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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

Demographers estimate that the fraction of people who will still be residing in a particular town t years after they arrive is given by the function f(t) = e−0.02t. If the current population is
In Exercises 5 through 18, sketch the given region R and then find its area.R is the region bounded by the curve y = x2 − 2x and the x axis.
In Exercises 59 through 62, solve the given initial value problem.dx/dt = e−2t, where x = 4 when t = 0
Records indicate that t hours past midnight, the temperature at the local airport was f(t) = −0.3t2 + 4t + 10 degrees Celsius. What was the average temperature at the airport between 9:00 A.M. and
In Exercises 37 through 44, sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed. L Vy+1 -1J-Vy+1 f(x, y) dy dx
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. S [1₂² 10 1-2 (2x + 3y) dy dx
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.In a learning experiment, a subject is first given x minutes to examine a list of facts. The fact sheet is
In Exercises 41 through 46, the demand functions for a pair of commodities are given. Use partial derivatives to determine whether the commodities are substitute, complementary, or neither. D₁ =
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. Lo fex exy dy dx -x-y
In Exercises 45 through 54, use a double integral to find the area of R.R is the triangle with vertices (−4, 0), (2, 0), and (2, 6).
Suppose a person of age A years has weight w in kilograms (kg) and height h in centimeters (cm). Then, the Harris-Benedict equations say that the daily basal energy expenditure in kilocalories is
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.Define the livable space of a building to be the volume of space in the building where a person 6 feet tall
Output in a certain factory is given by the Cobb-Douglas production functionwhere K is the capital in $1,000 and L is the size of the labor force, measured in worker-hours.a. Use your calculator to
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.The beautiful patterns on the wings of butterflies have long been a subject of curiosity and scientific
Pediatricians and medical researchers sometimes use the following empirical formula* relating the surface area S (m2) of a person to the person’s weight W (kg) and height H (cm):a. Find S(15.83,
The thin lens formula in optics says that the focal length L of a thin lens is related to the object distance do and image distance di by the equationIf L remains constant while do and di are allowed
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. S. SAVT 0 xV1 - y dx dy
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.Certain malignant tumors that do not respond to conventional methods of treatment such as surgery or
In Exercises 41 through 46, the demand functions for a pair of commodities are given. Use partial derivatives to determine whether the commodities are substitute, complementary, or neither. D₁ =
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.Tom, Dick, and Mary are participating in a cross-country relay race. Tom will trudge as fast as he can
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. [Lx -1 xe²y dy dx
A jewelry box is constructed by partitioning a box with a square base as shown in the accompanying figure. If the box is designed to have volume 800 cm3, what dimensions should it have to minimize
Suppose the jewelry box in Exercise 48 is designed so that the material in the top costs twice as much as the material in the bottom and sides and three times as much as the material in the interior
Having escaped from Blabba’s village before being detected the spy sneaks into Scelerat’s château. He enters a room and the door slams shut behind him. He begins to feel warm and realizes he is
In Exercises 45 through 54, use a double integral to find the area of R.R is the triangle with vertices (0, −1), (−2, 1), and (2, 1).
A bicycle dealer has found that if 10-speed bicycles are sold for x dollars each and the price of gasoline is y cents per gallon, approximately F(x, y) bicycles will be sold each month,
A particle of mass m in a rectangular box with dimensions x, y, and z has ground state energywhere k is a physical constant. In Exercise 52 of Section 7.3, you were asked to minimize the ground state
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. блу² 1 X2 + 1 ITA -1 dy dx
A manufacturer estimates that the annual output at a certain factory is given byunits, where K is the capital expenditure in units of $1,000 and L is the size of the labor force in worker-hours.a.
Dumping and other material handling operations near a landfill may result in contaminated particles being emitted into the surrounding air. To estimate such particulate emission, the following
One of Poiseuille’s laws† says that the speed of blood V (cm/sec) flowing at a distance r (cm) from the axis of a blood vessel of radius R (cm) and length L (cm) is given bywhere P (dynes/cm2) is
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. "e "e In (xy) dy dx
At a certain factory, the daily output is Q(K, L) = 60K1/2L1/3 units, where K denotes the capital investment measured in units of $1,000 and L the size of the labor force measured in worker-hours.
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = 1/2x2 and y = 2x.
A machine based on wind energy generally converts the kinetic energy of moving air to mechanical energy by means of a device such as a rotating shaft, as in a windmill. Suppose we have wind of
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = √x and y = x2.
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. (1-x SS x( y − 1)² dy dx 10 0
A rectangular building is to be constructed of material that costs $31 per square foot for the roof, $27 per square foot for the sides and the back, and $55 per square foot for the facing and glass
A grocer’s daily profit from the sale of two brands of cat food iscents, where x is the price per can of the first brand and y is the price per can of the second. Currently the first brand sells
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = x2 − 4x + 3 and the x axis.
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = x2 + 6x + 5 and the x axis.
In Exercises 44 through 51, evaluate the double integral. You may need to exchange the order of integration. fife 0 ex dy dx
In Exercises 50 through 52, assume that the required extreme value is a relative extremum.A farmer wishes to fence off a rectangular pasture along the bank of a river. The area of the pasture is to
A person’s intelligence quotient (IQ) is measured by the functionwhere a is the person’s actual age and m is his or her mental age.a. Find I(12, 11) and I(16, 17).b. Sketch the graphs of several
In Exercises 50 through 52, assume that the required extreme value is a relative extremum.Suppose you wish to construct a rectangular box with a volume of 32 ft3. Three different materials will be
In Exercises 50 through 52, assume that the required extreme value is a relative extremum.A particle of mass m in a rectangular box with dimensions x, y, and z has ground state energywhere k is a
In Exercises 52 and 53, evaluate the given double integral for the specified region R.where R is the rectangle with vertices (−1, 0), (2, 0), (2, 3), and (−1, 3). 6x²y dA, JR
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = ln x, y = 0, and x = e.
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = x, y = ln x, y = 0, and y = 1.
In Exercises 73 through 76, evaluate the double integral over the specified region R. Choose the order of integration carefully. R dA; R: Vy≤ x ≤ 1,0 ≤ y ≤ 1
In an electric circuit with two resistors of resistance R1 and R2 connected in parallel, the total resistance R is given by the formulaShow that 1 || -| R R₁ + 1 R₂
A bicycle dealer has found that if 10-speed bicycles are sold for x dollars apiece and the price of gasoline is y cents per gallon, then approximatelybicycles will be sold each month. If the price of
The ideal gas law says that for n moles of an ideal gas, PV = nRT, where P is the pressure exerted by the gas, V is the volume of the gas, T is the temperature of the gas, and R is a constant (the
Each of Exercises 81 through 85 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.Use the formula obtained in Exercise 84
A manufacturer estimates that when x units of a particular commodity are sold domestically and y units are sold to foreign markets, the profit is given byhundred dollars. If monthly domestic sales
For the soft drink can in Exercise 77, the surface area is given by S = 2πR2 + 2πRH. Use calculus to estimate the change in surface that results if:a. The radius is increased from 3 to 4 cm while
A consultant determines that the density of people positively influenced by an advertising campaign in the region shown in the accompanying figure isp(x, y) = x ln y thousand people per square mile
A soft drink can is a cylinder H cm tall with radius R cm. Its volume is given by the formula V = πR2H. A particular can is 12 cm tall with radius 3 cm. Use calculus to estimate the change in volume
Each of Exercises 81 through 85 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.Suppose y = h(x) is a differentiable
At a certain factory, output Q is related to inputs x and y by the expression Q(x, y) = 2x3 + 3x2y + y3 If 0 ≤ x ≤ 5 and 0 ≤ y ≤ 7, what is the average output of the factory?
Sandy Gibbons is an investment advisor with a new product she has been actively promoting. She estimates that in the region shown in the accompanying region, the density of interest is p(x, y) =
Each of Exercises 81 through 85 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.A soft drink can is H centimeters (cm)
Each of Exercises 81 through 85 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.A cylindrical silo set on a concrete
Use the formula obtained in Exercise 84 to find the slope of the level curve x2y + 2y3 − 2e−x = 14 at the point (0, 2). What is the equation of the tangent line to the level curve at this
The likelihood that a person with a contagious disease will infect others in a social situation may be assumed to be a function f(s) of the distance s between individuals. Suppose contagious
Each of Exercises 81 through 85 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.A rectangular garden that is 30 yards
A community is laid out as a rectangular grid in relation to two main streets that intersect at the city center. Each point in the community has coordinates (x, y) in this grid, for −10 ≤ x ≤
Repeat Exercise 80 for V(x, y) = (300 + x + y)e−0.01x and the region −1 ≤ x ≤ 1, −1 ≤ y ≤ 1.Data from Exercises 80A community is laid out as a rectangular grid in relation to two main
Repeat Exercise 80 for V(x, y) = 400xe−y and the region R: 0 ≤ y ≤ x, 0 ≤ x ≤ 1.Data from Exercises 80A community is laid out as a rectangular grid in relation to two main streets that
A health official wants to estimate the number of people susceptible to a new strain of influenza. Examinations conducted in the test region R shown in the accompanying figure suggest that the
A box has the shape of the solid bounded above by the plane 3x + 4y + 2z = 12 below by the xy plane, and on the sides by the planes x = 0 and y = 0, where x, y, and z are in inches. Find the volume
A map of a small regional park is a rectangular grid, bounded by the lines x = 0, x = 4, y = 0, and y = 3, where units are in miles. It is found that the elevation above sea level at each point (x,
A building is to have a curved roof above a rectangular base. In relation to a rectangular grid, the base is the rectangular region −30 ≤ x ≤ 30, −20 ≤ y ≤ 20, where x and y are measured
In a psychological experiment, x units of stimulus A and y units of stimulus B are applied to a subject, whose performance on a certain task is then measured by the function P(x, y) = 10 +
Recall from Exercise 47, Section 7.1, that the surface area of a person’s body may be estimated by the empirical formula S(W, H) = 0.0072W0.425H0.725 where W is the person’s weight in kilograms,
A storage bin is to be constructed in the shape of the solid bounded above by the surface z = 20 − x2 − y2 below by the xy plane, and on the sides by the plane y = 0 and the parabolic
The population density is f(x, y) = 2,500e−0.01x−0.02y people per square mile at each point (x, y) within the triangular region R with vertices (−5, −2), (0, 3), and (5, −2). Find the total
A state legislature decides to create a new district from the region shown in the accompanying figure. It is found that the population density function for the region is p(x, y) = ye−0.2x thousand
The population density is f(x, y) = 1,000y2e−0.01x people per square mile at each point (x, y) within the region R bounded by the parabola x = y2 and the vertical line x = 4. Find the total
In Exercises 96 through 98, use double integration to find the required quantity. In some cases, you may need to use the numeric integration feature of your calculator.Find the area of the region
In Exercises 29 through 58, factor the given polynomial using integer coefficients.2x2 − x − 15
In Exercises 29 through 58, factor the given polynomial using integer coefficients.12x2 − x − 20
In Exercises 96 through 98, use double integration to find the required quantity. In some cases, you may need to use the numeric integration feature of your calculator.Find the average value ofthe
In Exercises 1 through 16, use L’Hôpital’s rule to evaluate the given limit if the limit is an indeterminate form. lim (1 + 2x)¹/x x→0
In Exercises 1 through 16, use L’Hôpital’s rule to evaluate the given limit if the limit is an indeterminate form. x³ - 3x² lim 4 x-0 3x² + 2x
In Exercises 35 through 42, solve the given equation for n. (Assume a > 0 and a ≠ 1.) 1 (a")5 ,10 ||
In Exercises 96 through 98, use double integration to find the required quantity. In some cases, you may need to use the numeric integration feature of your calculator.Find the volume of the solid
In Exercises 35 through 42, solve the given equation for n. (Assume a > 0 and a ≠ 1.) u_D/D 1 2 a D
In Exercises 35 through 42, solve the given equation for n. (Assume a > 0 and a ≠ 1.)a2an = 1/a
In Exercises 37 through 42, factor the given polynomial using integral coefficients.4x2 + 12x + 9
In Exercises 29 through 58, factor the given polynomial using integer coefficients.x7 − x5
In Exercises 35 through 42, solve the given equation for n. (Assume a > 0 and a ≠ 1.) n (a^)3 1 Va
In Exercises 35 through 42, solve the given equation for n. (Assume a > 0 and a ≠ 1.)(a3)n = a12
In Exercises 37 through 42, factor the given polynomial using integral coefficients.12x2 + 5x − 3
In Exercises 29 through 58, factor the given polynomial using integer coefficients.x3 + 2x2 + x
In Exercises 43 through 76, simplify the given expression as much as possible. Assume a, b, and c are positive real numbers. 2.314 a²c² b
In Exercises 37 through 42, factor the given polynomial using integral coefficients.x3 + 3x2 − x − 3
In Exercises 29 through 58, factor the given polynomial using integer coefficients.2x3 − 8x2 − 10x
In Exercises 37 through 42, factor the given polynomial using integral coefficients.x4 − 5x2 + 4
In Exercises 29 through 58, factor the given polynomial using integer coefficients.x4 + 5x3 − 14x2

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