but are expected to be declining at the rate oft months from now. The store is profitable as long as the sales level is above $8,000 per
Suppose it has been determined that the marginal revenue associated with the production of x units of a particular commodity is R'(x) = 240 − 4x dollars per unit. What is the revenue function R(x)?
In Exercises 45 through 50, f(x) and g(x) are functions that are continuous on the interval −3 ≤ x ≤ 2 and satisfyIn each case, use this information along with rules for definite integrals to
In Exercises 47 through 50, use an appropriate substitution to solve the given separable differential equation. dy dx II. 2-1² 好
After initiating an advertising campaign, Lazaro Jimenez, the manager of a satellite dish provider in an urban area, estimates that t months from now, the number of new subscribers N(t) will be
A square cabin with a plot of land adjacent to a lake is shown in the accompanying figure. If a coordinate system is set up as indicated, with distances given in yards, the lakefront boundary of the
In Exercises 51 through 54, an initial population P0 is given along with a renewal rate R(t) and a survival function S(t). In each case, use the given information to find the population at the end of
An experiment is conducted with two bacterial colonies, each of which initially has a population of 100,000. In the first colony, a mild toxin is introduced that restricts growth so that only 50 new
Use integration to show that a right circular cone of height h and top radius r has volume 1 V v==arth 3
In Exercises 51 through 58, find the area of the region R that lies under the given curve y = f(x) over the indicated interval a ≤ x ≤ b.Under y = √x(x + 1), over 0 ≤ x ≤ 4
In Exercises 45 through 50, f(x) and g(x) are functions that are continuous on the interval −3 ≤ x ≤ 2 and satisfyIn each case, use this information along with rules for definite integrals to
Use integration to show that a sphere of radius r has volume V || 4 3 T 3
In Exercises 47 through 50, use an appropriate substitution to solve the given separable differential equation. dx dt In t xt
At a certain factory, the marginal cost is 3(q − 4)2 dollars per unit when the level of production is q units.a. Express the total production cost in terms of the overhead (the cost of producing 0
In Exercises 45 through 50, f(x) and g(x) are functions that are continuous on the interval −3 ≤ x ≤ 2 and satisfyIn each case, use this information along with rules for definite integrals to
In Exercises 47 through 50, use an appropriate substitution to solve the given separable differential equation. dy dx 2-y (x + 1)²
In Exercises 47 through 50, use an appropriate substitution to solve the given separable differential equation. dy dx ху 2x 1
In Exercises 51 through 58, find the area of the region R that lies under the given curve y = f(x) over the indicated interval a ≤ x ≤ b.Under y = x4, over −1 ≤ x ≤ 2
In Exercises 47 through 50, sketch the Lorenz curve y = L(x) and find the corresponding Gini index.L(x) = 0.75x2 + 0.25x
The territory occupied by a certain community is bounded on one side by a river and on all other sides by mountains, forming the shaded region shown in the accompanying figure. If a coordinate system
In Exercises 45 through 50, f(x) and g(x) are functions that are continuous on the interval −3 ≤ x ≤ 2 and satisfyIn each case, use this information along with rules for definite integrals to
A manufacturer estimates marginal revenue to be R'(q) = 100q−1/2 dollars per unit when the level of production is q units. The corresponding marginal cost has been found to be 0.4q dollars per
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) = 2x 2' 1 +
In Exercises 47 through 50, sketch the Lorenz curve y = L(x) and find the corresponding Gini index.L(x) = 0.3x2 + 0.7x
In Exercises 45 through 50, f(x) and g(x) are functions that are continuous on the interval −3 ≤ x ≤ 2 and satisfyIn each case, use this information along with rules for definite integrals to
In Exercises 43 through 46, solve the given separable initial value problem. dy dx √√ y = 1 when x = 1
An annuity pays a continuous income stream of M dollars per year into an account that pays interest at an annual rate r compounded continuously for a term of T years.a. Show that the future value of
In Exercises 47 through 50, sketch the Lorenz curve y = L(x) and find the corresponding Gini index.L(x) = x1.2
A manufacturer estimates that the marginal cost of producing q units of a certain commodity is C'(q) = 3q2 − 24q + 48 dollars per unit. If the cost of producing 10 units is $5,000, what is the cost
The cost of producing x units of a new product is C(x) = 3x√x + 10 hundred dollars. What is the average cost of producing the first 81 units?
In Exercises 45 through 50, f(x) and g(x) are functions that are continuous on the interval −3 ≤ x ≤ 2 and satisfyIn each case, use this information along with rules for definite integrals to
The marginal revenue derived from producing q units of a certain commodity is R'(q) = 4q − 1.2q2 dollars per unit. If the revenue derived from producing 20 units is $30,000, how much revenue should
In Exercises 47 through 50, sketch the Lorenz curve y = L(x) and find the corresponding Gini index.L(x) = x3/2
A tumor has approximately the same shape as the solid formed by rotating the region under the curveabout the x axis, where x and y are measured in centimeters. Use integration to estimate the volume
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)=x√x² + 5;
A company determines that if L worker-hours of labor are employed, then Q units of a particular commodity will be produced, where Q(L) = 500L2/3a. What is the average production as labor varies from
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
After t hours on the job, one factory worker is producing Q'1(t) = 60 − 2(t − 1)2 units per hour, while a second worker is producing Q'2(t) = 50 − 5t units per hour.a. If both arrive on the job
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. Su -2/5 du
A tumor has approximately the same shape as the solid formed by rotating the region under the curve y = √x (A − x)about the x axis, where x and y are measured in centimeters, for some
Having ransomed his superior in Exercise 49 of Section 3.5, the spy returns home, only to learn that his best friend, Sigmund (“Siggy”) Leiter, has been murdered. The police say that Siggy’s
In Exercises 43 through 46, solve the given separable initial value problem.dy/dx = ex+y; y = 0 when x = 0
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) = xe4−x2;
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. fxe* dx
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
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
A tumor has approximately the same shape as the solid formed by rotating the region under the curve y = x(4 − x) about the x axis, where x and y are measured in centimeters. Use integration to

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