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mathematics
calculus with applications

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

For a certain drug, the rate of reaction in appropriate units is given bywhere t is time (in hours) after the drug is administered. Find the total reaction to the drug over the following time periods.(a) From t = 1 to t = 12(b) From t = 12 to t = 24 R'(t) = = 5 t+1 2 t +1
Let the expected number of cells in a culture that have an x percent probability of undergoing cell division during the next hour be denoted by n(x).(a) Give an integral representing the number of cells that have less than a 60% chance of dividing during the next hour.(b) Let n(x) = √5x + 1 give
A model for describing the population of adult flour beetles involves evaluating the integralwhere g(x) is the per-unit-abundance growth rate for a population of size x. The researchers consider the simple case in which g(x) = a - bx for positive constants a and b. Find the integral in this case.
Under certain conditions, the number of cancer cells N(t) at time t increases at a rate N′(t) = Aekt, where A is the rate of increase at time 0 (in cells per day) and k is a constant.(a) Suppose A = 50, and at 5 days, the cells are growing at a rate of 250 per day. Find a formula for the number
The rate at which a substance grows is given by R′(x) = 150e0.2x, where x is the time (in days). What is the total accumulated growth during the first 3.5 days?
If the rate of excretion of a biochemical compound is given by ƒ′(t) = 0.01e -0.01t, the total amount excreted by time t (in minutes) is ƒ(t).(a) Find an expression for ƒ(t).(b) If 0 units are excreted at time t = 0, how many units are excreted in 10 minutes?
After long study, tree scientists conclude that a eucalyptus tree will grow at the rate of 0.6 + 4/(t + 1)3 ft per year, where t is time (in years).(a) Find the number of feet that the tree will grow in the second year.(b) Find the number of feet the tree will grow in the third year.
An oil tanker is leaking oil at the rate given (in barrels per hour) bywhere t is the time (in hours) after the tanker hits a hidden rock (when t = 0).(a) Find the total number of barrels that the ship will leak on the first day.(b) Find the total number of barrels that the ship will leak on the
The marginal profit of a small fast-food stand is given, in thousands of dollars, bywhere x is the sales volume in thousands of hamburgers. The “profit” is -$1000 when no hamburgers are sold. Find the profit function. P'(x) = √x + 12/₁
The marginal profit in dollars on Brie cheese sold at a cheese store is given by P′(x) = x150x2 + 30x2, where x is the amount of cheese sold, in hundreds of pounds. The “profit” is -$40 when no cheese is sold.(a) Find the profit function.(b) Find the profit from selling 200 lb of Brie cheese.
Consider the function ƒ(x) = x(x2 + 3)7.Use the Fundamental Theorem of Calculus to evaluate -5 J-5 f(x) dx.
Pollution from a factory is entering a lake. The rate of concentration of the pollutant at time t is given by P′(t) = 140t5/2, where t is the number of years since the factory started introducing pollutants into the lake. Ecologists estimate that the lake can accept a total level of pollution of
The approximate rate of change in the number of design patent applications received in the United States in recent years is given by p′(t) = 0.01056t + 1.94, where t represents the number of years since 2000, and p represents the number of design patents (in thousands). In 2010, the number of
A worker new to a job will improve his efficiency with time so that it takes him fewer hours to produce an item with each day on the job, up to a certain point. Suppose the rate of change of the number of hours it takes a worker in a certain factory to produce the xth item is given by H′(x) = 20
Let g(t) = t4 and define ƒ(x) = ∫10ex2 dx = 1.46265 and ∫20ex2 dx = 16.45263. Use this information to find(a)(b) Verify that ƒ′(1) = g(x). The fact thatis true for all continuous functions g is an alternative version of the Fundamental Theorem of Calculus.(c) Let us verify
Use the Fundamental Theorem to show that the following are true. rb f(x) dx = -√ "f(x b 6 f(x) dx
Find the demand function for each marginal revenue function. Recall that if no items are sold, the revenue is 0.R′(x) = 600 - 5e0.0002x
Use Exercise 53 to find ∫4-1 ƒ(x) dx, givenExercise 53Use the Fundamental Theorem to show that the following are true. f(x) = 2x + 3 if x ≤ 0 3 if x > 0. X 4
Find the demand function for each marginal revenue function. Recall that if no items are sold, the revenue is 0.R′(x) = 500 - 0.15√x
Find the demand function for each marginal revenue function. Recall that if no items are sold, the revenue is 0.R′(x) = 50 - 5x2/3
Find the cost function for each marginal cost function.C′(x) = 1.2x(ln 1.2); 2 units cost $9.44
Use the Fundamental Theorem to show that the following are true. b ·b kf(x) dx = k f(x) dx a a
Find the cost function for each marginal cost function.C′(x) = 5x - 1/x; 10 units cost $94.20
An epidemic is growing in a region according to the ratewhere N(t) is the number of people infected after t days.(a) Find a formula for the number of people infected after t days, given that 37 people were infected at t = 0.(b) Use your answer from part (a) to find the number of people infected
Find the cost function for each marginal cost function.C′(x) = x + 1/x2; 2 units cost $5.50
The graph of ƒ(x), shown here, consists of two straight line segments and two quarter circles. Find the value of ∫016 ƒ(x) dx. y 3 0 2 y = f(x) 5 8 16 X
Find the cost function for each marginal cost function.C′(x) = x2/3 + 2; 8 units cost $58
Find the cost function for each marginal cost function.C′(x) = x1/2; 16 units cost $45
Is the equation you wrote for Exercise 49 still true(a) If b is replaced by d? (b) If b is replaced by g?Exercise 49Assume ƒ(x) is continuous for g ≤ x ≤ c as shown in the figure. Write an equation relating the three quantities [F(x) dx. [F(x) dx. [F(x) dx. a g y b d Q y = f(x) X
Find the cost function for each marginal cost function.C′(x) = 0.03e0.01x; fixed cost is $8
Assume ƒ(x) is continuous for g ≤ x ≤ c as shown in the figure. Write an equation relating the three quantities [F(x) dx. [F(x) dx. [F(x) dx. a g y b d Q y = f(x) X
Find the area of each shaded region. 0 -2 -4 1 y = +2 2 In x X e 3 X
A population grows at a rate P′(t) = 500te -t2/5, where P1t2 is the population after t months.(a) Find a formula for the population size after t months, given that the population is 2000 at t = 0.(b) Use your answer from part (a) to find the population size after 3 months.
Find the cost function for each marginal cost function.C′(x) = 0.2x2 + 5x; fixed cost is $10
According to data from the American Hospital Association, the rate of change in the number of hospital outpatient visits, in millions, in the United States each year from 1980 to the present can be approximated by ƒ′(t) = 0.001483t(t - 1980)0.75, where t is the year.(a) Using the fact that in
Find the cost function for each marginal cost function.C′(x) = 4x - 5; fixed cost is $8
According to data from the Bureau of Transportation Statistics, the rate of change in the number of local transit vehicles (buses, light rail, etc.), in thousands, in the United States from 1970 to the present can be approximated by ƒ′(t) = 4.0674 * 10-4t(t - 1970)0.3, where t is the
The rate of growth of the profit (in millions of dollars) from a new technology is approximated by P′(t) = te-t2, where t represents time measured in years. The total profit in the third year that the new technology is in operation is $10,000.(a) Find the total profit function.(b) What happens to
A marshy region used for agricultural drainage has become contaminated with selenium. It has been determined that flushing the area with clean water will reduce the selenium for a while, but it will then begin to build up again. A biologist has found that the percent of selenium in the soil x
Many populations grow according to the logistic equationwhere r is a constant involving the rate of growth and N is the carrying capacity of the environment, beyond which the population decreases. Show that the graph of x has an inflection point where x = N/2. dx dt rx(N − x),
The equation ƒ(x)g(N) - m - s(x) = 0 describes the growth rate of phytoplankton at equilibrium, where x is the phytoplankton cell size, ƒ is the maximum growth rate, N is the nutrient concentration, g represents the nutrient limitation experienced by phytoplankton, m is the mortality rate (a
In studying how a predator forages a prey, an important concept is the functional response, defined aswhere R is the prey population density, C is the attack rate constant, h is the handling time for a single prey, and t is the optimal foraging time, which is itself a function of R. A type-2
A simple biochemical reaction with three molecules has solutions that oscillate toward a steady state when positive constants a and b are below the curve b - a = (b + a)3. Find the largest possible value of a for which the reaction has solutions that oscillate toward a steady state.
The dentinal formation of molars in mice has been studied by researchers in Copenhagen. They determined that the growth curve that best fits dentinal formation for the first molar is M(t) = 1.3386309 - 0.4321173t + 0.0564512t2 -0.0020506t3 + 0.0000315t4 - 0.0000001785t5, 5 ≤ t ≤ 51, where t is
The surface of the skin is made up of a network of intersecting lines that form polygons. Researchers have discovered a functional relationship between the age of a female and the number of polygons per area of skin according to P(t) = 237.09 - 8.0398t + 0.20813t2 - 0.0027563t3 + 0.000013016t4, 0
From information given in a recent business publication, we constructed the mathematical modelto represent the miles per gallon used by a certain car at a speed of s mph. Find the absolute maximum miles per gallon and the absolute minimum and the speeds at which they occur. M(s) = 1 45 s²+2s 20,
The number of salmon swimming upstream to spawn is approximated by S(T) = -T3 + 3T2 + 360T + 5000, 6 ≤ T ≤ 20, where T represents the temperature of the water in degrees Celsius. Find the water temperature that produces the maximum number of salmon swimming upstream.
A water trough 2 ft across, 4 ft long, and 1 ft deep has ends in the shape of isosceles triangles. (See the figure.) It is being filled with 3.5 ft3 of water per minute. Find the rate at which the depth of water in the tank is changing when the water is 1/3 ft deep. 1 ft 4 ft 2 ft
A 50-ft ladder is placed against a building. The top of the ladder is sliding down the building at the rate of 2 ft per minute. Find the rate at which the base of the ladder is slipping away from the building at the instant that the base is 30 ft from the building.
Researchers have determined that the crown length of first molars in fetuses is related to the postconception age of the tooth as L(t) = -0.01t2 + 0.788t - 7.048, where L(t) is the crown length (in millimeters) of the molar t weeks after conception. Find the maximum length of the crown of first
A large weather balloon is being inflated with air at the rate of 0.9 ft3 per minute. Find the rate of change of the radius when the radius is 1.7 ft.
The position of a particle at time t is given by s. Find the velocity ds/dt.2s2 + √st - 4 = 3t
Because of the time that many people spend indoors, there is a concern about the health risk of being exposed to harmful fungi that thrive in buildings. The risk appears to increase in damp environments. Researchers have discovered that by controlling both the temperature and the relative humidity
A piece of wire 12 ft long is cut into two pieces. One piece is made into a circle and the other piece is made into a square. Let the piece of length x be formed into a circle. We allow x to equal 0 or 12, so all the wire may be used for the square or for the circle.Where should the cut be made in
The position of a particle at time t is given by s. Find the velocity ds/dt.s3 - 4st + 2t3 - 5t = 0
A piece of wire 12 ft long is cut into two pieces. One piece is made into a circle and the other piece is made into a square. Let the piece of length x be formed into a circle. We allow x to equal 0 or 12, so all the wire may be used for the square or for the circle.Where should the cut be made in
Approximate the volume of coating on a sphere of radius 4 in. if the coating is 0.02 in. thick.
A boat moves north at a constant speed. A second boat, moving at the same speed, pursues the first boat in such a way that it always points directly at the first boat. When the first boat is at the point (0, 1), the second boat is at the point (6, 2.5), with the positive y-axis pointing north. It
A piece of wire 12 ft long is cut into two pieces. One piece is made into a circle and the other piece is made into a square. Let the piece of length x be formed into a circle. We allow x to equal 0 or 12, so all the wire may be used for the square or for the circle.Show that the side of the square
For a certain sports utility vehicle, M(s) = -0.015s2 + 1.31s - 7.3, 30 ≤ s ≤ 60, represents the miles per gallon obtained at a speed of s mph. Find the absolute maximum miles per gallon and the absolute minimum and the speeds at which they occur.
A square has an edge of 9.2 in., with a possible error in the measurement of ± 0.04 in. Estimate the possible error in the area of the square.
A mathematician is surfing in Long Beach, New York. He is standing on the shore and wants to paddle out to a spot 40 ft from shore; the closest point on the shore to that spot is 40 ft from where he is now standing. (See the figure.) If he can walk 5 ft per second along the shore and paddle 3 ft
UPS has the following rule regarding package dimensions. The length can be no more than 108 in., and the length plus the girth (twice the sum of the width and the height) can be no more than 130 in. If the width of a package is 4 in. more than its height and it has the maximum length plus girth
Each graph below gives the cost as a function of production level. Use the method of graphical optimization to estimate the production level that results in the minimum cost per item produced. Cost f(x) 400 200 0 10 + 20 30 Production 40 50 X
Repeat Exercise 67, but the closest point on the shore to the desired destination is now 25 ft from where he is standing.Exercise 67A mathematician is surfing in Long Beach, New York. He is standing on the shore and wants to paddle out to a spot 40 ft from shore; the closest point on the shore to
Suppose a migrating bird flies at a velocity v, and suppose the amount of time the bird can fly depends on its velocity according to the function T(v).(a) If E is the bird’s initial energy, then the bird’s effective power is given by kE/T, where k is the fraction of the power that can be
The city park department is planning an enclosed play area in a new park. One side of the area will be against an existing building, with no fence needed there. Find the dimensions of the maximum rectangular area that can be made with 900 m of fence.
Each graph below gives the profit as a function of production level. Use graphical optimization to estimate the production level that gives the maximum profit per item produced. Profit f(x) 400 200 0 + 100 + 200 Production 300 X
Suppose dots and dashes are transmitted over a telegraph line so that dots occur a fraction p of the time (where 0 < p < 1) and dashes occur a fraction 1 - p of the time. The information content of the telegraph line is given by I(p), where I(p) = -p In p (1 - p) ln(1 - p). -(a) Show that I'
A hunter is at a point along a river bank. He wants to get to his cabin, located 3 miles north and 8 miles west. (See the figure.) He can travel 5 mph along the river but only 2 mph on this very rocky land. How far upriver should he go in order to reach the cabin in minimum time? 3 Cabin 8 Hunter
Repeat Exercise 53, but assume the cabin is 19 miles north and 8 miles west.Exercise 53A hunter is at a point along a river bank. He wants to get to his cabin, located 3 miles north and 8 miles west. (See the figure.) He can travel 5 mph along the river but only 2 mph on this very rocky land. How
A local group of scouts has been collecting aluminum cans for recycling. The group has already collected 12,000 lb of cans, for which they could currently receive $7.50 per hundred pounds. The group can continue to collect cans at the rate of 400 lb per day. However, a glut in the aluminum market
In Exercises, find the equation of the tangent line at the given value of x on each curve.y4(1 - x) + xy = 2; x = 1
The U.S. Postal Service stipulates that any boxes sent through the mail must have a length plus girth totaling no more than 108 in. (See the figure.) Find the dimensions of the box with maximum volume that can be sent through the U.S. mail, assuming that the width and the height of the box are
Each graph below gives the profit as a function of production level. Use graphical optimization to estimate the production level that gives the maximum profit per item produced. Profit f(x)↑ 1000 800 600 400 200 0 200 400 600 Production 800 1000 X
Researchers found the demand for cheese in Mexico for a particular year can be estimated by the implicit equation ln q = D - 0.44 ln p, where p represents the price of a unit of cheese and D represents a constant that can be calculated uniquely for a particular year. Here q represents the annual
The demand for butter in Mexico for a particular year can be estimated by the general function ln q = D - 0.47 ln p, where p represents the price of a unit of butter and D represents a constant that can be calculated uniquely for a particular year. Here q represents the annual per capita butter
The stage of a new local theater has a rectangular opening, which is 20 m wide and 10 m high. A curtain is to be draped into triangles on the sides, extending from the top to the floor, with a semicircle in the center, as shown in the figure.(a) If the radius of the semicircle is r, show that the
Researchers found the demand for milk in Mexico for a particular year can be estimated by the implicit equation ln q = C - 0.678 ln p, where p represents the price of a unit of fluid milk and C represents a constant that can be calculated uniquely for a particular year. Here q represents the annual
Suppose the demand function for a product is given by q = A/pk, where A and k are positive constants. For what values of k is the demand elastic? Inelastic?
Researchers have found a correlation between respiratory rate and body mass in the first three years of life. This correlation can be expressed by the function log R(w) = 1.83 - 0.43 log(w), where w is the body weight (in kilograms) and R(w) is the respiratory rate (in breaths per minute).(a) Find
A friend observes that the goal is to minimize the cost per item, and that the derivative ƒ′(x) has units of cost per item. To minimize the derivative would require setting its derivative, that is, the second derivative, equal to 0. Therefore, the sought-after point is the inflection point.
A circle of pollution is spreading from a broken underwater waste disposal pipe, with the radius increasing at the rate of 4 ft per minute. Find the rate of change of the area of the circle when the radius is 7 ft.
A study of administrative intensity (the number of managers divided by the number of workers) found that the profit P of a company depended on the administrative intensity i according to the formula P(i) = (p - r)ASaitkbX1-c - W0 X - W1Xi - C, where p is the selling price of a product; r is the
The relationship between the number of species in a genus (x) and the number of genera (y) comprising x species is given by xya = k, where a and k are constants. Find dy/dx.
Find dy/dx by implicit differentiation for the following.y ln x + 2 = x3/2y5/2
Each graph below gives the cost as a function of production level. Use the method of graphical optimization to estimate the production level that results in the minimum cost per item produced. Cost f(x)+ 1000 800 600 400 200 0 2 4 ||||||||| 10 6 8 Production 12 14 X
For the Beverton-Holt model for cod in the previous exercise, consider harvesting at a rate proportional to each year’s population, so that the population next year is given bywhere r is a positive constant of proportionality. Find the value of S that maximizes ƒ(S), and describe in words what
In 1 year, a health food manufacturer produces and sells 240,000 cases of vitamins. It costs $2 to store 1 case for 1 year and $15 to produce each batch. Find the number of batches that should be produced annually.
The demand equation for a certain product is 2p2 + q2 = 1600, where p is the price per unit in dollars and q is the number of units demanded.(a) Find and interpret dq/dp. (b) Find and interpret dp/dq.
Cells in the human body are not necessarily spherical; some cancer cells are cylindrical. Suppose that the volume of such a cell is 67 μm3. Find the radius and height of the cell with the minimum surface area.
A company produces 128,000 cases of soft drink annually. It costs $1 to store 1 case for 1 year and $10 to produce 1 lot. Find the number of lots that should be produced annually.
A recent article described the population ƒ(S) of cod in the North Sea next year as a function of this year’s population S (in thousands of tons) by various mathematical models.where a, b, and c are constants.(a) Find a replacement of variables in the Ricker model above that will make it the
For a certain product, cost C and revenue R are given as follows, where x is the number of units sold (in hundreds)Cost: C² = x² + 100√x + 50Revenue: 900(x - 5)² + 25R² = 22,500(a) Find and interpret the marginal cost dC/dx at x = 5.(b) Find and interpret the marginal revenue dR/dx at x = 5.
A store sells 180,000 cases of a product annually. It costs $12 to store 1 case for 1 year and $20 to place a reorder. Find the number of cases that should be ordered each time.
Find the minimum value of the average cost for the given cost function on the given intervals.C(x) = 81x2 + 17x + 324 on the following intervals.(a) 1 ≤ x ≤ 10(b) 10 ≤ x ≤ 20
The Lambert W function w = f(x) is defined implicitly by the relation x = wew for w ≥ 1. Among its many uses are finding the impact time for a projectile facing air resistance. Find dw/dx when w = 1.
Repeat Exercise 47, but assume a pigeon needs 10/9 as much energy to fly over water as over land.Exercise 47Homing pigeons avoid flying over large bodies of water, preferring to fly around them instead. (One possible explanation is the fact that extra energy is required to fly over water because
A store sells 20,000 batteries annually. It costs 15¢ to store 1 battery for 1 year and $12 to place a reorder. Find the number of batteries that should be ordered each time.
Find the minimum value of the average cost for the given cost function on the given intervals.C(x) = x3 + 37x + 250 on the following intervals.(a) 1 ≤ x ≤ 10(b) 10 ≤ x ≤ 20

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