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All Textbook Solutions for Calculus and Its Applications (11th Edition)

Find each antiderivative using Table 1. 1x29dx Find each antiderivative using Table 1. 125x2dx 6E 7E 8E 9E 10E Find each antiderivative using Table 1. 11. Find each antiderivative using Table 1. In(45x)dx,x0 Find each antiderivative using Table 1. 13. Find each antiderivative using Table 1. x3e2xdx Find each antiderivative using Table 1. 15. Find each antiderivative using Table 1. 5x4Inxdx,x0 Find each antiderivative using Table 1. 17. Find each antiderivative using Table 1. 18. Find each antiderivative using Table 1. 10dxx(57x)2 Find each antiderivative using Table 1. 25x(7x+2)dx Find each antiderivative using Table 1. 54x21dx Find each antiderivative using Table 1. 22. Find each antiderivative using Table 1. 4m2+16dm Find each antiderivative using Table 1. 24. Find each antiderivative using Table 1. 25. Find each antiderivative using Table 1. (Inx)4dx;x0 Find each antiderivative using Table 1. exx3dx Find each antiderivative using Table 1. 28. Find each antiderivative using Table 1. x1+2xdx Find each antiderivative using Table 1. 30. Supply from marginal supply. Stellar Lawn Care introduces a new kind of lawn seeder. It finds that its marginal supply for the seeder satisfies S(x)=100x(20x)2,0x19, Where S(x) is the quantity purchased when the price is x thousand dollars per seeder. Find S(x), given that the company sells 2000 seeders when the price is 19 thousand dollars. S(x)=100[2020x+In(20x)]Learning rate. The rate of change of the probability that an employee learns a task on a new assembly line is p(t)=1t(2+t)2, Where p(t) is the probability of learning the task after t months. Find p(t) given that p=0.8267whent=2. p(t)=12(2+t)+14In(t2+t)+0.8750 Evaluate using Table 1 or www.wolframalpha.com. 33. Evaluate using Table 1 or www.wolframalpha.com. 34. Evaluate using Table 1 or www.wolframalpha.com. dxx34x2+4x Evaluate using Table 1 or www.wolframalpha.com. 36. Evaluate using Table 1 or www.wolframalpha.com. e2xdx96ex+e2x Evaluate using Table 1 or www.wolframalpha.com. 38. Prove Formula 6 in Table 1 using integration by parts.40. Prove Formula 18 in Table 1 using the substitution. These review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect. Concept Reinforcement Match each term in column A with the most appropriate graph in column B. 1. Column A Column B Consumer Surplus (d) a. a) These review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect. Concept Reinforcement Match each term in column A with the most appropriate graph in column B. 2. Column A Column B Producer Surplus (e) a. b) These review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect. Concept Reinforcement Match each term in column A with the most appropriate graph in column B. Column A Column B Exponential Distribution [5,4] (f) a. c)These review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect. Concept Reinforcement Match each term in column A with the most appropriate graph in column B. 4. Column A Column B Standard Normal Distribution (b) a. d) These review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect. Concept Reinforcement Match each term in column A with the most appropriate graph in column B. 5. Column A Column B Uniform Distribution (c) a. e) These review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect. Concept Reinforcement Match each term in column A with the most appropriate graph in column B. 6. Column A Column B Solid of Revolution (a) a. f) Classify each statement as either true or false. 7. The consumer surplus is the extra amount a consumer pays a tax on a purchased product. Classify each statement as either true or false. The accumulated present value of an investment is the value of the investment as a tax-deductible gift to a nonprofit charity.9RE 10RE Classify each statement as either true or false. 11. If f is a probability density function over and x is a continuous random variable over then the mean value of is always. Classify each statement as either true or false. If y=f(x) is a solution of y=x2+3x, then y=f(x)+C is also a solution.Let be the price, in dollars per unit, that consumers are willing to pay x units of an items, and be the price, in dollars per unit, that producers are willing to accept for x units. 13. Find the equilibrium point. Let D(x)=(x6)2 be the price, in dollars per unit, that consumers are willing to pay x units of an items, and S(x)=x2+12 be the price, in dollars per unit, that producers are willing to accept for x units. Find the consumer surplus at the equilibrium point.15RE 16RE 17RE 18RE 19RE 20RE 21RE Physical Science: Depletion of iron ore. Would reserves of iron ore in 2013 were estimated to be 81 billion metric tons. (Source: minerals.usgs.gov.) Assuming that the growth rate in Exercise 21 continues and no new reserves are discovered, when will world reserves of iron ore be depleted?Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 11x2dx Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 1e4xdx Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 25. 26RE Business: waiting time. Sharif arrives at a random time at a doctors office where the waiting time t to see a doctor is no more than 25 min. The probability density function for t is f(t)=125, for 0t25. Find the probability that Sharif will have to wait no more than 15 min to see a doctor.28RE 29RE 30RE 31RE Given the probability density function f(x)=6x(1x), over [0,1] Find each of the following. The standard deviation Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(0x1.85)Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(1.74x1.43)35RE 36RE 37RE 38RE 39RE 40RE 41RE 42RE Solve each differential equation. 43. 44RE 45RE 46RE 47RE 48RE 49RE 50RE 51RE 52RE 53RE 54RE 55RE 1T 2T 3T 4T 5T 6T 7T 8. Business: accumulated present value of a continuous income stream. Bruce Kent wants to have $25,000 in 5 yr for a down payment on a house. Find the amount he needs to save, at dollars per year, at 4.125%, compounded continuously, to achieve the desired future value. Business: contract buyout. Guy Laplace signs a 6-yr contract to play professional hockey at a salary of $475,000 per year. After 2 yr, his team offers to buy out the remainder of his contract. What is the least amount Guy should accept, if the going interest rate is 5.1%, compounded continuously?Business: future value of a noncontinuous income stream. Sonia signs a contract that will pay her an income of R(t)=(100,000+10,000t) where t is in years and 0t8. If she invests this money at 5%, compounded continuously, what is the future value of the income stream?Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 11. Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 12. 13T Business: times of telephone calls. A telephone company determines that the length of a phone call, t, in times, is an exponentially distributed random variable with probability density function f(t)=2e2t,0t Find the probability that a phone call will last between 1 min and 2 min.15T Given the probability density function over find each of the following. 16. Given the probability density function over find each of the following. 17. The mean Given the probability density function f(x)=14x over [1,3], find each of the following. The variance Given the probability density function over find each of the following. 19. The standard variation Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(0x1.3)Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. 21. Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. 22. Business: price distribution. The price per pound p of wild salmon at various stores in a certain city is normally distributed with mean =16 and standard deviation =2.50. What is the probability that the price at a randomly selected store is least $17.25 per pound?24. Business: price distribution. If the per pound p of wild salmon is normally distributed with mean and standard deviation , what is the lowest price in the top 15% of salmon prices? Find the volume generated by rotating the area bounded by each set of equations around the x-axis. y=1x,x=1,x=5 26T 27T Business: grain storage. A grain silo is a circular cylinder with a conical roof. The roof can be modeled by rotating the area under the graph of y=300.5x, where 0x10, around the y-axis. Find the volume of the silo, assuming the floor of the cylinder lies on the x-axis and both x and y are measured in feet.29T 30T Solve each differential equation. dydt=6y;y=11 when t=0 32T 33T Solve each differential equation. y=4y+xy Economics: elasticity. Find the demand function q=D(x), given the elasticity condition E(x)=4 for all x0.36. Business: stock growth. The growth rate of Fabric Industries stock, in dollars per month, can be modeled by where V is the value of a share, in dollars, after t months; the limiting value of the stock; k is a constant; and . a. Express the solution in terms of L and k. b. If determine k to the nearest hundredth. c. Rewrite in terms of t and k using the value of k found in part (b). d. Use the equation in part (c) to find the value of the stock after 12 months. e. In how many months will the value be $30? 37T 38T 39T 1ETE 2ETE Now consider the bottle shown at the right. To find the bottles volume in a similar manner, we turn it on its side, use a measuring device to take vertical measurements of the radii, and proceed as we did with the urn. The table of measurements is as follows. x (in inches) y (in inches) 0 1.125 1 1.275 2 1.250 3 1.275 4 1.275 5 1.125 6 1.000 7 0.875 8 0.750 9 0.500 10 0.500 Using regression, fit a quadratic polynomial function to the data. Label this function y1.4ETE 5ETE 6ETE Now consider the bottle shown at the right. To find the bottles volume in a similar manner, we turn it on its side, use a measuring device to take vertical measurements of the radii, and proceed as we did with the urn. The table of measurements is as follows. x (in inches) y (in inches) 0 1.125 1 1.275 2 1.250 3 1.275 4 1.275 5 1.125 6 1.000 7 0.875 8 0.750 9 0.500 10 0.500 To determine the volume of the bottles base, extend the x-axis 0.015 unit to the left of 0. The total volume of the plastic that forms the bottle can then be found by first integrating y3=(y2)2 over [0.015,0] to find the volume of the base and then integrating y4=(y2y1)2 over [0,10] to find the volume of the side. What is the total volume of plastic used to form the bottle?In Exercises 1-14, is the price, in dollars per unit, that consumers will pay for x units of an item, is the price, in dollars per unit, that producers will accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. 1. a. (a) b. (b) $1.50 c. (c) $1 In Exercises 1-14, is the price, in dollars per unit, that consumers will pay for x units of an item, is the price, in dollars per unit, that producers will accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. 2. a. (a) b. (b) $15 c. (c) $9 In Exercises 1-14, D(x) is the price, in dollars per unit, that consumers will pay for x units of an item, S(x) is the price, in dollars per unit, that producers will accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. D(x)=(x3)2,S(x)=x2+2x+1 a. (a) (1,4) b. (b) $2.33 c. (c) $1.67 In Exercises 1-14, is the price, in dollars per unit, that consumers will pay for x units of an item, is the price, in dollars per unit, that producers will accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. 4. a. (a) b. (b) $3.33 c. (c) $1.67 In Exercises 1-14, D(x) is the price, in dollars per unit, that consumers will pay for x units of an item, S(x) is the price, in dollars per unit, that producers will accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. D(x)=(x8)2,S(x)=x2 a. (a) (4,16) b. (b) $85.33 c. (c) $42.67 In Exercises 1-14, D(x) is the price, in dollars per unit, that consumers will pay for x units of an item, S(x) is the price, in dollars per unit, that producers will accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. D(x)=(x6)2,S(x)=x2 a. (a) (3,9) b. (b) $36 c. (c) $18 In Exercises 1-14, D(x) is the price, in dollars per unit, that consumers will pay for x units of an item, S(x) is the price, in dollars per unit, that producers will accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. D(x)=880030x,S(x)=7000+15x a. (a) (40,7600) b. (b) $24,000 c. (c) $12,000 8E 9E 10E In Exercises 1-14, D(x) is the price, in dollars per unit, that consumers will pay for x units of an item, S(x) is the price, in dollars per unit, that producers will accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. D(x)=1800x+1,S(x)=2x+1 a. (a) (899,60) b. (b) $50,460 c. (c) $17,941.33 In Exercises 1-14, is the price, in dollars per unit, that consumers will pay for x units of an item, is the price, in dollars per unit, that producers will accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. 12. a. (a) b. (b) 1000 c. (c) $333.33 In Exercises 1-14, D(x) is the price, in dollars per unit, that consumers will pay for x units of an item, S(x) is the price, in dollars per unit, that producers will accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. D(x)=13x, for 0x13; S(x)=x+17 a. (a) (8,5) b. (b) $32 c. (c) $3.40 In Exercises 1-14, D(x) is the price, in dollars per unit, that consumers will pay for x units of an item, S(x) is the price, in dollars per unit, that producers will accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. D(x)=(x4)2,S(x)=x2+2x+8 a. (a) (0.8,10.24) b. (b) $2.22 c. (c) $0.98 Business: Consumer and Producer Surplus. Beth enjoys skydiving and is willing to pay p dollars per jump, where p=D(x)=7.5x260.5x+254. a. Find Beths consumer surplus if she makes 2 jumps. b. Suppose the supply function for Aero Skydiving Center is given by p=S(x)=15x+95. Find the producer surplus if the centers sell Beth 2 jumps. c. Find the equilibrium point and the consumer and producer surpluses at this point. Assume that Beth makes no more than 5 jumps. d. Explain what the equilibrium point represents to both Beth and Aero Skydiving Center.16. Business: Consumer and Producer Surplus. Chris is enjoying a day at the Splashorama Water Park. His demand function for sliding down EI Monstro Water Slide is given by , where x is the number of slides and p is the price, in dollars per slide, that he is willing to pay. a. Find Chris’s consumer surplus if he buys 3 slides b. Suppose the supply function for Splashorama Water Park for the EI Monstro Water Slide is given by. Find the producer surplus if Chris is sold 3 slides. c. Find the equilibrium point and the consumer and producer surpluses at this point. Assume that Chris buys no more than 6 slides. Equilibrium point: (4, 6.5), consumer surplus: $27.47, producer surplus: $2.40 d. Explain what the equilibrium point represents to both Chris and to the Splashorama Water Park. For Exercises 17 and 18, follow the directions given for Exercises 1-14. D(x)=ex+4.5,S(x)=ex5.5 a. (a) (5, $0.61) b. (b) $86.36 c. (c) $2.45 For Exercises 17 and 18, follow the directions given for Exercises 1-14. 18. a. (a) (7, $7) b. (b) $1.71 c. (c) $24.50 Explain why both consumers and producers feel good when consumer and producer surpluses exist.Research consumer and producer surpluses in an economics book. Write a brief description of what each represents.For Exercises 21 and 22, graph each pair of demand and supply functions. Then: a. Find the equilibrium point using the intersect feature or another feature that will allow you to find this point of intersection. b. Graph and identify the regions of both consumer and producer surpluses. c. Find the consumer surplus. d. Find the producer surplus. 21. For Exercises 21 and 22, graph each pair of demand and supply functions. Then: a. Find the equilibrium point using the intersect feature or another feature that will allow you to find this point of intersection. b. Graph y=D(xE) and identify the regions of both consumer and producer surpluses. c. Find the consumer surplus. d. Find the producer surplus. D(x)=1513x,S(x)=2x3 For all exercises in this exercise set, use a graphing calculator. Find the future value P of each amount invested for time period t at interest rate k, compounded continuously. 1. 2E 3E For all exercises in this exercise set, use a graphing calculator. Find the future value P of each amount invested for time period t at interest rate k, compounded continuously. 4. For all exercises in this exercise set, use a graphing calculator. Find the present value of each amount P due t years in the future and invested at interest rate k, compounded continuously. 5. For all exercises in this exercise set, use a graphing calculator. Find the present value P0 of each amount P due t years in the future and invested at interest rate k, compounded continuously. P=100,000,t=8yr,k=4 For all exercises in this exercise set, use a graphing calculator. Find the present value of each amount P due t years in the future and invested at interest rate k, compounded continuously. 7. For all exercises in this exercise set, use a graphing calculator. Find the present value P0 of each amount P due t years in the future and invested at interest rate k, compounded continuously. P=2,000,000,t=20yr,k=3.5 For all exercises in this exercise set, use a graphing calculator. Find the accumulated future value of each continuous income stream at rate R(t) for the given time T and interest rate k, compounded continuously. R(t)=50,000,T=22yr,k=5 For all exercises in this exercise set, use a graphing calculator. Find the accumulated future value of each continuous income stream at rate R(t) for the given time T and interest rate k, compounded continuously. R(t)=125,000,T=20yr,k=6 For all exercises in this exercise set, use a graphing calculator. Find the accumulated future value of each continuous income stream at rate for the given time T and interest rate k, compounded continuously. 11. For all exercises in this exercise set, use a graphing calculator. Find the accumulated future value of each continuous income stream at rate R(t) for the given time T and interest rate k, compounded continuously. R(t)=50,000,T=22yr,k=2.75 For all exercises in this exercise set, use a graphing calculator. Find the accumulated present value of each continuous income stream at rate R(t), for the given time T and interest rate k, compounded continuously. R(t)=250,000,T=18yr,k=4 For all exercises in this exercise set, use a graphing calculator. Find the accumulated present value of each continuous income stream at rate for the given time T and interest rate k, compounded continuously. 14. For all exercises in this exercise set, use a graphing calculator. Find the accumulated present value of each continuous income stream at rate for the given time T and interest rate k, compounded continuously. 15. For all exercises in this exercise set, use a graphing calculator. Find the accumulated present value of each continuous income stream at rate R(t), for the given time T and interest rate k, compounded continuously. R(t)=520,000,T=25yr,k=6 For all exercises in this exercise set, use a graphing calculator. Find the accumulated present value of each continuous income stream at rate for the given time T and interest rate k, compounded continuously. 17. For all exercises in this exercise set, use a graphing calculator. Find the accumulated present value of each continuous income stream at rate for the given time T and interest rate k, compounded continuously. 18. For all exercises in this exercise set, use a graphing calculator. Find the accumulated present value of each continuous income stream at rate R(t), for the given time T and interest rate k, compounded continuously. R(t)=(2000t+7),T=30yr,k=4.5 For all exercises in this exercise set, use a graphing calculator. Find the accumulated present value of each continuous income stream at rate R(t), for the given time T and interest rate k, compounded continuously. R(t)=t2,T=40yr,k=4.25 Present value of a trust. In 18 yr, Maggie Oaks is to receive $200,000 under the terms of a trust established by her grandparents. Assuming an interest rate of 3.8%, compounded continuously, what is the present value of Maggies trust?22. Present value of a trust. In 16 yr, Claire Beasley is to receive $180,000 under the terms of a trust established by her aunt. Assuming an interest rate of 4.2%, compounded continuously, what is the present value of Claire’s trust? 23. Salary Value. At age 35, Rochelle earns her MBA and accepts a position as vice president of an asphalt company. Assume that she will retire at the age of 65, having received an annual salary of $95,000, and that the interest rate is 5%, compounded continuously. a. What is the accumulated present value of her position? b. What is the accumulated future value of her position? 24. Salary Value. At age 25, Del earns his CPA and accepts a position in an accounting firm. Del plans to retire at the age of 65, having received an annual salary of $125,000. Assume an interest rate of 4%, compounded continuously. a. What is the accumulated present value of his position? b. What is the accumulated future value of his position? 25. Future value of an inheritance. Upon the death of his uncle, David receives an inheritance of $50,000, which he invests for 16 yr at 4.3%, compounded continuously. What is the future value of the inheritance? 26. Future value of an inheritance. Upon the death of his aunt, Burt receives an inheritance of $80,000, which he invests for 20 yr at 3.9%, compounded continuously. What is the future value of the inheritance? 27. Decision-Making. A group of entrepreneurs is considering the purchase of a fast-food franchise. Franchise A predicts that it will bring in a constant revenue stream of $80,000 per year for 10 yr. Franchise B predicts that it will bring in a constant revenue stream of $95,000 per year for 8 yr. Based on a comparison of accumulated present values, which franchise is the better buy, assuming the interest rate is 4.1%, compounded continuously, and both franchises have the same purchase price? 28. Decision-Making. A group of entrepreneurs is considering the purchase of a fast-food franchise. Franchise A predicts that it will bring in a constant revenue stream of $120,000 per year for 8 yr. Franchise B predicts that it will bring in a constant revenue steam of $112,000 per year for 10 yr. Based on a comparison of accumulated present values, which franchise is the better buy, assuming the interest rate is 5.4%, compounded continuously, and both franchises have the same purchase price? Decision-Making. An athlete attains free agency and is looking for a new team. The Bronco Crunchers offer a salary of $100,000t for 8 yr, and the Doppler Radars offer a salary of $83,000t for 9 yr, where t is in years. a. Based on the accumulated present values of the salaries, which team has the better offer, assuming an interest rate of 4.2%, compounded continuously? b. What signing bonus should the team with the lower offer give to equalize the offers? The difference between the accumulated present values of the two offers, or $57,309.30 30. Capital Outlay. Chrome solutions determines that the rate of revenue coming in from a new machine is in dollars per year, for 8 yr, after which the machine will be replaced. The company learns that an alternative machine will yield revenue at a rate of . a. Find the accumulated present value of the income stream from each machine at an interest rate of 5.8%, compounded continuously. b. Find the difference in the accumulated present values. 31. Trust Fund. Bob and Ann MacKenzie have a new grandchild, Brenda, and want to create a trust fund for her that will yield $250,000 on her 24th birthday. a. What lump sum should they deposit now at 5.8%, compounded continuously, to achieve $250,000? b. The amount in part (a) is more than they can afford, so they decide to invest a constant amount, dollars per year. Find such that the accumulated future value of the continuous money stream is $250,000, assuming an interest rate of 5.8%, compounded continuously. 32. Trust Fund. Ted and Edith Markey have a new grandchild, Kurt, and want to create a trust a trust fund for him that will yield $1,000,000 on his 22nd birthday. a. What lump sum should they deposit now at 4.2%, compounded continuously, to achieve $1,000,000? b. The amount in part (a) is more than they can afford, so they decide to invest a constant amount, dollars per year. Find such that the accumulated future value of the continuous money stream is $1,000,000, assuming an interest rate of 4.2%, compounded continuously. 33. Early Retirement. Lauren Johnson signs a 10-yr contract as a loan officer for a bank, at a salary of $84,000 per year. After 7 yr, the bank offers her early retirement. What is the least amount the bank should offer Lauren, assuming an interest rate of 4.7%, compounded continuously? 34. Early Sports Retirement. Tory Johnson signs a 10-yr contract to play for a football team at a salary of $5,000,000 per year. After 6 yr, his skills deteriorate, and the team offers to buy the rest of his contract. What is the least amount Tory should accept for the buyout, assuming an interest rate of 4.9%, compounded continuously? Disability Insurance Settlement. A movie stuntman who receives an annual salary of $180,000 per year is injured and can no longer work. Through a settlement with an insurance company, he is granted a continuous income stream of $120,000 per year for 20 yr. The stuntman invests the money at 4% compounded continuously. a. Find the accumulated future value of the continuous income stream. Round your answer to the nearest $10. b. Thinking that he might not live 20 yr, the stuntman negotiates a flat sum payment from the insurance company, which is the accumulated present value of the continuous income stream. What is that amount? Round your answer to the nearest $10.Disability Insurance Settlement. Dale was a furnace maintenance employee who earned a salary of $70,000 per year before he was injured on the job. Through a settlement with is employers insurance company, he is granted a continuous income stream of $40,000 per year for 25 yr. Dale invests the money at 5%, compounded continuously. a. Find the accumulated future value of the continuous income stream. Round your answer to the nearest $10. b. Thinking that he might not live for 25 yr, Dale negotiates a flat sum payment from the insurance company, which is the accumulated present value of the continuous stream plus $100,000. What is that amount? Round your answer to the nearest $10.37. Lottery Winnings and Risk Analysis. Lucky Larry wins $1,000,000 in a state lottery. The standard way in which a state pays such lottery winnings is at a constant rate of $50,000 per year for 20 yr. a. If Lucky invests each payment from the state at 4.4%, compounded continuously, what is the accumulated future value of the income stream? Round your answer to the nearest $10. b. What is the accumulated present value of the income stream 4.4%, compounded continuously? This amount represents what the state has to invest at the start of its lottery payments, assuming the 4.4% interest rate holds. c. The risk for Lucky is that he doesn’t know how long he will live or what the future interest rate will be; it might drop or rise, or it could vary considerably over 20 yr. This is the risk he assumes in accepting payments of $50,000 a year over 20 yr. Lucky has taken a course in business calculus so he is aware of the formulas for accumulated future value and present value. He calculates the accumulated present value of the income stream for interest rates of 3%, 4%, and 5%. What values does he obtain? d. Lucky thinks “a bird in the hand (present value) is worth two in the bush (future value)” and decides to negotiate with the state for immediate payment of his lottery winnings. He asks the state for $750,000. They offer $600,000. Discuss the pros and cons of each amount. Lucky finally accepts $675,000. Is this a good decision? Why or why not? Negotiating a sports contract. Gusto Stick is a professional baseball player who has just become a free agent. His attorney begins negotiations with an interested team by asking for a contract that provides Gusto with an income stream given by R1(t)=800,000+340,000t, over 10 yr, where t is in years. (Round all answers to the nearest $100.) a. What is the accumulated future value of the offer, assuming an interest rate of 5%, compounded continuously? b. What is the accumulated present value of the offer, assuming an interest rate of 5%, compounded continuously? c. The team counters by offering an income stream given by R2(t)=600,000+210,000t. What is the accumulated present value of this counteroffer? d. Gusto comes back with a demand for an income stream given by R3(t)=1,000,000+250,000t. What is the accumulated present value of this income stream? e. Gusto signs a contract for the income stream in part (d) but decides to live on $500,000 each year, investing the rest at 5%, compounded continuously. What is the accumulated future value of the remaining income, assuming an interest rate of 5%, compounded continuously?39. Accumulated Present Value. The Wilkinsons want to have $100,000 in 10 yr for a down payment on a retirement home. Find the continuous money stream, dollars per year, that they need to invest at 4.33%, compounded continuously, to generate $100,000. 40. Accumulated Present Value. Tania wants to have $20,000 in 5 yr for her dream vacation. Find the continuous money stream, dollars per year that she needs to invest at 5.125%, compounded, to generate $20,000. 41. Demand for Natural Gas. In 2013 world consumption of natural gas was approximately 117.2 trillion cubic feet and was growing exponentially at about 1.24% per year(Source: U.S. Energy Information Administration.). If the demand continues to grow at this rate, how many cubic feet of natural gas will the gas the world use from 2015 to 2025? 42. Demand for aluminum ore (bauxite). In 2013, bauxite production was approximately 232 million metric tons, and the demand was growing exponentially at a rate of 2.6% per year (Source: minerals.usgs.gov.). If the demand continues to grow at this rate, how many metric tons of bauxite will the world use from 2015 to 2030? Depletion of Natural Gas. The world reserves of natural gas were approximately 6597 trillion cubic feet in 2013. (Source: www.cia.com.) Assuming that the growth described in Exercise 41 continues and that no new reserves are found, when will the world reserves of natural gas be depleted?44. Depletion of aluminum ore (bauxite). In 2013, the world reserves of bauxite were about 65 billion metric tons (Source: U.S. Geological Survey summaries, Jan. 2014.). Assuming that the growth described in Exercise 42 continues and that no new reserves are discovered, when will the world reserves of bauxite be depleted? 45. Demand for and depletion of oil. In 2013, annual world demand for crude oil was approximately 33.3 billion barrels, and it was projected to increase by 1.5% per year. (Sources: Based on information from www.iea.gov and www.cia.gov.) a. Assuming an exponential growth model, predict the demand in 2020. b. World reserves of crude oil in 2013 were approximately 1635 billion barrels. Assuming that no new oil reserves are found, when will the reserves be depleted? The model can be applied to calculate the buildup of a radioactive material that is being released into the atmosphere at a constant annual rate. Some of the material decays, but more continues to be released. The amount present at time T is given by the integral above, where P is the amount released per year and k is the half-life. 46. Radioactive Buildup. Plutonium-239 has a decay rate of approximately 0.003% per year. Suppose plutonium-239 is released into the atmosphere for 20 yr at a constant rate of 1 lb per year. How much plutonium-239 will be present in the atmosphere after 20 yr? The model can be applied to calculate the buildup of a radioactive material that is being released into the atmosphere at a constant annual rate. Some of the material decays, but more continues to be released. The amount present at time T is given by the integral above, where P is the amount released per year and k is the half-life. 47. Radioactive Buildup. Cesium-137 has a decay rate of 2.3% per year. Suppose cesium-137 is released into the atmosphere for 20 yr at a rate of 1 lb per year. How much cesium-137 will be present in the atmosphere after 20 yr? 48E The capitalized cost, c, of an asset over its lifetime is the total of the initial cost and the present value of all maintenance expenses that will occur in the future. It is computed with the formula where is the initial cost of the asset, L is lifetime (in years), k is the interest rate (compounded continuously), and is the annual cost of maintenance. Find the capitalized cost under each set of assumptions. 49. 50E 51E The capitalized cost, c, of an asset over its lifetime is the total of the initial cost and the present value of all maintenance expenses that will occur in the future. It is computed with the formula c=c0+0Lm(t)ektdt, where c0 is the initial cost of the asset, L is lifetime (in years), k is the interest rate (compounded continuously), and m(t) is the annual cost of maintenance. Find the capitalized cost under each set of assumptions. How would you explain the concepts of present value and accumulated present value to a friend who has not studied this chapter?Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 1. 2E 3E 4E Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 03e3xdx Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 6. Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 7. Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 1dxx4 Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 9. 10E 11E Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 12. 13E 14E 15E 16E Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 0xexdx Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 1lnxdx 19E Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 0Qektdt,k0 Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 21. Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 22. Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. tdt 24E 25. Find the area, if it is finite, of the region under the graph of over. 26. Find the area, if it is finite, of the region under the graph of over. 27. Find the area, if it is finite, of the region under the graph of over. Find the area, if it is finite, of the region under the graph of y=1/(3x2)3 over [6,).29. Total Profit from Marginal Profit. Myna’s Fashions determines that its marginal profit, in dollars, from producing x shawls is given by Suppose it were possible for this firm to make infinitely many shawls. What would its total profit be? 30. Total Profit from Marginal Profit. Find the total profit in Exercise 29 if where. 31E Total Production. A firm determines that it can produce tires at the rate r(t)=(2106)e0.42t, where t is in years. Assuming that the firm endures forever (it never gets tired), how many tires can it make?Accumulated Present Value. Find the accumulated present value of an investment for which there is a perpetual continuous money flow of $3600 per year at an interest rate of 5%, compounded continuously.34. Accumulated Present Value. Find the accumulated present value of an investment for which there is a perpetual continuous money flow of $3500 per year at an interest rate of 4%, compounded continuously. Accumulated Present Value. Find the accumulated present value of an investment for which there is a perpetual continuous money flow of $5000 per year, assuming continuously compounded interest at a rate of 3.7%.Accumulated Present Value. Find the accumulated present value of an investment for which there is a perpetual continuous money flow of 2000e0.01t per year, assuming continuously compounded interest at a rate of 5%.The capitalized cost, c, of an asset for an unlimited is the total of the initial cost and the present value of all maintenance expenses that will occur in the future. It is computed with the formula where is the initial cost of the asset, k is the interest rate (compounded continuously), and is the annual cost of maintenance. Find the capitalized cost under each set of assumptions. 37. The capitalized cost, c, of an asset for an unlimited is the total of the initial cost and the present value of all maintenance expenses that will occur in the future. It is computed with the formula where is the initial cost of the asset, k is the interest rate (compounded continuously), and is the annual cost of maintenance. Find the capitalized cost under each set of assumptions. 38. Radioactive Buildup. Plutonium has a decay rate of 0.003% per year. Suppose a nuclear accident causes plutonium to be released into the atmosphere perpetually at the rate of 1 lb per year. What is the limiting value of the radioactive buildup?Radioactive Buildup. Cesium-137 has a decay rate of 2.3% per year. Suppose a nuclear accident causes cesium-137 to be released into the atmosphere perpetually at the rate of 1 lb per year. What is the limiting value of the radioactive buildup?In the treatment of prostate cancer, radioactive implants are often used. The implants are left in the patient and never removed. The amount of energy that is transmitted to the body from the implant is measured in rem units and is given by where k is the decay constant for the radioactive material, t is the number of years since the implant, a is the time (in years) until the rem measurement is made, and is the initial rate at which energy is transmitted. (Source: www.cancer.gov. Use this information for Exercises 41 and 42. 41. Suppose the treatment uses iodine-125, which has a half-life of 60.1 days. a. Find the decay rate, k, of iodine-125 b. How much energy (measured in rems) is transmitted in the first month if the initial rate of transmission is 10 rems per year? c. What is the total amount of energy that the implant will transmit to the body? In the treatment of prostate cancer, radioactive implants are often used. The implants are left in the patient and never removed. The amount of energy that is transmitted to the body from the implant is measured in rem units and is given by where k is the decay constant for the radioactive material, t is the number of years since the implant, a is the time (in years) until the rem measurement is made, and is the initial rate at which energy is transmitted. (Source: www.cancer.gov. Use this information for Exercises 41 and 42. 42. Suppose the treatment uses palladium-103, which has a half-life of 16.99 days. a. Find the decay rate, k, of palladium-103. b. How much energy (measured in rems) is transmitted in the first month if the initial rate of transmission is 10 rems per year? c. What is the total amount of energy that the implant will transmit to the body? 43E 44E Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 45. Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 46. Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. 0xex2dx Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. xex2dx Suppose an oral dose of a drug is taken. Over, time, the drug is assimilated in the body and excreted though the urine. The total amount of the drug that has passed through the body in T hours is given by where is the rate of excretion of the drug. A typical rate-of-excretion function is where and t is the time, in hours. Use this information for Exercises 49-50. 49. Find and interpret the answer. That is, what does the integral represent? Suppose an oral dose of a drug is taken. Over, time, the drug is assimilated in the body and excreted though the urine. The total amount of the drug that has passed through the body in T hours is given by 0TE(t)dt, where E(t) is the rate of excretion of the drug. A typical rate-of-excretion function is E(t)=tekt, where k0 and t is the time, in hours. Use this information for Exercises 49-50. A physician prescribes a dosage of 100 mg. Find k.51. Consider the functions and . Suppose you get to a paint store to buy paint to cover the region under each graph over. Discuss whether you could be successful, and explain why or why not. Suppose you own a building that yields a continuous series of rental payments and you decide to sell the building. Explain how you would use the concept of the accumulated present value of a perpetual continuous money flow to determine a fair selling price.53E 54E 55. Find and explain the error in the following calculation: Approximate each integral. 141+x2dx 57E 58E Graph the function E and shade the area under the curve for the situation in Exercises 49 and 50.1E 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E Find k such that each function is a probability density function over the given interval. Then write the probability density function. f(x)=k(2x),[0,2]Find k such that each function is a probability density function over the given interval. Then write the probability density function. 20. Find k such that each function is a probability density function over the given interval. Then write the probability density function. 21. Find k such that each function is a probability density function over the given interval. Then write the probability density function. f(x)=kx,[1,3]23E Find k such that each function is a probability density function over the given interval. Then write the probability density function. 24. A dart is thrown at a number line in such a way that it always lands in [0,10]. Let x represent the number the dart hits. Suppose the probability density function for x is given by f(x)=150x, for 0x10 Find P(2x6), the probability that the dart lands in [2,6].26E 27E 28E 29. Transportation planning. Refer to Example 7. A transportation planner determines that the average distance between cars on a certain highway is 100 ft. Find the probability that the distance between two successive cars, chosen at random, is at most 40 ft.

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