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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
In Exercises 43 through 46, the first derivative f'(x) of a certain function f(x) is given. In each case,(a) Find intervals on which f is increasing and decreasing.(b) Find intervals on which the graph of f is concave up and concave down.(c) Find the x coordinates of the relative extrema and
In Exercises 35 through 44, use calculus to sketch the graph of the given function.f(x) = 3x5 − 5x3 + 4
The number of worker-hours W required to distribute new telephone books to x% of the households in a certain community is modeled by the functiona. Sketch the graph of W(x).b. Suppose only 1,500 worker-hours are available for distributing telephone books. What percentage of households do not
In Exercises 43 through 46, the first derivative f'(x) of a certain function f(x) is given. In each case,(a) Find intervals on which f is increasing and decreasing.(b) Find intervals on which the graph of f is concave up and concave down.(c) Find the x coordinates of the relative extrema and
Let R(x) be the revenue obtained from the production and sale of x units of a commodity, and let C(x) be the total cost of producing the x units. Show that the ratiois optimized when the relative rate of change of revenue equals the relative rate of change of cost. Would you expect this optimum to
What is the maximum possible volume of a cylindrical can with no top that can be made from 27π square inches of metal?
A cruise line estimates that when each deluxe balcony stateroom on a particular cruise is priced at p thousand dollars, then q tickets for staterooms will be demanded by travelers, where q = 300 − 0.7p2.a. Find the elasticity of demand for the stateroom tickets.b. When the price is $8,000 (p = 8)
A cylindrical can (with top) is to be constructed using a fixed amount of metal. Use calculus to derive a simple relationship between the radius and height of the can having the greatest volume.
In Exercises 45 through 48, the derivative of a function f(x) is given. In each case, find the critical numbers of f(x) and classify each as corresponding to a relative maximum, a relative minimum, or neither. f'(x) = x(2 - x) x² + x + 1
The percentage of codling moths* that survive the pupa stage at a given temperature T (degrees Celsius) is modeled by the formulaFind the temperatures at which the greatest and smallest percentage of moths survive. P(T) = -1.427² +68T - 746 for 20 ≤ T ≤ 30
Income elasticity of demand is defined to be the percentage change in quantity purchased divided by the percentage change in real income.a. Write a formula for income elasticity of demand E in terms of real income I and quantity purchased Q.b. In the United States, which would you expect to be
Suppose that the demand equation for a certain commodity is q = a/pm, where a and m are positive constants. Show that the elasticity of demand is equal to m for all values of p. Interpret this result.
Lamar is an artist who has been commissioned to construct an ornate window. The window is to be in the form of an equilateral triangle surmounted on a rectangle, and the entire window is to have a perimeter of 20 feet. The rectangular part will be made of clear glass and will admit twice as much
A manufacturer estimates that if each shipment of raw materials contains x units, the total cost in dollars of obtaining and storing the year’s supply of raw materials will bea. Find all vertical and horizontal asymptotes of the graph of C(x).b. Note that as x gets larger and larger, the term
In Exercises 45 through 48, the derivative of a function f(x) is given. In each case, find the critical numbers of f(x) and classify each as corresponding to a relative maximum, a relative minimum, or neither. f'(x) (x + 1)²(4-3x)³ (x² + 1)²
In Exercises 45 through 48, the derivative of a function f(x) is given. In each case, find the critical numbers of f(x) and classify each as corresponding to a relative maximum, a relative minimum, or neither.f'(x) = x2(4 − x2)
In Exercises 43 through 46, the first derivative f'(x) of a certain function f(x) is given. In each case,(a) Find intervals on which f is increasing and decreasing.(b) Find intervals on which the graph of f is concave up and concave down.(c) Find the x coordinates of the relative extrema and
A national consumers’ association determines that x years after its founding in 1998, it will have P(x) members, whereP(x) = 100(2x3 − 45x2 + 264x)a. At what time between 2000 and 2013 was the membership largest? Smallest?b. What were the largest and smallest membership levels between 2008 and
Oil from an offshore rig located 3 miles from the shore is to be pumped to a location on the edge of the shore that is 8 miles east of the rig. The cost of constructing a pipe in the ocean from the rig to the shore is 1.5 times more expensive than the cost of construction on land. How should the
Carla is a carpenter who has been hired to make a closed box with a square base and volume of 250 cubic meters. The material for the top and bottom of the box costs $2 per square meter, and the material for the sides costs $1 per square meter. Can Carla construct the box for less than $300?
A business manager determines that t months after production begins on a new product, the number of units produced will be P million per month, wherea. Find P'(t) and P"(t).b. Sketch the graph of P(t).c. What happens to production in the long run (as t → ∞)? P(t) = t 2 (t + 1)²
Vernon, a construction worker, must carry a pipe around a corner as shown in the accompanying figure. What is the length of the longest pipe Vernon can carry horizontally around the corner? 3 ft Pipe 4 ft
Sketch the graph of a function that has all the following properties:a. f'(x) > 0 when x < −1 and when x > 3b. f'(x) < 0 when −1 < x < 3c. f"(x) < 0 when x < 2d. f"(x) > 0 when x > 2
A poll indicates that x months after a particular candidate for public office declares her candidacy, she will have the support of S(x) percent of the voters, whereIf the election is held in November, when should the politician announce her candidacy? Should she expect to win if she needs at least
Through its franchised stations, an oil company gives out 16,000 road maps per year. The cost of setting up a press to print the maps is $100 for each production run. In addition, production costs are 6 cents per map and storage costs are 20 cents per map per year. The maps are distributed at a
It is noon, and the spy is back from space (see Exercise 69 in Section 2.2) and driving a jeep through the sandy desert in the tiny principality of Alta Loma. He is 32 kilometers from the nearest point on a straight paved road. Down the road 16 kilometers is an abandoned power plant where a group
After a presidential election, the proportion h(p) of seats in the House of Representatives won by the party of the winning presidential candidate may be modeled by the cube rulewhere p is the proportion of the popular vote received by the winning presidential candidate.a. Find h'(p) and h"(p).b.
In Exercises 49 through 52, the graph of a derivative function y = f'(x) is given. Describe the function f (x), and sketch a possible graph of y = f (x). 2 y = f'(x) X
The strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from a wooden log of diameter 15 inches. 15 in. W h
An electronics firm uses 600 cases of components each year. Each case costs $1,000. The cost of storing one case for a year is 90 cents, and the ordering fee is $30 per shipment. How many cases should the firm order each time to keep total cost at a minimum? (Assume that the components are used at
A manufacturing firm receives raw materials in equal shipments arriving at regular intervals throughout the year. The cost of storing the raw materials is directly proportional to the size of each shipment, while the total yearly ordering cost is inversely proportional to the shipment size. Show
In Exercises 49 through 52, the graph of a function f is given. In each case, sketch a possible graph for f'. 0 y 1 2 3 4 X
In Exercises 49 through 52, the graph of a function f is given. In each case, sketch a possible graph for f'. y लगी # 0 12 23 3 4 5 6 X
In Exercises 49 through 52, the graph of a derivative function y = f'(x) is given. Describe the function f (x), and sketch a possible graph of y = f (x). -3 + 1 y = f'(x) 5 -X
In Exercises 45 through 48, the derivative of a function f(x) is given. In each case, find the critical numbers of f(x) and classify each as corresponding to a relative maximum, a relative minimum, or neither.f'(x) = x3(2x − 7)2(x + 5)
In a learning model, two responses (A and B) are possible for each of the series of observations. If there is a probability p of getting response A in any individual observation, the probability of getting response A exactly n times in a series of m observations is F(p) = pn(1 − p)m − n. The
Sketch the graph of a function f that has all the following properties:a. The graph has discontinuities at x = −1 and at x = 3b. f'(x) > 0 for x < 1, x ≠1c. f'(x) < 0 for x > 1, x ≠ 3d. f"(x) > 0 for x < −1 and x > 3 and f"(x) 0 for −1 < x < 3e. f(0) = 0 = f
A manufacturer of motorcycles estimates that if x thousand dollars are spent on advertising, then for x > 0,cycles will be sold.a. Sketch the graph of the sales function M(x).b. What level of advertising expenditure results in maximum sales? What is the maximum sales level? M(x) = 2,300
Physiologists define the flow F of air in the trachea by the formula F = SA, where S is the speed of the air and A is the area of a cross section of the trachea.a. Assume the trachea has a circular cross section with radius r. Use the formula for the speed of air in the trachea during a cough given
A fundamental problem in crystallography is the determination of the packing fraction of a crystal lattice, which is the fraction of space occupied by the atoms in the lattice, assuming that the atoms are hard spheres. When the lattice contains exactly two different kinds of atoms, it can be shown
Body reaction to drugs is often modeled§ by an equation of the formwhere D is the dosage and C (a constant) is the maximum dosage that can be given. The rate of change of R(D) with respect to D is called the sensitivity.a. Find the value of D for which the sensitivity is the greatest. What is the
The total cost of producing x units of a certain commodity is C(x) thousand dollars, where C(x) = x3 − 20x2 + 179x + 242a. Find A(x), whereis the average cost function.b. For what values of x is A(x) increasing? For what values is it decreasing?c. For what level of production x is average cost
The total cost of producing x units of a certain commodity is given bySketch the cost curve and find the marginal cost. Does marginal cost increase or decrease with increasing production? C(x) = √5x + 2 + 3.
Refer to Exercise 53. What is the largest volume of a cylindrical parcel that can be sent by fourth-class mail?Data from Exercises 53According to postal regulations, the girth plus length of parcels sent by fourth-class mail may not exceed 108 inches. What is the largest possible volume of a
According to postal regulations, the girth plus length of parcels sent by fourth-class mail may not exceed 108 inches. What is the largest possible volume of a rectangular parcel with two square sides that can be sent by fourth-class mail? X Girth = 4x -x-
According to the results† of Tucker and Schmidt-Koenig, the energy expended by a certain species of parakeet is given bywhere v is the bird’s velocity (in km/hr).What velocity minimizes energy expenditure? E(v) = 1 [0.074(v — 35)² +22] V
A manufacturing firm receives an order for q units of a certain commodity. Each of the firm’s machines can produce n units per hour. The setup cost is s dollars per machine, and the operating cost is p dollars per hour.a. Derive a formula for the number of machines that should be used to keep
In Exercises 49 through 52, the graph of a derivative function y = f'(x) is given. Describe the function f (x), and sketch a possible graph of y = f (x). -3 y = f'(x) 2 -X
In Exercises 49 through 52, the graph of a derivative function y = f'(x) is given. Describe the function f (x), and sketch a possible graph of y = f (x). -2, 0 y = f'(x) 2 4 5 -X
In Exercises 49 through 52, the graph of a function f is given. In each case, sketch a possible graph for f'. y 0 1 2 3 4 5 6
In a model‡ developed by C. J. Pennycuick, the power P required by a bird to maintain flight is given by the formulawhere v is the relative speed of the bird, w is the weight, ρ is the density of air, and S and A are positive constants associated with the bird’s size and shape. What relative
The population of a bacterial colony increases at an increasing rate for 1 hour, after which it continues to increase but at a rate that gradually decreases toward zero. Sketch a possible graph for the population P(t) as a function of time t.
A manufacturer finds that in producing x units per day (for 0 < x < 100), three different kinds of cost are involved:a. A fixed cost of $1,200 per day in wagesb. A production cost of $1.20 per day for each unit producedc. An ordering cost of 100/x2 dollars per day Express the total cost as a
Hal Cooper is given an injection of a particular drug at noon, and samples of blood are taken at regular intervals to determine the concentration of drug in his system. It is found that the concentration increases at an increasing rate with respect to time until 1 P.M., and for the next 3 hours,
If air resistance is neglected, it can be shown that the stream of water emitted by a fire hose will have heightfeet above a point located x feet from the nozzle, where m is the slope of the nozzle and v is the velocity of the stream of water as it leaves the nozzle. Assume v is constant.a. Suppose
The profit obtained from producing x thousand units of a particular commodity each year is P(x) dollars, wherea. Find the marginal profit P'(x), and determine all values of x such that P'(x) = 0.b. Sketch the graph of marginal profit along with the graph of P(x) on the same coordinate plane.c. Find
It is known that a quantity of water that occupies 1 liter at 0ºC will occupyliters when the temperature is TºC, for 0 ≤ T ≤ 30.Use a graphing utility to graph V(T) for 0 ≤ T ≤ 10. The density of water is maximized when V(T) is minimized. At what temperature does this occur? V(T) -
Epidemiologists studying a contagious disease observe that the number of newly infected people increases at an increasing rate during the first 3 years of the epidemic. At that time, a new drug is introduced, and the number of infected people declines at a decreasing rate. Two years after its
A company estimates that if x thousand dollars are spent on marketing a certain product, then S(x) units of the product will be sold each month, wherea. How many units will be sold if no money is spent on marketing?b. Sketch the graph of S(x). For what value of x does the graph have an inflection
Let p = (10 − 3x)2 for 0 ≤ x ≤ 3 be the price at which x hundred units of a certain commodity will be sold, and let R(x) = xp(x) be the revenue obtained from the sale of the x units. Find the marginal revenue R'(x), and sketch the revenue and marginal revenue curves on the same graph. For
A useful model for the production p(x) of blood cells involves a function of the formwhere x is the number of cells present, and A, B, and m are positive constants.†a. Find the rate of blood production R(x) = p'(x), and determine where R(x) = 0.b. Find the rate at which R(x) is changing with
To produce x units of a particular commodity, a monopolist has a total cost ofand total revenue of R(x) = xp(x), where p(x) = 5 − 2x is the price at which the x units will be sold. Find the profit function P(x) = R(x) − C(x), and sketch its graph. For what level of production is profit
In designing airplanes, an important feature is the so-called drag factor; that is, the retarding force exerted on the plane by the air. One model measures drag by a function of the formwhere v is the velocity of the plane and A and B are constants. Find the velocity (in terms of A and B) that
Studies show that when environmental factors impose an upper bound on the possible size of a population P(t), the population often tends to grow in such a way that the percentage rate of change of P(t) satisfieswhere A and B are positive constants. Where does the graph of P(t) have an inflection
Given the function f(x) = 2x3 + 3x2 − 12x − 7, complete these steps:a. Graph using [−10, 10]1 by [−10, 10]1 and [−10, 10]1 by [−20, 20]2.b. Fill in this table:c. Approximate the x intercepts and y intercept to two decimal places.d. Find the relative maximum and relative minimum
Repeat Exercise 73 for the function f(x) = 3.7x4 − 5.03x3 + 2x2 − 0.7Data from Exercises 73Given the function f(x) = 2x3 + 3x2 − 12x − 7, complete these steps:a. Graph using [−10, 10]1 by [−10, 10]1 and [−10, 10]1 by [−20, 20]2.b. Fill in this table:c. Approximate the x intercepts
Let f(x) = x4 + x. Show that even though f"(0) = 0, the graph of f has neither a relative extremum nor an inflection point where x = 0. Sketch the graph of f(x).
Let Before actually graphing this function, what do you think the graph looks like? Now use a graphing utility to sketch the graph. Were you right? f(x) = 4 + V9 - 2x -x²².
Sketch a graph of a function that has all the following properties:a. f'(0) = f'(1) = f'(2) = 0b. f'(x) < 0 when 0 < x < 1 c. f'(x) > 0 when x < 0, 1 < x < 2, and x > 2
Find constants a, b, and c so that the graph of the function f(x) = ax2 + bx + c has a relative maximum at (5, 12) and crosses the y axis at (0, 3).
Find constants a, b, c, and d so that the graph of the function f(x) = ax3 + bx2 + cx + d will have a relative maximum at (−2, 8) and a relative minimum at (1, −19).
Sketch a graph of a function that has all the following properties:a. f'(x) > 0 when x < −5 and when x > 1 b. f'(x) < 0 when −5 < x < 1 c. f(−5) = 4 and f(1) = −1
Use calculus to show that the graph of the quadratic function y = ax2 + bx + c is concave upward if a is positive and concave downward if a is negative.
Sketch a graph of a function that has all the following properties:a. f'(x) < 0 when x < −1 b. f'(x) > 0 when −1 < x < 3 and when x > 3 c. f'(−1) = 0 and f'(3) = 0
Suppose f(x) and g(x) are continuous functions with f'(c) = 0. If both f and g have an inflection point at x = c, does P(x) = f(x)g(x) have an inflection point at x = c? Either explain why this must always be true or find functions f(x) and g(x) for which it is false.
In Exercises 78 through 81, use a graphing utility to sketch the graph of f(x). Then find f'(x) and graph it on the same axes (screen) as f(x). Finally, use TRACE, ZOOM, or the ZERO command to find the values of x where f'(x) = 0.f(x) = (x2 + x − 1)3(x + 3)2
Sketch the graph of f(x) = 1 − x3/5.
Use calculus to prove that the relative extremum of the quadratic function f(x) = ax2 + bx + c occurs when x = -b/2a.Where is f(x) increasing and decreasing?
Sketch the graph of f(x) = (x − 1)2/5. Explain why f'(x) is not defined at x = 1.
In Exercises 78 through 81, use a graphing utility to sketch the graph of f(x). Then find f'(x) and graph it on the same axes (screen) as f(x). Finally, use TRACE, ZOOM, or the ZERO command to find the values of x where f'(x) = 0.f(x) = (1 − x1/2)1/2
Use calculus to prove that the relative extremum of the quadratic function y = (x − p)(x − q) occurs midway between its x intercepts.
In Exercises 78 through 81, use a graphing utility to sketch the graph of f(x). Then find f'(x) and graph it on the same axes (screen) as f(x). Finally, use TRACE, ZOOM, or the ZERO command to find the values of x where f'(x) = 0.f(x) = x4 + 3x3 − 9x2 + 4
In Exercises 78 through 81, use a graphing utility to sketch the graph of f(x). Then find f'(x) and graph it on the same axes (screen) as f(x). Finally, use TRACE, ZOOM, or the ZERO command to find the values of x where f'(x) = 0.f(x) = x5 − 7.6x3 + 2.1x2 − 5
Use a graphing utility to sketch the graph of f(x) = x3 + 3x2 − 5x + 11. Then sketch the graph of g(x) = (x + 1)3 + 3(x + 1)2 − 5(x + 1) + 11 on the same axes. What function h(x) would have a graph that is the same as that of f(x), only shifted upward by 2 units and to the left by 3 units?
Let f(x) = x3 − 6x2 + 5x − 11. Use a graphing utility to sketch the graph of f(x). Then, on the same axes, sketch the graph of g(x) = f(2x) = (2x)3 − 6(2x)2 − 5(2x) − 11 What (if any) relation exists between the two graphs?
Suppose a particular tissue culture has area A(t) at time t and a potential maximum area M. Based on properties of cell division, it is reasonable to assume that the area A grows at a rate jointly proportional to √A(t) and M − A(t); that is,where k is a positive constant.a. Let R(t) = A'(t) be
An epidemiologist determines that a particular epidemic spreads in such a way that t weeks after the outbreak, N hundred new cases will be reported, wherea. Find N'(t) and N"(t).b. At what time is the epidemic at its worst? What is the maximum number of reported new cases?c. Health officials
Madge, the manager of the Footloose sandal company, determines that t months after initiating an advertising campaign, S(t) hundred pairs of sandals will be sold, wherea. Find S'(t) and S"(t).b. At what time will sales be maximized? What is the maximum level of sales?c. Madge plans to terminate the
Leta. Use a graphing utility to sketch the graph of f(x). What happens at x = 1?b. Sketch the graph of g(x). Now what happens at x = 1? f(x) = x-1 2 X - 1 and let g(x) = x - 1.01 2 x - 1
The percentage of codling moth eggs* that hatch at a given temperature T (degrees Celsius) is given bySketch the graph of the hatching function H(T). At what temperature T (15 ≤ T ≤ 30) does the maximum percentage of eggs hatch? What is the maximum percentage? H(T) = -0.53T² + 25T - 209 for 15
The concentration of a drug t hours after being injected into the arm of a patient is given bySketch the graph of the concentration function. When does the maximum concentration occur? C(t) = 0.15t ² +0.81
Repeat Exercise 60 for the functionData from Exercises 60Let f(x) = x1/3(x − 4).a. Find f'(x), and determine intervals of increase and decrease for f(x). Locate all relative extrema on the graph of f(x).b. Find f"(x), and determine intervals of concavity for f(x). Find all inflection points on
Water is poured at a constant rate into the vase shown in the accompanying figure. Let h(t) be the height of the water in the vase at time t (assume the vase is empty when t = 0). Sketch a rough graph of the function h(t). In particular, what happens when the water level reaches the neck of the
Commissioners of a certain city determine that when x million dollars are spent on controlling pollution, the percentage of pollution removed is given bya. Sketch the graph of P(x).b. What expenditure results in the largest percentage of pollution removal? P(x) = 100 Vx 0.04x² + 12 2
Dom, the manager of a fishery, determines that t weeks after 300 fish of a particular species are released into a pond, the average weight of an individual fish (in pounds) for the first 10 weeks will be w(t) = 3 + t − 0.05t2 Dom further determines that the proportion of the fish that are still
A demographic study of a certain city indicates that P(r) hundred people live r miles from the civic center, wherea. What is the population at the city center?b. For what values of r is P(r) increasing? For what values is it decreasing?c. At what distance from the civic center is the population
The value V (in thousands of dollars) of an industrial machine is modeled bywhere N is the number of hours the machine is used each day. Suppose further that usage varies with time in such a way thatwhere t is the number of months the machine has been in operation.a. Over what time interval is the
Suppose that in a certain community, there will be M(r) thousand new houses built when the 30-year fixed mortgage rate is r%, wherea. Find M'(r) and M"(r).b. Sketch the graph of the construction function M(r).c. At what rate of interest r is the rate of construction of new houses minimized? M(r) 1
During a recession, Congress decides to stimulate the economy by providing funds to hire unemployed workers for government projects. Suppose that t months after the stimulus program begins, there are N(t) thousand people unemployed, wherea. What is the maximum number of unemployed workers? When
Find constants A, B, and C so that the function f(x) = Ax3 + Bx2 + C will have a relative extremum at (2, 11) and an inflection point at (1, 5). Sketch the graph of f.
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