Skip to main content

Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Mathematical Applications for the Management, Life, and Social Sciences

11E 12E 13E 14E 15E 16E 17E 18E 19E In Problems 19-24, a function and its first and second derivatives are given. Use these to find any horizontal and vertical asymptotes, critical points, relative maxima, relative minima, and points of inflection. Then sketch the graph of each function. 20. 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E For each function in Problems 29-34, complete the following steps. (a) Use a graphing calculator to graph the function in the standard viewing window. (b) Analytically determine the location of any asymptotes and extrema. (c) Graph the function in a viewing window that shows all features of the graph. State the ranges for x-values and y-values for your viewing window. 32. 33E 34E 35E 36E 37. Revenue A recently released film has its weekly revenue given by where R(t) is in millions of dollars and t is in weeks. (a) Graph R(t). (b) When will revenue be maximized? (c) Suppose that if revenue decreases for 4 consecutive weeks, the film will be removed from theaters and will be released as a video 12 weeks later. When will the video come out? 38E 39E 40. Profit An entrepreneur starts new companies and sells them when their growth is maximized. Suppose that the annual profit for a new company is given by where P is in thousands of dollars and x is the number of years after the company is formed. If she wants to sell the company before profits begin to decline, after how many years should she sell it? 41. Productivity The figure is a typical graph of worker productivity per hour P as a function of time t on the job. (a) What is the horizontal asymptote? (b) What is (c) What is the horizontal asymptote for (d) What is 42E 43. Females in the workforce For selected years from 1950 and projected to 2050, the table shows the percent of total U.S. workers who were female. Year % Female Year % Female 1950 29.6 2010 47.9 1960 33.4 2015 48.3 1970 38.1 2020 48.1 1980 42.5 2030 48.0 1990 45.2 2040 47.9 2000 46.6 2050 47.7 Source: U.S. Bureau of Labor Statistics Assume that these data can be modeled with the function where is the percent of the U.S. workforce that is female and t is the number of years past 1950. (a) Find . (b) Interpret your answer to part (a). (c) Does p(t) have any vertical asymptotes within its domain 0? (d) Whenever the model would be inappropriate. Determine whether the model is ever inappropriate for . In Problems 1-12, find the derivative of each function. 1. 2RE 3RE In Problems 1-12, find the derivative of each function. 4. 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE In Problems 15-20, find the indicated derivative. 19. Find 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29. Radioactive decay A breeder reactor converts stable uranium-238 into the isotope plutonium-239. The decay of this isotope is given by where A(t) is the amount of isotope at time t, in years, and is the original amount. This isotope has a half- life of 24,101 years (that is, half of it will decay away in 24,101 years). (a) At what rate is A(t) decaying at t = 24,101 years? (b) At what rate is A(t) decaying after 1 year? (c) Is the rate of decay at its half-life greater or less than after 1 year? 30RE 31RE 32RE 33RE 34RE 35RE 36RE 37. Elasticity Suppose the weekly demand function for a product is given by where p is the price in dollars and q is the number of tons demanded. (a) What is the elasticity of demand when the price is $36.79 and the quantity demanded is 10? (b) How will a price increase affect total revenue? 38RE 39RE 40RE 41RE In Problems 1-8, find the derivative of each function. 2T 3T 4T 5T In Problems 1-8, find the derivative of each function. 6. In Problems 1-8, find the derivative of each function. 7. In Problems 1-8, find the derivative of each function. 8. 9T 10T 11T 12. Suppose the weekly revenue and weekly cost (both in dollars) for a product are given by , respectively, where x is the number of units produced and sold. Find the rate at which profit is changing with respect to time when the number of units produced and sold is 50 and is increasing at a rate of 5 units per week 13T 14T 15T 16T 17T 19T 1. 2. If 3CP 4. Find . 1E 2E 3E 4E 5E Find the derivatives of the functions in Problems 1-10. 6. Find the derivatives of the functions in Problems 1-10. 7. Find the derivatives of the functions in Problems 1-10. 8. Find the derivatives of the functions in Problems 1-10. 9. Find the derivatives of the functions in Problems 1-10. 10. 11. Find . 12E In each of Problems 13-18, find the derivative of the function in part (a). Then find the derivative of the function in part (b) or show that the function in part (b) is the same function as that in part (a). 13. (a) (b) In each of Problems 13-18, find the derivative of the function in part (a). Then find the derivative of the function in part (b) or show that the function in part (b) is the same function as that in part (a). 14. (a) (b) In each of Problems 13-18, find the derivative of the function in part (a). Then find the derivative of the function in part (b) or show that the function in part (b) is the same function as that in part (a). 15. (a) (b) In each of Problems 13-18, find the derivative of the function in part (a). Then find the derivative of the function in part (b) or show that the function in part (b) is the same function as that in part (a). 16. (a) (b) In each of Problems 13-18, find the derivative of the function in part (a). Then find the derivative of the function in part (b) or show that the function in part (b) is the same function as that in part (a). 17. (a) (b) In each of Problems 13-18, find the derivative of the function in part (a). Then find the derivative of the function in part (b) or show that the function in part (b) is the same function as that in part (a). 18. (a) (b) 19. Find . 20E 21E 22E 23E 24. Find In Problems 25-38, find y'. In Problems 25-38, find y'. 26. In Problems 25-38, find y'. 27. 28E In Problems 25-38, find y'. 29. 30E In Problems 25-38, find y'. 31. 32E In Problems 25-38, find y'. 33. 34E 35E 36E 37E 38E 39E 40E 41E 42E 43. Marginal cost Suppose that the total cost (in dollars) for a product is given by where x is the number of units produced. (a) Find the marginal cost function. (b) Find the marginal cost when 200 units are produced and interpret your result. (c) Total cost functions always increase because producing more items costs more. What then must be true of the marginal cost function? Does it apply in this problem? 44. Investment If money is invested at the constant rate r, the time to increase the investment by a factor x is (a) At what rate is the time changing at (b) What happens to as x gets very large? Interpret this result. 45. Marginal revenue The total revenue, in dollars, from the sale of x units of a product is given by (a) Find the marginal revenue function. (b) Find the marginal revenue when 100 units are sold and interpret your result. 46. Supply Suppose that the supply of x units of a product at price p dollars per unit is given by (a) Find the rate of change of supply price with respect to the number of units supplied. (b) Find the rate of change of supply price when the number of units is 33. (c) Approximate the price increase associated with the number of units supplied changing from 33 to 34. 47. Demand The demand function for a product is given by p = 4000/ln (x + 10), where p is the price per unit in dollars when x units are demanded. (a) Find the rate of change of price with respect to the number of units sold when 40 units are sold. (b) Find the rate of change of price with respect to the number of units sold when 90 units are sold. (c) Find the second derivative to see whether the rate at which the price is changing at 40 units is increasing or decreasing. 48E 49E 50E 51E 52. Women in the workforce From 1950 and projected to 2050, the percent of women in the workforce can be modeled by where x is the number of years past 1940 (Source: U.S. Bureau of Labor Statistics). If this model is accurate, at what rate will the percent be changing in 2020? 1. If , find y’. 2. If , find y’. 3CP 4. If the sales of a product are given by , where x is the number of days after the end of an advertising campaign, what is the rate of change in sales 20 days after the end of the campaign? Find the derivatives of the functions in Problems 1-34. Find the derivatives of the functions in Problems 1-34. 2. 3E Find the derivatives of the functions in Problems 1-34. 4. Find the derivatives of the functions in Problems 1-34. 5. Find the derivatives of the functions in Problems 1-34. 6. Find the derivatives of the functions in Problems 1-34. 7. Find the derivatives of the functions in Problems 1-34. 8. 9E Find the derivatives of the functions in Problems 1-34. 10. Find the derivatives of the functions in Problems 1-34. 11. Find the derivatives of the functions in Problems 1-34. 12. 13E 14E 15E 16E Find the derivatives of the functions in Problems 1-34. 17. 18E 19E 20E 21E Find the derivatives of the functions in Problems 1-34. 22. Find the derivatives of the functions in Problems 1-34. 23. Find the derivatives of the functions in Problems 1-34. 24. Find the derivatives of the functions in Problems 1-34. 25. Find the derivatives of the functions in Problems 1-34. 26. 27E 28E Find the derivatives of the functions in Problems 1-34. 29. 30E 31E 32E 33E 34E 35E 36E 37E 38E In Problems 39-42, find any relative maxima and minima. Use a graphing utility to check your results. 39. 40E 41E 42E 43. Future value If $P is invested for n years at 10% compounded continuously, the future value that results after n years is given by the function (a) At what rate is the future value growing at any time (b) At what rate is the future value growing after 1 year (n = 1)? (c) Is the rate of growth of the future value after 1 year greater than 10%? Explain. 44. Future value The future value that accrues when $700 is invested at 9% compounded continuously is where t is the number of years. (a) At what rate is the money in this account growing when t = 4? (b) At what rate is it growing when t = 10? 45. Sales decay After the end of an advertising campaign, the sales of a product are given by where S is weekly sales in dollars and t is the number of weeks since the end of the campaign. (a) Find the rate of change of S (that is, the rate of sales decay). (b) From looking at the function and its derivative, explain how you know sales are decreasing. 46E 47. Marginal cost Suppose that the total cost in dollars of producing x units of a product is given by Find the marginal cost when 600 units are produced. *The mode occurs at the highest point on normal curses and equals the mean. 48. Marginal revenue Suppose that the revenue in dollars from the sale of x units of a product is given by Find the marginal revenue function. 49. Drugs in a bloodstream The percent concentration y of a certain drug in the bloodstream at any time t (in hours) is given by (a) What function gives the instantaneous rate of change of the concentration of the drug in the bloodstream? (b) Find the rate of change of the concentration after 1 hour. Give your answer to three decimal places. 50. Radioactive decay The amount of the radioactive isotope thorium-234 present at time t in years is given by (a) Find the function that describes how rapidly the isotope is decaying. (b) Find the rate of radioactive decay of the isotope after 10 years. 51. Pollution Pollution levels in Lake Sagamore have been modeled by the equation where x is the volume of pollutants (in cubic kilometers) and t is the time (in years). What is the rate of change of x with respect to time? 52E 53. National health care With U.S. Department of Health and Human Services data from 2002 and projected to 2024, the total public expenditures for health care H can be modeled by where t is the number of years past 2000 and H is in billions of dollars. If this model is accurate, at what rate will health care expenditures change in 2022? 54E 55E 56E 58. Blood pressure Medical research has shown that between heartbeats, the pressure in the aorta of a normal adult is a function of time in seconds and can be modeled by the equation (a) Use the derivative to find the rate at which the pressure is changing at any time t. (b) Use the derivative to find the rate at which the pressure is changing after 0.1 second. (c) Is the pressure increasing or decreasing? 59E 60E 61E 62. Carbon dioxide emissions Using U.S. Department of Energy data for selected years from 2010 and projected to 2032, the millions of metric tons of carbon dioxide emissions from biomass energy combustion in the United States can be modeled with the function where t is the number of years past 2010. (a) Find the function that models the rate of change of . Report this function as with 3 significant digits. (b) Find and interpret . 65. Disposable income per capita Using U.S. Energy Information Administration data for selected years from 2010 and projected to 2040, the U.S. real disposable income per capita (in thousands of dollars) can be modeled by where t is the number of years after 2010. (a) Write the function that models the rate of change of U.S. per capita real disposable income. Use 4 significant digits. (b) Predict the rate of change of U.S. per capita real disposable income in 2025. 66E Find the following: (b) (c) 2CP In Problems 1-6, find dy/dx at the given point without first solving for y. 1. In Problems 1-6, find at the given point without first solving for y. 2. In Problems 1-6, find dy/dx at the given point without first solving for y. 3. In Problems 1-6, find at the given point without first solving for y. 4. In Problems 1-6, find at the given point without first solving for y. 5. In Problems 1-6, find at the given point without first solving for y. 6. Find dy/dx for the functions in Problems 7-10. 7. Find for the functions in Problems 7-10. 8. Find for the functions in Problems 7-10. 9. Find for the functions in Problems 7-10. 10. 11. 12. 13. 14E 15E 16. 17. 18. If find . 19. 20E 21E 22E 23E 24E 25E For Problems 23-26, find the slope of the line tangent to the curve. 26. 27E 28E 29E 30E 31E 32E 33E 34. If ln find . 35. 36E 37. 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E 53E 54E 55. Advertising and sales Suppose that a company’s sales volume y (in thousands of units) is related to its advertising expenditures x (in thousands of dollars) according to Find the rate of change of sales volume with respect to advertising expenditures when x = 10 (thousand dollars). 56E 57. Production Suppose that a company can produce 12,000 units when the number of hours of skilled labor y and unskilled labor x satisfy Find the rate of change of skilled-labor hours with respect to unskilled-labor hours when x = 255 and y = 214. This can be used to approximate the change in skilled-labor hours required to maintain the same production level when unskilled-labor hours are increased by 1 hour. 58E 59. Demand If the demand function for q units of a product at $p per unit is given by find the rate of change of quantity with respect to price when p = $80. Interpret this result. 60E 61E 62E 63E 1. If V represents volume, write a mathematical symbol that represents “the rate of change of volume with respect to time.” 2CP 3. True or false: In solving a related-rates problem, we substitute all numerical values into the equation before we take derivatives. 1E In Problems 1-4, find using the given values. 2. 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13. The radius of a circle is increasing at a rate of 2 ft/min. At what rate is its area changing when the radius is 3 ft? (Recall that for a circle, 14E 15E 16E 17. Profit Suppose that the daily profit (in dollars) from the production and sale of x units of a product is given by At what rate per day is the profit changing when the number of units produced and sold is 100 and is in- creasing at a rate of 10 units per day?

Page: [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]