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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
A company has received an order from the city recreation department to manufacture 8,000 Styrofoam kickboards for its summer swimming program. The company owns several machines, each of which can produce 30 kickboards an hour. The cost of setting up the machines to produce these particular
For Exercises 27 through 36, findandIf the limiting value is infinite, indicate whether it is +∞ or −∞.f(x) = (1 + x2)3 X lim f(x) 00
For Exercises 27 through 36, findandIf the limiting value is infinite, indicate whether it is +∞ or −∞. X lim f(x) 00
For Exercises 27 through 36, findandIf the limiting value is infinite, indicate whether it is +∞ or −∞. X lim f(x) 00
In Exercises 21 through 36, write an equation for the line with the given properties. Through (1) 1 and (²3-1-) 3'4
In Exercises 29 through 42, list all the values of x for which the given function is not continuous. f(x) = x + 1 x-2
In Exercises 29 through 42, list all the values of x for which the given function is not continuous.f(x) = 3x2 − 6x + 9
For Exercises 27 through 36, findandIf the limiting value is infinite, indicate whether it is +∞ or −∞. X lim f(x) 00
In Exercises 30 through 33, supply and demand functions, S(x) and D(x), are given for a particular commodity in terms of the level of production x. In each case:(a) Find the value of xe for which equilibrium occurs and the corresponding equilibrium price pe.(b) Sketch the graphs of the supply and
For Exercises 27 through 36, findandIf the limiting value is infinite, indicate whether it is +∞ or −∞. X lim f(x) 00
For Exercises 27 through 36, findandIf the limiting value is infinite, indicate whether it is +∞ or −∞. X lim f(x) 00
In Exercises 30 through 33, supply and demand functions, S(x) and D(x), are given for a particular commodity in terms of the level of production x. In each case:(a) Find the value of xe for which equilibrium occurs and the corresponding equilibrium price pe.(b) Sketch the graphs of the supply and
In Exercises 29 through 42, list all the values of x for which the given function is not continuous. f(x) = 3x - 1 2х - 6
In Exercises 30 through 33, supply and demand functions, S(x) and D(x), are given for a particular commodity in terms of the level of production x. In each case:(a) Find the value of xe for which equilibrium occurs and the corresponding equilibrium price pe.(b) Sketch the graphs of the supply and
In Exercises 21 through 36, write an equation for the line with the given properties.Through (−2, 3) and (0, 5)
For Exercises 27 through 36, findandIf the limiting value is infinite, indicate whether it is +∞ or −∞. X lim f(x) 00
In Exercises 29 through 42, list all the values of x for which the given function is not continuous. f(x) = 3x + 3 x + 1
Producers will supply x units of a certain commodity to the market when the price is p = S(x) dollars per unit, and consumers will demand (buy) x units when the price is p = D(x) dollars per unit, wherea. Find the equilibrium production level xe and the equilibrium price pe.b. Draw the supply and
In Exercises 29 through 42, list all the values of x for which the given function is not continuous.f(x) = x5 − x3
In Exercises 37 and 38, the graph of a function f(x) is given. Use the graph to determineand lim f(x) x +∞0 00
In Exercises 29 through 42, list all the values of x for which the given function is not continuous. f(x) = x² - 1 x + 1
Julia can sell a certain product for $110 per unit. Total cost consists of a fixed overhead of $7,500 plus production costs of $60 per unit.a. Express Julia’s total revenue, total cost, and total profit in terms of x, the number of units sold. Sketch the total revenue and total cost functions on
In Exercises 30 through 33, supply and demand functions, S(x) and D(x), are given for a particular commodity in terms of the level of production x. In each case:(a) Find the value of xe for which equilibrium occurs and the corresponding equilibrium price pe.(b) Sketch the graphs of the supply and
In Exercises 37 and 38, the graph of a function f(x) is given. Use the graph to determineand lim f(x) x +∞0 00
In Exercises 21 through 36, write an equation for the line with the given properties.Through (1, 5) and (3, 5)
A company plans to construct a new building and parking lot on a rectangular plot of land 100 meters wide and 120 meters long. The building is to be 20 meters high and to have a rectangular footprint with perimeter 320 meters, as shown in the accompanying figure.a. Express the volume V(x) of the
A furniture manufacturer can sell dining room tables for $500 each. The manufacturer’s total cost consists of a fixed overhead of $30,000 plus production costs of $350 per table.a. How many tables must the manufacturer sell to break even?b. How many tables must the manufacturer sell to make a
In Exercises 21 through 36, write an equation for the line with the given properties.Through (1, 5) and (1, −4)
In Exercises 29 through 42, list all the values of x for which the given function is not continuous. f(x) 2 X
In Exercises 21 through 36, write an equation for the line with the given properties.Through (−2, 3) and parallel to the line x + 3y = 5
In Exercises 21 through 36, write an equation for the line with the given properties.Through (−1/2, 1) and perpendicular to the line 2x + 5y = 3
In Exercises 21 through 36, write an equation for the line with the given properties.Through (4, 1) and parallel to the line 2x + y = 3
In Exercises 29 through 42, list all the values of x for which the given function is not continuous. f(x) = x² - 2x + 1 x²2² - x - 2
A greeting card company can sell cards for $2.75 each. The company’s total cost consists of a fixed overhead of $12,000 plus variable costs of 35 cents per card.a. Express the company’s total revenue, total cost, and total profit in terms of x, the number of cards sold.b. How many cards must
When electric blenders are sold for p dollars apiece, manufacturers will supply P2/10 blenders to local retailers, while the local demand will be 60 − p blenders. At what market price will the manufacturers’ supply of electric blenders be equal to the consumers’ demand for the blenders? How
In Exercises 39 through 42, complete the table by evaluating f(x) at the specified values of x. Then use the table to estimate the indicated limit or show it does not exist. f(x) = x² = x; lim f(x) - x-2 1.9 1.99 | 1.999 X f(x) 2 2.0012.01 2.1
In Exercises 29 through 42, list all the values of x for which the given function is not continuous. f(x) = 옥 [x² if x ≤ 2 if x > 2 9
In Exercises 21 through 36, write an equation for the line with the given properties.Through (3, 5) and perpendicular to the line x + y = 4
In Exercises 29 through 42, list all the values of x for which the given function is not continuous. f(x) = = 2x + 3 6x1 if x ≤ 1 ifx>1
A manufacturer’s total cost consists of a fixed overhead of $5,000 plus production costs of $60 per unit.a. Express the total cost as a function of the number of units produced, and sketch its graph.b. Find the average cost function AC(x). What is the average cost of producing 20 units?
In Exercises 39 through 42, complete the table by evaluating f(x) at the specified values of x. Then use the table to estimate the indicated limit or show it does not exist. 1 f(x) = x; lim f(x) X x-0 X f(x) -0.09 -0.009 | 0 IX 0.0009 0.009 |0.09
Alicia, the manager of a plant, determines that when x% of her plant’s capacity is being used, the total cost of operation is C hundred thousand dollars, wherea. Find C(0) and C(100).b. Explain why Alicia cannot use the result of part (a) along with the intermediate value property to conclude
Producers will supply q units of a certain commodity to the market when the price is p = S(q) dollars per unit, and consumers will demand (buy) q units when the price is p = D(q) dollars per unit, wherefor constants a, b, c, and d.a. What can you say about the signs of the constants a, b, c, and d
A credit card company estimates that the average cardholder owed $7,853 in the year 2005 and $9,127 in 2010. Suppose average cardholder debt D grows at a constant rate.a. Express D as a linear function of time t, where t is the number of years after 2005. Draw the graph.b. Use the function in part
In Exercises 29 through 42, list all the values of x for which the given function is not continuous. f(x) = [2 - 3x x²2²x + 3 - x + 3 if x ≤ -1 if x > -1
A manufacturer estimates that it costs $75 to produce each unit of a particular commodity. The fixed overhead is $4,500.a. Express the total cost of production as a function of the number of units produced, and sketch the graph.b. Find the average cost function AC(x). What is the average cost of
In Exercises 39 through 42, complete the table by evaluating f(x) at the specified values of x. Then use the table to estimate the indicated limit or show it does not exist. - f(x) = X f(x) ³+1 x - 1 lim f(x) 1x1 0.9 0.99 0.999 1 IX 1.0011.01 | 1.1
In Exercises 29 through 42, list all the values of x for which the given function is not continuous. f(x) = 3x-2 [x² + x if x < 0 if x ≥ 0
In Exercises 39 through 42, complete the table by evaluating f(x) at the specified values of x. Then use the table to estimate the indicated limit or show it does not exist. f(x) = X f(x) x³ + 1 lim f(x) x + 1x1 1' -1.1-1.01|-1.001|-1 -0.999 -0.99 -0.9 1.1 -1.01-
It costs a publisher $74,200 to prepare a book for publication (typesetting, illustrating, editing, and so on); printing and binding costs are $5.50 per book. The book is sold to bookstores for $19.50 per copy.a. Make a table showing the cost of producing 2,000, 4,000, 6,000, and 8,000 books.b.
In certain situations, it is necessary to weigh the benefit of pursuing a certain goal against the cost of achieving that goal. For instance, suppose that to remove x% of the pollution from an oil spill, it costs C thousands of dollars, wherea. How much does it cost to remove 25% of the pollution?
In Exercises 43 through 50, find the indicated limit or show that it does not exist using the following facts about limits involving the functions f(x) and g(x): lim f(x) = 5 lim g(x): -2 and X-C and lim f(x) = −3 -3 lim g(x) = 4 X-00
Several different formulas have been proposed for determining the appropriate dose of a drug for a child in terms of the adult dosage. Suppose that A milligrams (mg) is the adult dose of a certain drug and C is the appropriate dosage for a child of age N years. Then Cowling’s rule says thatwhile
In Exercises 43 through 50, find the indicated limit or show that it does not exist using the following facts about limits involving the functions f(x) and g(x): lim f(x) = 5 lim g(x): -2 and X-C and lim f(x) = −3 -3 lim g(x) = 4 X-00
In Exercises 43 through 50, find the indicated limit or show that it does not exist using the following facts about limits involving the functions f(x) and g(x): lim f(x) = 5 lim g(x): -2 and X-C and lim f(x) = −3 -3 lim g(x) = 4 X-00
In Exercises 43 through 50, find the indicated limit or show that it does not exist using the following facts about limits involving the functions f(x) and g(x): lim f(x) = 5 lim g(x): -2 and X-C and lim f(x) = −3 -3 lim g(x) = 4 X-00
In Exercises 43 through 50, find the indicated limit or show that it does not exist using the following facts about limits involving the functions f(x) and g(x): lim f(x) = 5 lim g(x): -2 and X-C and lim f(x) = −3 -3 lim g(x) = 4 X-00
A car rental agency charges $75 per day plus 70 cents per mile.a. Express the cost of renting a car from this agency for 1 day as a function of the number of miles driven, and draw the graph.b. How much does it cost to rent a car for a l-day trip of 50 miles?c. The agency also offers a rental for a
In 1900, the life expectancy of a newborn child was 46 years, and by 2000, it had grown to 77 years. Over the same century, the life expectancy of a person at age 65 grew from 76 years to 83 years. Both of these life expectancies increased linearly with time between 1900 and 2000.a. Find the
If the air temperature on a given day is 80ºF, the heat index I(h) (also in ºF) can be approximated by the following function, where h is the relative humidity as a percentage:a. What is the heat index if the relative humidity is 30%? What if it is 90%?b. What relative humidity produces a heat
As an alternative to Friend’s rule and Cowling’s rule, pediatricians sometimes use the formulato estimate an appropriate drug dosage for a child whose surface area is S square meters, when the adult dosage of the drug is A milligrams (mg). In turn, the surface area of the child’s body is
Belinda, a Florida citrus grower, estimates that if 60 orange trees are planted, the average yield per tree will be 400 oranges. The average yield will decrease by 4 oranges per tree for each additional tree planted on the same acreage. Express Belinda’s total yield as a function of the number of
In Exercises 43 through 50, find the indicated limit or show that it does not exist using the following facts about limits involving the functions f(x) and g(x): lim f(x) = 5 lim g(x): -2 and X-C and lim f(x) = −3 -3 lim g(x) = 4 X-00
Several different formulas have been proposed for determining the appropriate dose of a drug for a child in terms of the adult dosage. Suppose that A milligrams (mg) is the adult dose of a certain drug and C is the appropriate dosage for a child of age N years. Then Cowling’s rule says thatwhile
Several different formulas have been proposed for determining the appropriate dose of a drug for a child in terms of the adult dosage. Suppose that A milligrams (mg) is the adult dose of a certain drug and C is the appropriate dosage for a child of age N years. Then Cowling’s rule says thatwhile
In biochemistry, the rate R of an enzymatic reaction is found to be given by the equationwhere Km is a constant (called the Michaelis constant), Rm is the maximum possible rate, and [S] is the substrate concentration.* Rewrite this equation so that y = 1/R is expressed as a function ofand sketch
Suppose air temperature on a certain day is 30ºF. Then the equivalent wind chill temperature (in F) produced by a wind with speed v miles per hour (mph) is given by*a. What is the windchill temperature when v = 20 mph? When v = 50 mph?b. What wind speed produces a windchill temperature of 0F?c. Is
To raise money, a service club has been collecting used bottles that it plans to deliver to a local glass company for recycling. Since the project began 80 days ago, the club has collected 24,000 pounds of glass for which the glass company currently offers 1 cent per pound. However, since bottles
The pupil of the human eye is roughly circular. If the intensity of light I entering the eye is proportional to the area of the pupil, express I as a function of the radius r of the pupil.
Tomas, the organizer of a sports event, estimates that if the event is announced x days in advance, the revenue obtained will be R(x) thousand dollars, whereThe cost of advertising the event for x days isC(x) thousand dollars, wherea. Find the profit function P(x) = R(x) − C(x), and sketch its
In Exercises 43 through 50, find the indicated limit or show that it does not exist using the following facts about limits involving the functions f(x) and g(x): lim f(x) = 5 lim g(x): -2 and X-C and lim f(x) = −3 -3 lim g(x) = 4 X-00
The accompanying graph shows the number of units in inventory at a certain business over a 2-year period. When is the graph discontinuous? What do you think is happening at those times? I (units in inventory) 3004 200+ 100- 0 6 12 18 24 -t (months)
The shape of a cancerous tumor is roughly spherical and has volumewhere r is the radius in centimeters.a. When first observed, the tumor has radius 0.73 cm, and 45 days later, the radius is 0.95 cm. By how much does the volume of the tumor increase during this period?b. After being treated with
It is estimated that t years from now the population of a certain suburban community will be p thousand people, whereAn environmental study indicates that the average level of carbon monoxide in the air will be c parts per million when the population is p thousand, whereWhat happens to the level of
On January 1, 2012, Sam started working for Acme Corporation with an annual salary of $48,000, paid each month on the last day of that month. On July 1, he received a commission of $2,000 for his work, and on September 1, his base salary was raised to $54,000 per year. Finally, on December 21, he
A ruptured pipe in a North Sea oil rig produces a circular oil slick that is y meters thick at a distance x meters from the rupture. Turbulence makes it difficult to directly measure the thickness of the slick at the source (where x = 0), but for x > 0, it is found thatAssuming the oil slick
In Exercises 43 through 50, find the indicated limit or show that it does not exist using the following facts about limits involving the functions f(x) and g(x): lim f(x) = 5 lim g(x): -2 and X-C and lim f(x) = −3 -3 lim g(x) = 4 X-00
The accompanying graph shows the amount of gasoline in the tank of a car over a 30-day period. When is the graph discontinuous? What do you think happens at these times? 10 Q (gals) 0 131 15 20 - 10 25 30 -1 (days)
If a hollow sphere of radius R is charged with one unit of static electricity, then the field intensity E(x) at a point P located x units from the center of the sphere satisfies:Sketch the graph of E(x). Is E(x) continuous for x > 0? 0 1 E(x)=2x² 1 X if 0 < x < R if x = R if x > R
Discuss the continuity of the functionon the open interval 0 --=-) X f(x) = x 1 +
A business manager determines that the total cost of producing x units of a particular commodity may be modeled by the function(dollars). The average cost isFindand interpret your result. C(x) = 7.5x + 120,000
In Exercises 56 and 57, find the values of the constant A so that the function f (x) will be continuous for all x. f(x) = JAx - 3 if x < 2 3x + 2x² if x ≥ 2
A business manager determines that t months after production begins on a new product, the number of units produced will be P thousand, whereWhat happens to production in the long run (as t → ∞)? P(t) = 6t² + 5t (t + 1)²
Discuss the continuity of the functionon the open interval 0 f(x) = fx²-3x if x < 2 4 + 2x if x ≥ 2
The concentration of drug in a patient’s bloodstream t hours after an injection is C(t) milligrams per milliliter wherea. What is the concentration of drug immediately after the injection (when t = 0)?b. By how much does the concentration change during the 5th hour? Does it increase or decrease
In 2010, the cost p(x) in cents of mailing a letter weighing x ounces wasSketch the graph of p(x) for 0 p(x) [44 61 78 if 0 < x≤ 1 if 1 < x≤ 2 if2 < x≤ 3.5
When starting a new job at a production facility, employees can be expected to assemble n items per hour after t weeks of work experience, whereEmployees are paid 20 cents for each item they assemble.a. Find an expression for the amount of money A(t) earned per hour by an employee with t weeks of
Two species coexist in the same ecosystem. Species I has population P(t) in t years, while Species II has population Q(t), both in thousands, where P and Q are modeled by the functionsfor all times t ≥ 0 for which the respective populations are nonnegative.a. What is the initial population of
In Exercises 52 through 56, you need to know that a cylinder of radius r and height h has volume V = πr2h and lateral (side) surface area S = 2πrh. A circular disk of radius r has area A = πr2.A closed cylindrical can has a radius r and height h.a. If the surface area S of the can is a constant,
Scott, an urban planner, models the population P(t) (in thousands) of his community t years from now by the functiona. What is the current population of the community?b. By how much does the population change during the 3rd year? Is the population increasing or decreasing over this time period?c.
In Exercises 52 through 56, you need to know that a cylinder of radius r and height h has volume V = πr2h and lateral (side) surface area S = 2πrh. A circular disk of radius r has area A = πr2.A soda can holds 12 fluid ounces (approximately 6.89π cubic inches). Express the surface area of the
Alicia, the manager of a plant, determines that when x% of the plant’s capacity is being used, the total cost of operation is C hundred dollars, whereThe company has a policy of rotating maintenance in an attempt to ensure that approximately 80% of capacity is always in use. What cost should
In Exercises 56 and 57, find the values of the constant A so that the function f (x) will be continuous for all x. f(x) = f1 - 3x Ax² + 2x-3 if x < 4 if x ≥ 4
In Exercises 52 through 56, you need to know that a cylinder of radius r and height h has volume V = πr2h and lateral (side) surface area S = 2πrh. A circular disk of radius r has area A = πr2.A cylindrical can is to hold 4π cubic inches of frozen orange juice. The cost per square inch of
In Exercises 52 through 56, you need to know that a cylinder of radius r and height h has volume V = πr2h and lateral (side) surface area S = 2πrh. A circular disk of radius r has area A = πr2.A closed cylindrical can has surface area 120π square inches. Express the volume of the can as a
An open box is to be made from a square piece of cardboard, 18 inches by 18 inches, by removing a small square from each corner and folding up the flaps to form the sides. Express the volume of the resulting box as a function of the length x of a side of the removed squares. Draw the graph, and
Show that the equationhas at least one solution for the interval 0 ≤ x ≤ 8. √x-8 +9x²/3 29
Investigate the behavior of when x is near to (a) 2 (b) −2. Does the limit exist at these values of x? Is the function continuous at these values of x? f(x) = 2x² - 5x + 2 x²2²-4
To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose the time required for the rat to traverse the maze on the nth trial was approximatelyminutes. What happens to the time of traverse as the
In Exercises 52 through 56, you need to know that a cylinder of radius r and height h has volume V = πr2h and lateral (side) surface area S = 2πrh. A circular disk of radius r has area A = πr2.A cylindrical can with no top has been made from 27π square inches of metal. Express the volume of the
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