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All Textbook Solutions for Applied Calculus

In Exercises 3548 the graph of f is given. Use the graph to compute the quantities asked for. [HINT: See Examples 45.] a. limx1f(x) b. limx1f(x) c. limx+f(x) d. limxf(x)In Exercises 3548 the graph of f is given. Use the graph to compute the quantities asked for. [HINT: See Examples 45.] a. limx2f(x) b. limx0+f(x) c. limx0f(x) d. limx0f(x) e. f(0) f. limxf(x)In Exercises 3548 the graph of f is given. Use the graph to compute the quantities asked for. [HINT: See Examples 45.] a. limx3f(x) b. limx1+f(x) c. limx1f(x) d. limx1f(x) e. f(1) f. limx+f(x)41E 42E 43E 44E 45E In Exercises 3548 the graph of f is given. Use the graph to compute the quantities asked for. [HINT: See Examples 45.] a. limx1f(x) b. limx0+f(x) c. limx0f(x) d. limx0f(x) e. f(0) f. limxf(x)In Exercises 3548 the graph of f is given. Use the graph to compute the quantities asked for. [HINT: See Examples 45.] a. limx1f(x) b. limx0+f(x) c. limx0f(x) d. limx0f(x) e. f(0) f. f(1)In Exercises 3548 the graph of f is given. Use the graph to compute the quantities asked for. [HINT: See Examples 45.] a. limx0f(x) b. limx1+f(x) c. limx0f(x) d. limx1f(x) e. f(0) f. f(1)Doctorates in Mexico The annual number of PhD graduates in Mexico in the natural sciences for 19902012 can be approximated by n(t)=890(1e0.05t), where t is time in years since 1990.2 Numerically estimate limx+n(t), and interpret the answer. [HINT: See Example 6.]Housing Starts The number s(t) ofhousing starts for single-family homes in the United States each year from 2006 through 2013 can be approximated by s(t)=500e0.05(x5)1100thousandunits where t is time in years since 2006.3 Numerically estimate limx+s(t), and interpret the answer. [HINT: See Example 6.]51E Funding for NASA up to 1966 (Compare Exercise 51.) The percentage of the U.S. federal budget allocated to NASA from 1958 to 1966 can also be modeled by p(t)=4.51.07(t8)2percentagepoints (t is time in years since 1958). a. Numerically estimate limx+p(t), andinterpret the answer. [HINT: See Example 6.] b. How does your answer to part (a) compare with actual current funding for NASA?Scientific Research: 19832003 The number of research articles per year, in thousands, in the prominent journal Physical Review written by researchers in Europe during 19832003 can be modeled by A(t)=7.01+5.4(1.2)t, where t is time in years (t=0 represents1983).6 Numerically estimate limAt+(t), andinterpret the answer.Scientific Research: 19832003 The percentage of research articles in the prominent journal Physical Review written by researchers in the United States during 19832003 can be modeled by A(t)=361+0.6(0.7)t, where t is time in years (t=0 represents1983).7 Numerically estimate limAt+(t),, and interpret the answer.SAT Scores by Income The following bar graph shows U.S. math SAT scores as a function of household income:8 These data can be modeled by S(x)=573133(0.987)x, where S(x) is the average math SAT score of students whose household income is x thousand dollars per year. Numerically estimate limSx+(x), and interpret the answer.SAT Scores by Income The following bar graph shows U.S. critical reading SAT scores as a function of household income:9 These data can be modeled by S(x)=550136(0.985)x, where S(x) is the average critical reading SAT score of students whose household income is x thousand dollar per year. Numerically estimate limx+S(x), and interpret the answer.Flash Crash The graph shows a rough representation of what happened to the Russell 1000 Growth Index Fund (IWF) stock price on the day of the U.S. stock market crash at 2:45 pm on May 6, 2010, the Flash Crash (t is the time of the day in hours, and r(t) is the price of the stock in dollars).10 a. Compute the following (if a limit does not exist, say why): limt14.75r(t), limt14.75+r(t), limt14.75r(t), r(14.75). b. What do the answers to part (a) tell you about the IWF stock price?Flash Crash The graph shows a rough representation of the (aggregate) market depth11 of the stocks comprising the SP 500 on the day of the U.S. stock market crash at 2:45 pm on May 6, 2010, the Flash Crash (t is the time of the day in hours, and m(t) is the market depth in millions of shares). a. Compute the following (if a limit does not exist, say why): limt14.75m(t), limt14.75+m(t), limt14.75m(t), m(14.75). b. What do the answers to part (a) tell you about the market depth?Home Prices The following graph shows the values of the home price index12 for 20002014 together with a mathematical model I extrapolating the data: Estimate and interpret limt+I(t),Home Prices: Optimist Projection The following graph shows the values of the home price index for 20002014 together with another mathematical model I extrapolating the data: Estimate and interpret limt+I(t).61E 62E 63E 64E 65E Describe the method of evaluating limits graphically. Give at least one disadvantage of this method.67E 68E What is wrong with the following statement? Because f(a) isnot defined, limxaf(x) does not exist. Illustrate your claim with an example.70E 71E 72E 73E 74E In Exercises 114 the graph of a function f is given. Determine whether f is continuous on its domain. If it is not continuous on its domain, say why. [HINT: See Quick Examples 14.]In Exercises 114 the graph of a function f is given. Determine whether f is continuous on its domain. If it is not continuous on its domain, say why. [HINT: See Quick Examples 14.]In Exercises 114 the graph of a function f is given. Determine whether f is continuous on its domain. If it is not continuous on its domain, say why. [HINT: See Quick Examples 14.]In Exercises 114 the graph of a function f is given. Determine whether f is continuous on its domain. If it is not continuous on its domain, say why. [HINT: See Quick Examples 14.]In Exercises 114 the graph of a function f is given. Determine whether f is continuous on its domain. If it is not continuous on its domain, say why. [HINT: See Quick Examples 14.]In Exercises 114 the graph of a function f is given. Determine whether f is continuous on its domain. If it is not continuous on its domain, say why. [HINT: See Quick Examples 14.]In Exercises 114 the graph of a function f is given. Determine whether f is continuous on its domain. If it is not continuous on its domain, say why. [HINT: See Quick Examples 14.]In Exercises 114 the graph of a function f is given. Determine whether f is continuous on its domain. If it is not continuous on its domain, say why. [HINT: See Quick Examples 14.]In Exercises 114 the graph of a function f is given. Determine whether f is continuous on its domain. If it is not continuous on its domain, say why. [HINT: See Quick Examples 14.]In Exercises 114 the graph of a function f is given. Determine whether f is continuous on its domain. If it is not continuous on its domain, say why. [HINT: See Quick Examples 14.]In Exercises 114 the graph of a function f is given. Determine whether f is continuous on its domain. If it is not continuous on its domain, say why. [HINT: See Quick Examples 14.]12E 13E 14E In Exercises 15 and 16, identify which (if any) of the given graphs represent functions that are continuous on their domains. [HINT: See Quick Examples 14.] (A) (B) (C) (D) (E) (F)In Exercises 15 and 16, identify which (if any) of the given graphs represent functions that are continuous on their domains. [HINT: See Quick Examples 14.] (A) (B) (C) (D) (E) (F)17E In Exercises 1724, the graph of a function f is given. Deter- mine whether, at the given point a, f is continuous, discontinuous, or singular. [HINT: See Quick Examples 5 and 6.] a=0 In Exercises 1724, the graph of a function f is given. Deter- mine whether, at the given point a, f is continuous, discontinuous, or singular. [HINT: See Quick Examples 5 and 6.] a=1 In Exercises 1724, the graph of a function f is given. Deter- mine whether, at the given point a, f is continuous, discontinuous, or singular. [HINT: See Quick Examples 5 and 6.] a=0 In Exercises 1724, the graph of a function f is given. Deter- mine whether, at the given point a, f is continuous, discontinuous, or singular. [HINT: See Quick Examples 5 and 6.] a=1 In Exercises 1724, the graph of a function f is given. Deter- mine whether, at the given point a, f is continuous, discontinuous, or singular. [HINT: See Quick Examples 5 and 6.] a=2 In Exercises 1724, the graph of a function f is given. Deter- mine whether, at the given point a, f is continuous, discontinuous, or singular. [HINT: See Quick Examples 5 and 6.] a=1 In Exercises 1724, the graph of a function f is given. Deter- mine whether, at the given point a, f is continuous, discontinuous, or singular. [HINT: See Quick Examples 5 and 6.] a=2 In Exercises 2532, use a graph of f or some other method to determine what, if any, value to assign to f(a) to make f continuous at x=a. [HINT: See Example 2.] f(x)=x22x+1x1;a=1 In Exercises 2532, use a graph of f or some other method to determine what, if any, value to assign to f(a) to make f continuous at x=a. [HINT: See Example 2.] f(x)=x2+3x+2x+1;a=1 In Exercises 2532, use a graph of f or some other method to determine what, if any, value to assign to f(a) to make f continuous at x=a. [HINT: See Example 2.] f(x)=x3x2x;a=0 In Exercises 2532, use a graph of f or some other method to determine what, if any, value to assign to f(a) to make f continuous at x=a. [HINT: See Example 2.] f(x)=x23xx+4;a=4 In Exercises 2532, use a graph of f or some other method to determine what, if any, value to assign to f(a) to make f continuous at x=a. [HINT: See Example 2.] f(x)=33x2x;a=0 In Exercises 2532, use a graph of f or some other method to determine what, if any, value to assign to f(a) to make f continuous at x=a. [HINT: See Example 2.] f(x)=x1x31;a=1 In Exercises 2532, use a graph of f or some other method to determine what, if any, value to assign to f(a) to make f continuous at x=a. [HINT: See Example 2.] f(x)=1exx;a=0 In Exercises 2532, use a graph of f or some other method to determine what, if any, value to assign to f(a) to make f continuous at x=a. [HINT: See Example 2.] f(x)=1+ex1ex;a=0 In Exercises 3342, use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. [HINT: See Example 1.] f(x)=x In Exercises 3342, use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. [HINT: See Example 1.] f(x)=xx 35E 36E In Exercises 3342, use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. [HINT: See Example 1.] f(x)={x+2ifx02x1ifx0 In Exercises 3342, use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. [HINT: See Example 1.] f(x)={1xifx1x1ifx1 In Exercises 3342, use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. [HINT: See Example 1.] f(x)={xxifx00ifx=0 In Exercises 3342, use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. [HINT: See Example 1.] f(x)={1x2ifx00ifx=0 In Exercises 3342, use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. [HINT: See Example 1.] f(x)={x+2ifx02x+2ifx0 In Exercises 3342, use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. [HINT: See Example 1.] f(x)={1xifx1x+1ifx1 Multiple choice: If f is defined on all real numbers and limaxf(x) doesnot exist, then f is (A) singular (B) discontinuous (C) continuous at a.Multiple choice: If f is defined on all real numbers except a and limaxf(x) doesnot exist, then f is (A) singular (B) discontinuous (C) continuous at a.Multiple choice: If f is defined only at a, then f is (A) singular (B) discontinuous (C) continuous at a.Multiple: choice If f is defined everywhere except at a, then f is (A) singular (B) discontinuous (C) continuous at a.If a function is continuous on its domain, is it continuous at every real number? Explain.True or false? The graph of a function that is continuous on its domain is a continuous curve with no breaks in it. Explain your answer.True or false? The graph of a function that is continuous at every real number is a continuous curve with no breaks in it. Explain your answer.50E 51E 52E 53E 54E 55E 56E In Exercises 14, complete the given sentence. The closed-form function f(x)=1x1 iscontinuous for all x expect . [HINT: See Quick Example 3.]In Exercises 14, complete the given sentence. The closed-form function f(x)=1x21 iscontinuous for all x expect . [HINT: See Quick Example 3.]In Exercises 14, complete the given sentence. The closed-form function f(x)=x+1 has x=3 inits domain. Therefore, limx3x+1=. [HINT: See Quick Example 1.]4E In Exercises 520, determine whether the given limit leads to a determinate or indeterminate form. Evaluate the limit if it exists, or say why if not. [HINT: See Example 2 and Quick Examples 814.] limx060x4 In Exercises 520, determine whether the given limit leads to a determinate or indeterminate form. Evaluate the limit if it exists, or say why if not. [HINT: See Example 2 and Quick Examples 814.] limx02x2x2 7E In Exercises 520, determine whether the given limit leads to a determinate or indeterminate form. Evaluate the limit if it exists, or say why if not. [HINT: See Example 2 and Quick Examples 814.] limx02x2 9E 10E 11E In Exercises 520, determine whether the given limit leads to a determinate or indeterminate form. Evaluate the limit if it exists, or say why if not. [HINT: See Example 2 and Quick Examples 814.] limx+60+ex2ex In Exercises 520, determine whether the given limit leads to a determinate or indeterminate form. Evaluate the limit if it exists, or say why if not. [HINT: See Example 2 and Quick Examples 814.] limx0x33x3 14E In Exercises 520, determine whether the given limit leads to a determinate or indeterminate form. Evaluate the limit if it exists, or say why if not. [HINT: See Example 2 and Quick Examples 814.] limxx33x6 16E 17E 18E 19E 20E 21E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx0(2x4) [HINT: See Example 1(a).]23E 24E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx1x+1x In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx4(x+x)In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx8(xx3)In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx1x2x+1 In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limh1(h2+2h+1)In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limh0(h34)In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limh32 In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limh05 In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limh0h2h+h2 [HINT: See Example 1(b).]In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limh0h2+hh2+2h [HINT: See Example 1(b).]In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx1x22x+1x2x In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx1x2+3x+2x2+x In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx2x38x2 38E 39E 40E 41E 42E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx2+x2+8x2+3x+2 44E 45E 46E 47E 48E 49E 50E 51E 52E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx+3x2+10x12x25x [HINT: See Example 4.]54E 55E 56E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx+10x2+300x+15x+2 58E 59E 60E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limxx51,000x42x5+10,000 In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limxx6+3,000x3+1,000,0002x6+1,000x3 In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx10x2+300x+15x+2 In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx2x4+20x31,000x3+6 In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx10x2+300x+15x3+2 66E 67E 68E 69E 70E 71E 72E 73E 74E In Exercises 75-88, identify all singular points of discontinuity of the given function, [HINT: See Example 3.] f(x)=1x3 76E 77E 78E 79E 80E In Exercises 75-88, identify all singular points of discontinuity of the given function, [HINT: See Example 3.] f(x)={x21ifx0x2+1ifx0 In Exercises 75-88, identify all singular points of discontinuity of the given function, [HINT: See Example 3.] f(x)={x1ifx1x1ifx1 In Exercises 75-88, identify all singular points of discontinuity of the given function, [HINT: See Example 3.] g(x)={x+2ifx02x+2if0x2x2+2ifx2 84E In Exercises 75-88, identify all singular points of discontinuity of the given function, [HINT: See Example 3.] h(x)={x+2ifx00ifx=02x+2ifx0 In Exercises 75-88, identify all singular points of discontinuity of the given function, [HINT: See Example 3.] h(x)={1xifx11ifx=1x+2ifx1 87E 88E Processor Speeds The processor speeds, in megahertz (MHz), of Intel processors during the period 19962010 can be approximated by the following function of time t in years since the start of 1990: v(t)={440t2,200if6t153,800if15t20. a. Compute limt15v(t) and limt15+v(t), and interpret each answer. [HINT: See Example 3.] b. Is the function v continuous at t=15? According to the model, was there any abrupt change in processor speeds during the period 19962010?90E 91E Movie Advertising The percentage of movie advertising as a share of newspapers total advertising revenue from 1995 to 2004 can be approximated by p(t)={0.07t+6.0ift40.3t+17.0ift4, Where t is time in years since 1995. a. Compute limt4p(t) and limt4+p(t), and interpret each answer. [HINT: See Example 3.] b. Is the function f continuous at t=4? What does the answer tell you about movie advertising expenditures?93E Law Enforcement in the 1980s and 1990s Refer to Exercise 93. Total spending on police, courts, and prisons in the period 19821999 could be approximated by P(t)=1.745t+29.84billondollars(2t19)C(t)=1.097t+10.65billondollars(2t19),J(t)=1.919t+12.36billondollars(2t19) respectively, where t is time in years since 1980. Compute limt+P(t)P(t)+C(t)+J(t) totwo decimal places, and intercept the result. [HINT: See Example 4.]SAT Scores by Income The following bar graph shows U.S.math SAT scores as a function of household income: These data can be modeled by S(x)=57333e0.0131x where S(x) is the average math SAT score of students whose household income is x thousand dollars per year. Calculate limx+S(x), and interpret the answer.SAT Scores by Income The following bar graph shows U.S. critical reading SAT scores as a function of household income: These data can be modeled by S(x)=550136e0.0151x where S(x) is the average math SAT score of students whose household income is x thousand dollars per year. Calculate limx+S(x), and interpret the answer.97E 98E 99E Acquisition of Language The percentage q(t) ofchildren who can speak in sentences of five or more words by the age of t months can be approximated by the equation q(t)=100(15.271017t12)(t30). If p is the function referred to in the preceding exercise, Calculate limx+[p(t)q(t)], andinterpret the result. [HINT: See Example 4.]101E 102E 103E 104E Why was the following marked wrong? What is the correct answer? limx3x327x3=00undefined WRONG Why was the following marked wrong? What is the correct answer? limx1x1x22x+1=00undefinedWRONG Your friend Karin tells you that f(x)=1/(x2)2 cannotbe a closed-form function because it not continuous at x=2. Comment on her assertion.108E 109E 110E 111E 112E What is wrong with the following statement? If f(x) isspecified algebraically and f(a) is defined, then limxaf(x) existsand equals f(a). How can it be corrected?What is wrong with the following statement? If f(x) isspecified algebraically and f(a) isnot defined, then limxaf(x) does not exist.115E 116E 117E 118E In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] Interval [1, 3] x 0 1 2 3 f(x) 3 5 2 1 2E In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] Interval: [3,1] x 3 2 1 0 f(x) 2.1 0 1.5 0 4E In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] Interval: [2,6] t(months) 2 4 6 R(t)(million) 20.2 24.3 20.1 In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] Interval: [1,3] x(kilos) 1 2 3 C(x)() 2.20 3.30 4.00 In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] Interval: [5,5.5] p() 5.00 5.50 6.00 q(p)(items) 400 300 150 In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] Interval: [0.1,0.2] t(hours) 0 1.0 0.2 D(t)(miles) 0 3 6 In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] Interval: [2,5] [HINT: See Example 2.] Apple Computer Stock Price ($)In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] Interval: [1,5] [HINT: See Example 2.] Cisco System Stock Price ($)In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] Interval: [0,4] Unemployment (%) Budget deficit (% of GNP)In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] Interval: [0,4] Inflation (%) Beget deficit (% of GNP)In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] f(x)=x23;[1,3] [HINT: See Example 3.]In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] f(x)=2x2+4;[1,2] [HINT: See Example 3.]In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] f(x)=2x+4;[2,0]In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] f(x)=1x;[1,4]In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] f(x)=x22+1x;[2,3]In Exercises 118, calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. [HINT: See Example 1.] f(x)=3x2x2;[3,4]In Exercises 1924, calculate the average rate of change of the given function f over the intervals [a,a+h], where h=1,0.1,0.01,0.001,and0.0001. (Technology is recommended for the cases h=0.01,0.001,and0.0001.) [HINT: See Example 4.] f(x)=2x2;a=0 20E In Exercises 1924, calculate the average rate of change of the given function f over the intervals [a,a+h], where h=1,0.1,0.01,0.001,and0.0001. (Technology is recommended for the cases h=0.01,0.001,and0.0001.) [HINT: See Example 4.] f(x)=1x;a=2 22E 23E 24E World Military Expenditure The following table shows total military and arms trade expenditure in 2000, 2005, and 2010:32 Yeart(yearsince2000) 0 6 12 MilitaryExpenditureC(t)(billion) 1,100 1,450 1,750 Compute and interpret the average rate of change of C(t) (a) over the period 20062012 (that is, [6,12] ) and (b) over the period [0,12]. Be sure to state the units of measurement. [HINT: See Example 1.]Education Expenditure The following table shows education expenditure in the United States as a percentage of total federal spending in 2009, 2015, and 2019:33 Yeart(yearsince2000) 9 15 19 PercentageP(t) 25 27 26 Compute and interpret the average rate of change of P(t) (a) over the period 20092019 (that is, [15,19] ) and (b) over the period [15,19]. Be sure to state the units of measurement.Crude Oil Production: Mexico The following table shows daily crude oil production by Pemex, Mexicos national oil company, for 20082013:34 Yeart(yearsince2009) 0 1 2 3 4 5 DailyProductionp(t) 3.16 2.97 2.95 2.94 2.91 2.92 a. Compute the average rate of change of p(t) over the period 20102013. Interpret the result. [HINT: See Example 1.] b. Which of the following is true? From 2008 to 2013 the three-year average rate of change of oil production by Pemex (A) increased in value. (B) decreased in value. (C) increased then decreased in value. (D) decreased then increased in value. [HINT: See Example 2.]Offshore Crude Oil Production: Mexico The following table shows daily offshore crude oil production by Pemex, Mexicos national oil company, for 20082013:35 Yeart(yearsince2008) 0 1 2 3 4 5 DailyOffshoreProductions(t)(millionbarrels) 2.25 2.01 1.94 1.90 1.90 1.90 a. Use the data in the table to compute the average rate of change of s(t) over the period 20082013. Interpret the result. b. Which of the following is true? From 2008 to 2013 the two-year average rate of change of offshore crude oil production of Pemex (A) increased in value. (B) decreased in value. (C) increased then decreased in value. (D) decreased then increased in value.Subprime Mortgages during the Housing Crisis The following graph shows the approximate percentage P(t) ofmortgages issued in the United States that were subprime (normally classified as risky):36 Subprime mortgages a. Use the graph to estimate, to one decimal place, the average rate of change of P(t) withrespect to t over the interval 30, 64, and interpret the result b. Over which 2-year period(s) was the average rate of change of P(t) thegreatest? [HINT: See Example 2.]Subprime Mortgage Debt during the Housing Crisis The following graph shows the approximate value V(t) ofsubprime (normally classified as risky) mortgage debt outstanding in the United States:37 Subprime debt outstanding Year (t) a. Use the graph to estimate, to one decimal place, the average rate of change of V(t) withrespect to t over the interval 2,6, and interpret the result. b. Over which 2-year period(s) was the average rate of change of V(t) theleast? [HINT: See Example 2.]Immigration to Ireland The following graph shows the approximate number (in thousands) of people who immigrated to Ireland during the period 20102014 (t is time in years since 2010):32 During which 2-year interval(s) was the magnitude of the average rate of change of I(t) (a) greatest (b) least? Interpret your answers by referring to the rates of change.Emigration from Ireland The following graph shows the approximate number (in thousands) of people who emigrated from Ireland during the period 20102014 (t is time in years since 2010): During which 2-year interval(s) was the magnitude of the average rate of change of E(t) (a) greatest (b) least? Interpret your answers by referring to the rates of change.Science Research in the United States The following table shows the number of science research articles authored by U.S researchers during the period 19802010: yeart(yearsince1980) 0 5 10 15 20 25 30 ArticalesN(t)(thousands) 170 200 220 260 252 290 340 a. Find the interval(s) over which the average rate of change of N was the greatest. What was that rate of change? Interpret your answer. b. The percentage change of N over the interval [a,b] is defined to be Percentage change of N=ChangeinNFirstvalue=N(b)N(a)N(a) Compute the percentage change of N over the interval [0,30] and also the average rate of change. Interpret the answers.Science Research in Europe The following table shows the number of science research articles authored by researchers in the European Union during the period 1980-2010: yeart(yearsince1980) 0 5 10 15 20 25 30 ArticalesN(t)(thousands) 140 170 190 260 300 340 430 a. Find the interval(s) over which the average rate of change of N was the least positive. What was that rate of change? Interpret your answer. b. The percentage change of N over the interval [a,b] is defined to be Percentage change of N=ChangeinNFirstvalue=N(b)N(a)N(a) Compute the percentage change of N over the interval [10,30] and also the average rate of change. Interpret the answers.College Basketball: Men The following chart shows the number of NCAA mens college basketball teams in the United States during the period 20002010: Mens basketball teams Year (t) a. On average, how fast was the number of mens college basketball teams growing over the 4-year period beginning in 2002? b. By inspecting the chart, determine whether the 3-year average rates of change increased or decreased beginning in 2005. [HINT: See Example 2.]College Basketball: Women The following chart shows the number of NCAA womens college basketball teams in the United States during the period 20002010:43 Mens basketball teams Year (t) a. On average, how fast was the number of womens college basketball teams growing over the 4-year period beginning in 2004? b. By inspecting the graph, find the 3-year period over which the average rate of change was largest.Funding for the Arts State governments in the United States spend between $1 and $2 per person on the arts and culture each year. The following chart shows the data for 20022010, together with the regression line:44 State government funding for the arts Year a. Over the period [2,6] the average rate of change of state government funding for the arts was (A) less than (B) greater than (C) approximately equal to the rate predicted by the regression line. b. Over the period [3,10] the average rate of change of state government funding for the arts was (A) less than (B) greater than (C) approximately equal to the rate predicted by the regression line. c. Over the period [4,8] the average rate of change of state government funding for the arts was (A) less than (B) greater than (C) approximately equal to the rate predicated by the regression line. b. Estimate, to two significant digits, the average rate of change of per capita state government funding for the arts over the period [2,10]. (Be care full to state the unit of measurement.) How does it compare to the slope of the regression line?Funding for the Arts The U.S. federal government spends between $6 and $7 per person on the arts and culture each year. The following chart shows the data for 20022010, together with the regression line: Federal funding for the arts Year a. Over the period [4,10] the average rate of change of federal government funding for the arts was (A) less than (B) greater than (C) approximately equal to the rate predicted by the regression line. b. Over the period [2,7] the average rate of change of federal government funding for the arts was (A) less than (B) greater than (C) approximately equal to the rate predicted by the regression line. c. Over the period [3,10] the average rate of change of federal government funding for the arts was (A) less than (B) greater than (C) approximately equal to the rate predicted by the regression line. d. Estimate, to one significant digit, the average rate of change of per capita federal government funding for the arts over the period [2,10]. (Be careful to state the units of measurement.) How does it compare to the slope of the regression line?Market Index Joe Downs runs a small investment company from his basement. Every week, he publishes a report on the success of his investments, including the progress of the Joe Downs Index. At the end of one particularly memorable week, he reported that the index for that week had the value I(t)=1,000+1,500t800t2+100t3 points, where t represents the number of business days into the week; t ranges from 0 at the beginning of the week to 5 at the end of the week. The graph of I is shown below: I(Joe Down Index) On average, how fast and in which direction was the index changing over the first two business days (the interval [0,2] )?[HINT: See Example 3.]42E Crude Oil Prices The price per barrel of crude oil in constant 2008 dollars can be approximated by P(t)=0.45t212t+105dollar(0t28), Where t is time in years since the start of 1980.48 a. What, in constant 2008 dollars, was the average rate of change of the price of oil from the start 1980(t=1) to the start of 2006t=26? [HINT: See Example 3.] b. Your answer to part (a) is quite small. Can you conclude that the price of oil hardly changed all over the 25-year period 1981-2006? Explain.Median Home Prices The median home price in the United States over the period January 2010January 2015 can be approximated by P(t)=4.5t215t+180thousanddollar(0t5), where t is time in years since the start of 2010. a. What was the average rate of change of the median home price from the start of 2012 to the start of 2014? b. What, if anything, does your answer to part (a) say about the median home price in 2013? Explain.End of the Earth In 5 billion year the Sun will have run out of hydrogen fuel and begin to expand into a red giant, eventually engulfing the Earth and causing it to spiral into the core of the Sun 7.5 billion years from now. The following graph50 shows the expanding radius of the red giant Sun (in red) and the radius of the Earths orbit about the Sun (in green) during its final three and a half million years of existence. The radii are measured in astronomical units (AU; one AU is the current radius of the Earths orbit around the Sun, approximately 93 million miles), and time is measured in millions of years. r (AU) The curve representing the Suns radius has equation r=0.037t2+0.02t+0.4. (t=4 marks the end of the red giant expansion phase.) Exercises 45and 46are based on this curve. a. Calculate the rate of change of the radius of the Sun over the successive intervals [0,1],[1,2],[2,3],[3,4]. b. The successive rates of change are a linear function of t. What is the slope of that linear function? How fast will the rate of change of the Suns radius be increasing in the final 4 million years?46E The 2003 SARS Outbreak In the early stages of the SARS (severe acute respiratory syndrome) epidemic in 2003 the number of reported cases could be approximated by A(t)=167(1.18)t(0t20) t days after March 17, 2003 (the first day for which statistics were reported by the World Health Organization). a. What was the average rate of change of A(t) fromMarch 17 to March 23? Interpret the result. b. Which of the following is true? For the first 20 days of the epidemic, the number of reported cases (A) increased at a faster and faster rate. (B) increased at a slower and slower rate. (C) decreased at a faster and faster rate. (D) decreased at a slower and slower rate. [HINT: See Example 2.]48E 49E The 2014 Ebola Outbreak Repeat Exercise 49 using the following model for the total number of reported deaths from Ebola: D(t)=90.52e0.60t(0t6), where t is time in months since April 1, 2014.52 Ecology Increasing numbers of manatees (sea sirens) have been killed by boats off the Florida coast. The following graph shows the relationship between the number of boats registered in Florida and the number of manatees killed each year: Boats (100,000) The regression curve shown is given by f(x)=3.55x230.2x+81menatees(4.5x8.5), where x is the number of boats (in hundreds of thou- sands) registered in Florida in a particular year and f(x) is the number of manatees killed by boats in Florida that year. a. Compute the average rate of change of f over the intervals [5,6]and[7,8]. b. What does the answer to part (a) tell you about the manatee deaths per boat?52E 53E 54E Describe three ways we have used to determine the average rate of change of f over an interval [a,b]. Which of the three ways is least precise? Explain.If f is a linear function of x with slope m, what is its average rate of change over any interval [a,b]?Is the average rate of change of a function over [a,b] affectedby the values of the function between a and b? Explain.If the average rate of change of a function over [a,b] iszero, this means that the function is constant over that interval-right?Sketch the graph of a function whose average rate of change over [0,2] isnegative but whose average rate of change over [1,3] is positive.Sketch the graph of a function whose average rate of change over [0,2] is positive but whose average rate of change over [0,1] is negative.61E 62E 63E A certain function f has the property that its average rate of change over the interval [1,1+h] (for positive h) increases as h decreases. Which of the following graphs could be the graph of f? (A) (B) (C)65E 66E 67E 68E In Exercises 14, estimate the derivative from the table of average rates of change. [HINT: See discussion at the beginning of the section.] Estimate f(5). h 1 0.1 0.01 0.001 0.0001 Avg.RateofChangeoffover[5,5+h] 12 6.4 6.04 6.004 6.0004 h 1 0.1 0.01 0.001 0.0001 Avg.RateofChangeoffover[5+h,5] 3 5.6 5.96 5.996 5.9996 In Exercises 14, estimate the derivative from the table of average rates of change. [HINT: See discussion at the beginning of the section.] 1. Estimate f(5). Estimate g(7). h 1 0.1 0.01 0.001 0.0001 Avg.RateofChangeofgover[7,7+h] 4 4.8 4.98 4.998 4.9998 h 1 0.1 0.01 0.001 0.0001 Avg.RateofChangeofgover[7+h,7] 5 5.3 5.03 5.003 5.0003 In Exercises 14, estimate the derivative from the table of average rates of change. [HINT: See discussion at the beginning of the section.] Estimate r(6). h 1 0.1 0.01 0.001 0.0001 Avg.RateofChangeofrover[6,6+h] 5.4 5.498 5.4998 5.499982 5.49999822 h 1 0.1 0.01 0.001 0.0001 Avg.RateofChangeofrover[6+h,6] 7.52 6.13 5.5014 5.5000144 5.500001444 In Exercises 14, estimate the derivative from the table of average rates of change. [HINT: See discussion at the beginning of the section.] Estimate s(0). h 1 0.1 0.01 0.001 0.0001 Avg.RateofChangeofsover[0,h] 2.52 5.498 5.4998 5.499982 5.49999822 h 1 0.1 0.01 0.001 0.0001 Avg.RateofChangeofsover[h,0] 0.4 0.598 0.5998 0.599982 0.59999822 Consider the functions in Exercises 58 as representing the value of an ounce of palladium in U.S. dollars as a function of the time t in days. Find the average rates of change of R(t) over the time intervals [t,t+h], where t is as indicated and h=0,0.1, and 0.01 days. Hence, estimate the instantaneous rate of change of R at time t, specifying the units of measurement. (Use smaller values of h to check your estimates.) [HINT: See Example 1.] R(t)=60+50tt2;t=5 Consider the functions in Exercises 58 as representing the value of an ounce of palladium in U.S. dollars as a function of the time t in days. Find the average rates of change of R(t) over the time intervals [t,t+h], where t is as indicated and h=0,0.1, and 0.01 days. Hence, estimate the instantaneous rate of change of R at time t, specifying the units of measurement. (Use smaller values of h to check your estimates.) [HINT: See Example 1.] R(t)=60t2t2;t=3 Consider the functions in Exercises 58 as representing the value of an ounce of palladium in U.S. dollars as a function of the time t in days. Find the average rates of change of R(t) over the time intervals [t,t+h], where t is as indicated and h=0,0.1, and 0.01 days. Hence, estimate the instantaneous rate of change of R at time t, specifying the units of measurement. (Use smaller values of h to check your estimates.) [HINT: See Example 1.] R(t)=270+20t3;t=1 Consider the functions in Exercises 58 as representing the value of an ounce of palladium in U.S. dollars as a function of the time t in days. Find the average rates of change of R(t) over the time intervals [t,t+h], where t is as indicated and h=0,0.1, and 0.01 days. Hence, estimate the instantaneous rate of change of R at time t, specifying the units of measurement. (Use smaller values of h to check your estimates.) [HINT: See Example 1.] R(t)=200+50tt3;t=2 In Exercises 912 the function gives the cost to manufacture x items. Find the average cost per unit of manufacturing h more items (i.e., the average rate of change of the total cost) at a production lev h=0 el of x, where x is as indicated and and 1. Hence, estimate the instantaneous rate of change of the total cost at the given production level x, specifying the units of measurement. (Use smaller values of h to check your estimates.) [HINT: See Example 1.] C(x)=10,000+5xx210,000;x=1,000 In Exercises 912 the function gives the cost to manufacture x items. Find the average cost per unit of manufacturing h more items (i.e., the average rate of change of the total cost) at a production level of x, where x is as indicated and h=0 and 1. Hence, estimate the instantaneous rate of change of the total cost at the given production level x, specifying the units of measurement. (Use smaller values of h to check your estimates.) [HINT: See Example 1.] C(x)=20,000+7xx220,000;x=10,000 In Exercises 912 the function gives the cost to manufacture x items. Find the average cost per unit of manufacturing h more items (i.e., the average rate of change of the total cost) at a production level of x, where x is as indicated and h=0 and 1. Hence, estimate the instantaneous rate of change of the total cost at the given production level x, specifying the units of measurement. (Use smaller values of h to check your estimates.) [HINT: See Example 1.] C(x)=15,000+100x+1,000x;x=100 In Exercises 912 the function gives the cost to manufacture x items. Find the average cost per unit of manufacturing h more items (i.e., the average rate of change of the total cost) at a production level of x, where x is as indicated and h=0 and 1. Hence, estimate the instantaneous rate of change of the total cost at the given production level x, specifying the units of measurement. (Use smaller values of h to check your estimates.) [HINT: See Example 1.] C(x)=20,000+50x+10,000x;x=100 In Exercises 1316 the graph of a function is shown together with the tangent line at a point P. Estimate the derivative of f at the corresponding x value. [HINT: See Quick Example 3.]In Exercises 1316 the graph of a function is shown together with the tangent line at a point P. Estimate the derivative of f at the corresponding x value. [HINT: See Quick Example 3.]In Exercises 1316 the graph of a function is shown together with the tangent line at a point P. Estimate the derivative of f at the corresponding x value. [HINT: See Quick Example 3.]In Exercises 1316 the graph of a function is shown together with the tangent line at a point P. Estimate the derivative of f at the corresponding x value. [HINT: See Quick Example 3.]In Exercises 1722, say at which labeled point the slope of the tangent is (a) greatest and (b) least (in the sense that 7 is less than 1). [HINT: See Quick Example 3.]18E 19E In Exercises 1722, say at which labeled point the slope of the tangent is (a) greatest and (b) least (in the sense that 7 is less than 1). [HINT: See Quick Example 3.]In Exercises 1722, say at which labeled point the slope of the tangent is (a) greatest and (b) least (in the sense that 7 is less than 1). [HINT: See Quick Example 3.]In Exercises 1722, say at which labeled point the slope of the tangent is (a) greatest and (b) least (in the sense that 7 is less than 1). [HINT: See Quick Example 3.]In each of Exercises 2326, three slopes are given. For each slope, determine at which of the labeled points on the graph the tangent line has that slope. a. 0 b. 4 c. 1

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