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55-56 The velocity function (in meters per second) is givenfor a particle moving along a line. Find (a) the displacement and(b) the distance traveled by the particle during the given timeinterval.57-58 The acceleration function (in m/s 2 ) and the initial velocityare given for a particle moving along a line. Find (a) the velocity attime t and (b) the distance traveled during the given time interval.[g] aCt) = t + 4, v(O) = 5, 0,,; t"; to59. The linear density of a rod of length 4 m is given byp(x) = 9 + 2.[; measured in kilograms per meter, wherex is measured in meters from one end of the rod. Find the totalmass of the rod.60. Water flows from the bottom of a storage tank at a rate ofr(t) = 200 - 4t liters per minute, where 0,,; t ,,; 50. Findthe amount of water that flows from the tank during the first10 minutes.61. The velocity of a car was read from its speedometer at10-second intervals and recorded in the table. Use theMidpoint Rule to estimate the distance traveled by the car.t (5) v (km/h) t (5) V (km/h)0 0 60 9010 61 70 8520 83 80 8030 93 90 7540 88 100 7250 82meter). Find the increase in cost if the productionraised from 2000 meters to 4000 meters.level is64. Water flows into and out of a storage tank. A graph of the rateof change r(t) of the volume of water in the tank, in liters perday, is shown. If the amount of water in the tank at time t = 0is 25,000 L, use the Midpoint Rule to estimate the amount ofwater four days later.r2000/" -I'-..\0 1 2 3\ 4 t-100065. Economists use a cumulative distribution called a Lorenz curveto describe the distribution of income between households in agiven country. Typically, a Lorenz curve is defined on [0, lJwith endpoints (0, 0) and (I, 1), and is continuous, increasing,and concave upward. The points on this curve are determinedby ranking all households by income and then computing thepercentage of households whose income is less than or equalto a given percentage of the total income of the country. Forexample, the point (a/l 00, b/lOO) is on the Lorenz curve if thebottom a% of the households receive less than or equal to b%of the total income. Absolute equality of income distributionwould occur if the bottom a% of the households receive a% ofthe income, in which case the Lorenz curve would be the liney = x. The area between the Lorenz curve and the line y = xmeasures how much the income distribution differs fromabsolute equality. The coefficient of inequality is the ratio ofthe area between the Lorenz curve and the line y = x to thearea under y = x.f\..'162. Suppose that a volcano is erupting and readings of the rate r(t)at which solid materials are spewed into the atmosphere aregiven in the table. The time t is measured in seconds and theunits for r(t) are tonnes per second.t 0 1 2 3 4 5 6r(t) 2 10 24 36 46 54 60(a) Give upper and lower estimates for the total quantity Q(6)of erupted materials after 6 seconds.(b) Use the Midpoint Rule to estimate Q(6).63. The marginal cost of manufacturing x meters of a certainfabric is C'(x) = 3 - O.Olx + 0.000006x 2 (in dollars per(a) Show that the coefficient of inequality is twice the areabetween the Lorenz curve and the line y = x, that is, showthatcoefficient of inequality = 2 Sol [x - L(x)J dx(b) The income distribution for a certain country is representedby the Lorenz curve defined by the equationWhat is the percentageL(x) = f2x 2 + fixof total income received by the
ottom 50% of the households? Find the coefficient ofinequality.~ 66. On May 7, 1992, the space shuttle Endeavour was launchedon mission STS-49, the purpose of which was to install a newperigee kick motor in an Intelsat communications satellite.The table gives the velocity data for the shuttle betweenliftoff and the jettisoning of the solid rocket boosters.Event Time (s) Velocity (m/s)Launch 0 0Begin roll maneuver 10 56.4End roll maneuver 15 97.2Throttle to 89% 20 136.2Throttle to 67% 32 226.2Throttle to 104% 59 403.9Maximum dynamic pressure 62 440.4Solid rocket booster separation 125 1265.2The following exercises are intended only for those who havealready covered Chapter 7.67. f (sin x + sinh x) dx1//3 t2 - I71. f. -4--dtD t - 168. flD 2e x dx-10 sinh x + cosh x70. f2 (x - 1)3 dx1 x 2lIb] The area labeled B is three times the area labeled A. Expressb in terms of a.(a) Use a graphing calculator or computer to model thesedata by a third-degree polynomial.(b) Use the model in part (a) to estimate the height reachedby the Endeavour, 125 seconds after liftoff.WRITINGPROJECTNEWTON, LEIBNIZ, AND THE INVENTION OF CALCULUSWe sometimes read that the inventors of calculus were Sir Isaac Newton (1642-1727) andGottfried Wilhelm Leibniz (1646-1716). But we know that the basic ideas behind integrationwere investigated 2500 years ago by ancient Greeks such as Eudoxus and Archimedes, andmethods for finding tangents were pioneered by Pierre Fermat (1601-1665), Isaac Barrow(1630-1677), and others. Barrow-who taught at Cambridge and was a major influence onNewton-was the first to understand the inverse relationship between differentiation and integration.What Newton and Leibniz did was to use this relationship, in the form of the FundamentalTheorem of Calculus, in order to develop calculus into a systematic mathematical discipline. Itis in this sense that Newton and Leibniz are credited with the invention of calculus.Read about the contributions of these men in one or more of the given references and write areport on one of the following three topics. You can include biographical details, but the mainthrust of your report should be a description, in some detail, of their methods and notations. Inparticular, you should consult one of the sourcebooks, which give excerpts from the originalpublications of Newton and Leibniz, translated from Latin to English.• The Role of Newton in the Development• The Role of Leibniz in the Developmentof Calculusof Calculus• The Controversy between the Followers of Newton and Leibniz overPriority in the Invention of CalculusI. Carl Boyer and Uta Merzbach, A History of Mathematics (New York: Wiley, 1987),Chapter 19.
Page 2 and 3: 12. Speedometer readings for a moto
Page 4 and 5: § EXERCISESI. Evaluate the Riemann
Page 6: f2 f(x) dx + S: f(x) dx - [1 f(x) d
Page 9 and 10: 57-58 Evaluate the limit by first r
Page 11: 13. J (fJ - csc fJ cot fJ) dfJ 14.
Page 15 and 16: 5. f ( 4 )3 dx,1 + 2xfsec2(1lx)6. 2
Page 17 and 18: Determine whether the statement is
Page 19 and 20: 45. Let r(t) be the rate at which t

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