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All Textbook Solutions for Finite Mathematics and Applied Calculus (MindTap Course List)

44E 45E Bottled Water Sales The rate of U.S. per capita sales of bottled water for the period 2007-2014 could be approximated by s(t)=0.25t2t+29 gallons per year (0t7), where t is time in years since the start of 2007.27 Use a Riemann sum with n=5 to estimate the total U.S. per capita sales of bottled water from the start of 2008 to the start of 2012. (Round your answer to the nearest gallon.) [HINT: See Example 2.]47E Online Auctions The following graph shows the rate of change n(t) of the number of active eBay users, in millions of users per quarter, from the first quarter of 2008 through the first quarter of 2011. (t is time in quarters; t=1 represents the first quarter of 2008.)29 Use a left Riemann sum with four subdivisions to estimate the total change in the number of active users from the second quarter of 2008 to the second quarter of 2010 (the interval [2,10] ). [HINT: See Example 3.]Scientific Research Exercises 49 and 50 are based on the following figure, which shows the rate of publication of science research papers for researchers in the United States and the European Union in the years 1990-2010:30 Here, t is time in years since 1990, and n(t) is the publication rate in thousands of articles per year. a. Use both left and right Riemann sums with five subdivisions to estimate the total number of science research papers by researchers in the United States during the 20-year period shown. [Hint: See Example 3.] b. Use the answers from part (a) to obtain an estimate of 020n(t) dt for researchers in the United States. [HINT: See Example 4.] Interpret the result.50E Graduate Degrees: Women The following graph shows the approximate number f(t) of doctoral degrees per year awarded 10 women in the United State during 2005-2013. (t=0 represent the start of 2005. and each unit of the y-axis represents 10,000 degrees.)31 Use the graph to estimate the total number of doctoral degree awarded to women from the start of 2005 to the start of 2010. [HINT: See Example 4.]52E 53E 54E 55E 56E 57E 58E 59E 60E 61E 62E 63E 64E 65E 66E 67E 68E 69E 70E The Normal Curve The normal distribution curve, which models the distributions of data in a wide range of applications, is given by the function p(x)=12e(x)2/(22) where =3.14159265. and and are constants called the standard deviation and the mean, respectively. The graph of the normal distribution (when =1 and =2 ) is shown in the figure. Exercises 71 and 72 illustrate its use. Test Scores Enormous State University's Calculus I test scores are modeled by a normal distribution with =72.6 and =5.2. The percentage of students who obtained scores between a and b on the test is given by abp(x)dx. a. Use a Riemann sum with n=40 to estimate the percentage of students who obtained between 60 and 100 on the test. b. What percentage of students scored less than 30?72E 73E 74E 75E 76E 77E 78E 79E 80E 81E 82E Your cellphone company charges you c(t)=20t+100 dollars for the tth minute. You make a 60-minute phone call. What kind of (left) Riemann sum represents the total cost of the call? Explain.84E 85E 86E 87E 88E In Exercises 144, evaluate the integral. [HINT: See Example 2.] 11(x2+2)dx 2E 3E 4E 5E 6E 7E 8E 9E 10E In Exercises 144, evaluate the integral. [HINT: See Example 2.] 012exdx 12E 13E 14E 15E 16E 17E 18E 19E 20E In Exercises 144, evaluate the integral. [HINT: See Example 2.] [HINT: In Exercises 17-44, use a shortcut or see Example 4.] 022x+1dx 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E Profit: iPhones (Compare Exercise 43 in Section 13.3.) Assume that Apples marginal cost function for the manufacture of x 32GB iPhone 6s per hour at the Foxconn Technology Group is c(x)=1600.002x, andthat Apple sells iPhone 6s for an average wholesale price of $580. Determine the total additional hourly profit corresponding to an increase in production and sales from 10,000 to 20,000 iPhone 6s per hour.60E 61E 62E 63E 64E Fuel Consumption The way Professor Waner drives, he burns gas at the rate of 1et gallonseach hour, t hours after a fill-up. Find the number of gallons of gas he burns in the first 10 hours after a fill-up.Fuel Consumption The way Professor Costenoble drives, he burns gas at the rate of 1/(t+1) gallonseach hour, t hours after a fill-up. Find the number of gallons of gas he burns in the first 10 hours after a fill-up.Bottled Water Sales (Compare Exercise 45 in Section 13.3.) The rate of U.S. sales of bottled water for the period 20072014 can be approximated by s(t)=0.08t20.26+8.8billiongallonperyear(0t7) where t is time in years since the start of 2007. Use a definite integral to estimate the total U.S. sales of bottled water from the start of 2008 to the start of 2014. (Round your answer to the nearest billion gallons.)Bottled Water Sales (Compare Exercise 46 in Section 13.3.) The rate of U.S. per capita sales of bottled water for the period 20072014 can be approximated by s(t)=0.25t2t+29gallonsperyear(0t7), where t is time in years since the start of 2007. Use a definite integral to estimate the total U.S. per capita sales of bottled water from the start of 2008 to the start of 2012. (Round your answer to the nearest gallon.)69E 70E 71E Big Brother (Compare Exercise 66 in Section 13.3.) The number of wiretaps authorized each year by U.S. state courts from 1990 to 2015 can be approximated by w(t)=412e0.071t(0t25). (t is time in years since the start of 1990.) Compute 525w(t)dt. (Round your answer to the nearest 10.) Interpret the answer.73E 74E 75E Sales Annual sales of fountain pens in Littleville are 4,000 per year and are increasing by 10% per year. How many fountain pens will be sold over the next 5 years?77E 78E 79E Online Auctions The rate of change n(t) of the number of active eBay users could be approximated by n(t)=0.200t4+0.06t30.55t30.55t2+1.9t1millionusersperquarter(1t13) where t is time in quarters. (t=1 represents the first quarter of 2008.) Use technology to compute 513n(t)dt correctto the nearest whole number. Interpret your answer.81E 82E 83E 84E 85E 86E 87E 88E 89E 90E 91E 92E 93E 94E 95E 96E 97E 98E 99E 100E 1RE 2RE 3RE In Exercises 1-10, evaluate the given integral. log5xdx 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE In Exercises 27-32, decide whether the given integral converges. If the integral converges, compute its value. 02x(x21)1/3dx 31RE 32RE 33RE 34RE 35RE 36RE 37RE Spending on Shipping During the past 10 months, OHaganBooks.com shipped orders at a rate of about q(t)=25t+3,200packagespermonth(0t10). (t is the time in months since the beginning of the year.) During the same period the cost of shipping a package averaged approximately p(t)=4e0.04tdollarsperpackage. Use an integral to estimate, to the nearest thousand dollars, the total spent on shipping orders during the given period.39RE Investments OHaganBooks.com keeps its cash reserves in a hedge fund paying 6% compounded continuously. It starts a year with $1 million in reserves and does not withdraw or deposit any money. a. What is the average amount it will have in the fund over the course of 2 years? b. Find the 1-month moving average of the amount it has in the fund.41RE 42RE 43RE 44RE 45RE 46RE Acquisitions The Megabucks Corporation is considering buying OHaganBooks.com. It estimates OHaganBooks.coms revenue stream at $50 million per year, growing continuously at a 10% rate. Assuming an interest rate of 6%, how much is OHaganBooks.coms revenue for the next year worth now?48RE 49RE 50RE In Exercises 140, evaluate the integral using integration by parts where possible. [HINT: See the examples in the text.] 2xexdx 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E In Exercises 140, evaluate the integral using integration by parts where possible. [HINT: See the examples in the text.] (1x2)2xdx 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E In Exercises 140, evaluate the integral using integration by parts where possible. [HINT: See the examples in the text.] log3xdx 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E In Exercises 140, evaluate the integral using integration by parts where possible. [HINT: See the examples in the text.] 12xln(2x)dx In Exercises 140, evaluate the integral using integration by parts where possible. [HINT: See the examples in the text.] 12x2ln(3x)dx In Exercises 140, evaluate the integral using integration by parts where possible. [HINT: See the examples in the text.] 01xln(x+1)dx 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E 53E 54E 55E 56E Spending on Gasoline From the beginning of 2000 to the beginning of 2007 the United States consumed gasoline at a rate of about q(t)=2.0t+131billiongallonsperyear(0t7) (t is the year since 2000.)2 During the same period of price of gasoline was approximately p(t)=1.2e0.11tdollarspergallon Use an integral to estimate, to the nearest 10 billion dollars, the total spent on gasoline during the given period. [HINT: Rate of spending =p(t)q(t).]Spending on Gasoline From the beginning of 2007 to the beginning of 2014 the United States consumed gasoline at a rate of about q(t)=1.2t+141billiongallonsperyear(0t7) (t is the year since 2007.)2 During the same period of price of gasoline was approximately p(t)=3.52.38e0.5tdollarspergallon Use an integral to estimate, to the nearest 10 billion dollars, the total spent on gasoline during the given period. [HINT: Rate of spending =p(t)q(t).]Housing The following graph shows the annual number of housing starts in the United States during 20002008 together with a quadratic approximating model: s(t)=30t2+240t+800thousandhomesperyear(0t8) (t is the time in years since 2000.)4 At the same time, the homes being built were getting larger: a(t)=40t+2,000squarefeet. Use the given models to estimate the total housing area under construction over the given period. (Use integration by parts to evaluate the integral, and round your answer to the nearest billion square feet.) [HINT: Rate of change of area under construction =p(t)q(t).]Housing for Sale The following graph shows the number of housing starts for sale purposes in the United States during 20002008 together with a quadratic approximating model: s(t)=33t2+240t+700thousandhomesperyear(0t8) (t is the time in years since 2000.)5 At the same time, the homes being built were getting larger: The average area per home was approximately a(t)=40t+2,000squarefeet. Use the given models to estimate the total housing area under construction over the given period. (Use integration by parts to evaluate the integral, and round your answer to the nearest billion square feet.) [HINT: Rate of change of area under construction =s(t)a(t).]61E Bottled Water Sales The rate of U.S. sales of bottled water for the period 20002010 can be approximated by s(t)=0.18t2+3t+15gallonsperyear(0t10), where t is time in years since the start of 2000.7 After conducting a survey of sales in your state, you estimate that consumption in gyms accounts for a fraction. f(t)=0.2+0.04t. ofall bottled water consumed in year t. Assuming that your model is correct, estimate, to the nearest gallon, the total amount of bottled water consumed per capita in gyms from the start of 2005 to the start of 2010. [HINT: Rate of consumption =s(t)f(t). Also see Example 31.]Oil Production in Mexico The rate of crude oil production from 2008 to 2013 by Pemex, Mexicos national oil company, can be approximated by q(t)=6.2t2146t+1,910millionbarrelsperyear(8t13), where t is time in years since the start of 2000.8 During that time, the price of oil was approximately9 p(t)=48e0.046tdollarsperbarrel Obtain an expression for Pemexs total oil revenue R(x) from the start of 2008 to the start of year x as a function of x. (Do not simplify the answer; round all coefficients to three significant digits.) [HINT: Rate of revenue =p(t)q(t).]Oil Imports from Mexico The rate of crude oil imports to the United States from Mexico from 2009 to 2013 can be approximated by r(t)=14t2+292t1,100millionbarrelsperyear(9t13), where t is time in years since the start of 2000.10 During that time, the price of oil was approximately11 p(t)=48e0.046tdollarsperbarrel Obtain an expression for the total oil revenue R(x) Mexico earned from the United States from the start of 2009 to the start of year x as a function of x. (Do not simplify the answer; round all coefficients to three significant digits.) [HINT: Rate of revenue =p(t)r(t).]65E 66E 67E Integrals of Piecewise-Linear Functions Exercises 67 and 68 are based on the following formula that can be used to represent a piecewise-linear function as a closed-form function: {p(x)ifxaq(x)ifxa=p(x)+12[q(x)p(x)][1+|xa|xa] Such functions can then be integrated using the technique of Exercises 4552. Population: Mexico The rate of change of population in Mexico over 19502090 was approximately r(t)={0.05t+2.5if0t200.075t+5if20t40 million people per year, where t is time in years since 1950. a. Use the formula given before the exercise to represent r(t) asa closed-form function. [HINT: Use the formula with a=20.] b. Use the result of part (a) and a definite integral to estimate the total increase in population over the given 40-year period. [HINT: Break up the integral into two, and use the technique of Exercises 4552 to evaluate one of them.]69E 70E 71E 72E 73E 74E In Exercises 7380, indicate whether the given integral calls for integration by parts or substitution. (3x2x)e6x1dx 76E 77E 78E 79E 80E 81E 82E 83E 84E In Exercises 18, find the area of the shaded region. (We suggest that you use technology to check your answers.)2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E

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