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

42E 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.44E 45E Multiple: choice If f is defined everywhere except at a, then f is (A) singular (B) discontinuous (C) continuous at a.47E 48E 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.]3E 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 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.] limx0x31x3 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 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(x2+5)10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx14x2+1x In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx1x+1x 26E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx8(xx3)28E 29E 30E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limh32 32E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limh0h2h+h2 [HINT: See Example 1(b).]34E 35E 36E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx2x38x2 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E 53E 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 62E 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 In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx2x4+20x31,000x6+6 In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx+(4e3x+12)68E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx+255.3(33t)70E In Exercises 21-74, calculate the limit algebraically. If the limit does not exit, say why, limx+23t1+5.3et 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 82E 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 85E 86E 87E 88E 89E 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?Law Enforcement in the 1980s and 1990s The cost of fighting crime in the United States increased significantly during the period 19821999. Total spending on police and courts can be approximated by P(t)=1.745t+29.84billondollars(2t19)C(t)=1.097t+10.65billondollars(2t19), respectively, where t is time in years since 1980. Compute limt+P(t)C(t) totwo decimal places, and intercept the result. [HINT: See Example 4.]94E 95E 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 100E 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 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, 2] x 0 1 2 3 f(x) 1 3 2 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: [3,1] x 3 2 1 0 f(x) 2.1 0 1.5 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.] Interval: [1,1] x 2 1 0 1 f(x) 1.5 0.5 4 6.5 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 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)=x22;a=1 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.34E 35E 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.45E 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 The 2014 Ebola Outbreak In the first 6 months of the 2014 Ebola outbreak, the total number of reported cases could be approximated by C(t)=95.0e0.72t(0t6) Where t is time in months since April 1, 2014.51 a. Calculating the average rate of change of C(t) over the successive 2-mont periods [0,2],[1,3],[2,4],[3,5], and [4,6]. (Round answers to two decimal places.) b. What kind of model would best describe the successive rates of change obtained in part (a): linear, quadratic, or exponential? What does your answer tell you about the 2014 Ebola outbreak?50E 51E 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.56E 57E 58E 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.60E 61E 62E 63E 64E 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 4E 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 7E 8E 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 11E 12E 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.]16E 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.]21E 22E 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 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. 1 c. 1 25E 26E In Exercises 2730, find the approximate coordinates of all points (if any) where the slope of the tangent is (a) 0, (b) 1, and (c) 1. [HINT: See Quick Example 3.]28E 29E 30E 31E 32E 33E Which is correct? The derivative function f(x) tellsus (A) the slope of the tangent line at each of the points (x,f(x)). (B) the approximate slope of the tangent line at each of the points (x,f(x)). (C) the slope of the secant line through (x,f(x)) and (x+h,f(x+h)) for h=0.0001. (D) the slope of a certain secant line through each of the points (x,f(x)).Let f have the graph shown. a. The average rate of change of f over [2,4] is (A) greater than f(2). (B) less than f(2). (C) approximately equal to f(2). b. The average rate of change of f over [1,1] is (A) greater than f(0). (B) less than f(0). (C) approximately equal to f(0). c. Over the interval [0,2] theinstantaneous rate of change of f is (A) increasing. (B) decreasing. (C) neither. d. Over the interval [0,200] theinstantaneous rate of change of f is (A) increasing, then decreasing. (B) decreasing, then increasing. (C) always increasing. (D) always decreasing. e. when x=4,f(x) is (A) approximately 0 and increasing at a rate of about 0.7 units per unit of x. (B) approximately 0 and decreasing at a rate of about 0.7 units per unit of x. (C) approximately 0.7 and increasing at a rate of about 1 units per unit of x. (D) approximately 0.7 and decreasing at a rate of about 3 units per unit of x.36E 37E In Exercises 3740, use a quick approximation to estimate the derivative of the given function at the indicated point. [HINT: See Example 2(a).] f(x)=x31;x=3 In Exercises 3740, use a quick approximation to estimate the derivative of the given function at the indicated point. [HINT: See Example 2(a).] f(x)=x24x33;x=1 40E In Exercises 41-48, estimate the indicated derivative by any method. [HINT: See Example 2.] g(t)=1t5;estimateg(1)42E In Exercises 41-48, estimate the indicated derivative by any method. [HINT: See Example 2.] y=4x2;estimatedydx|x=2 In Exercises 41-48, estimate the indicated derivative by any method. [HINT: See Example 2.] y=1x2;estimatedydx|x=1 45E 46E In Exercises 41-48, estimate the indicated derivative by any method. [HINT: See Example 2.] R=1P;estimatedRdp|p=20 48E In Exercises 4954, (a) use any method to estimate the slope of the tangent to the graph of the given function at the point with the given x-coordinate, and (b) find an equation of the tangent line in part (a). In each case, sketch the curve together with the appropriate tangent line. [HINT: See Example 2(b).] f(x)=x3;x=1 50E 51E 52E In Exercises 4954, (a) use any method to estimate the slope of the tangent to the graph of the given function at the point with the given x-coordinate, and (b) find an equation of the tangent line in part (a). In each case, sketch the curve together with the appropriate tangent line. [HINT: See Example 2(b).] f(x)=x;x=4 54E In Exercises 55-58, estimate the given quantity. f(x)=ex;estimatef(0)56E 57E 58E In Exercises 5964, match the graph of f to the graph of f. (The graphs of f are shown after Exercise 64.) Graph of derivatives for Exercises 59-64: (A) (B) (C) (D) (E) (F)In Exercises 5964, match the graph of f to the graph of f. (The graphs of f are shown after Exercise 64.) Graph of derivatives for Exercises 59-64: (A) (B) (C) (D) (E) (F)In Exercises 5964, match the graph of f to the graph of f. (The graphs of f are shown after Exercise 64.) Graph of derivatives for Exercises 59-64: (A) (B) (C) (D) (E) (F)In Exercises 5964, match the graph of f to the graph of f. (The graphs of f are shown after Exercise 64.) Graph of derivatives for Exercises 59-64: (A) (B) (C) (D) (E) (F)In Exercises 5964, match the graph of f to the graph of f. (The graphs of f are shown after Exercise 64.) Graph of derivatives for Exercises 59-64: (A) (B) (C) (D) (E) (F)In Exercises 5964, match the graph of f to the graph of f. (The graphs of f are shown after Exercise 64.) Graph of derivatives for Exercises 59-64: (A) (B) (C) (D) (E) (F)In Exercises 6568 the graph of a function is given. For which x in the range shown is the function increasing? For which x is the function decreasing? [HINT: See Quick Example 6.]66E 67E 68E 69E 70E In Exercises 6972 the graph of the derivative of a function is given. For which x is the (original) function increasing? For which x is the (original) function decreasing? [HINT: See Quick Example 6.]72E 73E 74E 75E 76E Temperatures on Mars The air temperature one chilly spring morning at your time share condominium at the base of Olympus Mons, t hours after 6:00 am, was given by the function f(t)=5t2+5080 degreesFahrenheit (0t4).58 What was the temperature at 7:00 am, and how fast was it rising? (Use the method of Example 1(a)78E Demand Suppose the demand for a new brand of sneakers is given by q=5,000,000p, where p is the price per pair of sneakers in dollars and q is the number of pairs of sneakers that can be sold at price p. Find q(100), and estimate q(100). Interpret your answers. [HINT: See Example 1.]Demand Suppose the demand for an old brand of TV is given by q=100,000p+10, where p is the price per TV set in dollars and q is the number of TV sets that can be sold at price p. Find q(190), and estimate q(190). Interpret your answers. [HINT: See Example 1.]Oil Imports from Mexico The following graph shows approximate daily oil imports to the United States from Mexico.60 Also shown is the tangent line at the point corresponding to year 2011. a. Estimate the slope of the tangent line shown on the graph. What does the graph tell you about oil imports from Mexico in 2011? [HINT: Identify two points on the tangent line. Then see Quick Example 3.] b. According to the graph, is the rate of change of oil imports from Mexico increasing, decreasing, or increasing then decreasing? Why?Oil Production in Mexico The following graph shows approximate daily oil production by Pemex, Mexicos national oil company61. Also shown is the tangent line at the point corresponding to year 2010. Time (year since 2000) a. Estimate the slope of the tangent line shown on the graph. What does the graph tell you about oil production by Pemex in 2010? [HINT: Identify two points on the tangent line. Then see Quick Example 3.] b. According to the graph, is the rate of change of oil production by Pemex increasing or decreasing over the range [8,11]? Why?83E 84E 85E 86E Crude Oil Prices The price per barrel of crude oil in constant 2008 dollars can be approximated by P(t)=0.45t212t+105dollars(0t28), where t is time in years since the start of 1980. a. Compute the average rate of change of P(t) over the interval [0,28], and interpret your answer. [HINT: See Example 3 of section 10.4.] b. Estimate the instantaneous rate of change of P(t) at t=0 andinterpret your answer. [HINT: See Example 2 (a).] c. The answer to part (a) and part (b) have opposite signs. What does this indicate about the price of oil?Median Home Prices The median home price in the United States over the period January 2010January 2015 can be approximated by P(t)=0.4.5t215t+180thousanddollars(0t5), where t is time in years since the start of 2010.66 a. Compute the average rate of change of P(t) over the interval [1,5], and interpret your answer. [HINT: See Example 3 of section 10.4.] b. Estimate the instantaneous rate of change of P(t) at t=1 andinterpret your answer. [HINT: See Example 2 (a).] c. The answer to part (a) and part (b) have opposite signs. What does this indicate about the price?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 in which statistics were reported by the World Health Organization.) a. What, approximately, was the instantaneous rate of change of A(t)onMarch27(t=10)? Interpret the result. b. Which of the following is true? For the first 20 days of the epidemic, the instantaneous rate of change of the number of cases (A) increased. (B) decreased. (C) increased and then decreased. (D) decreased and then increased.90E 91E 92E 93E 94E 95E 96E 97E 98E 99E 100E 101E

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