[托马斯微积分.第11版.(附带习题答案)].review1

Math222Spring2009Exam1ReviewGetintothehabitofwritingupyoursolutionscarefullyandclearlyandexplainingyourreasoningbecauseontheexam,that'showyou'llbegraded.Correctanswerswithnocoherentjusticationwillbegivennocredit.Theformulasonthelastpageofthisreviewsheetwillalsobegiventoyouontheexam.1.Evaluatetheintegral,orshowthatitdoesn'texist.Ifitdoesexist,simplifyyouranswerasmuchaspossible.(a)Zt3t4+4dt(b)Zcos2xdx(c)Z1(x2)2(x1)dx(d)Ztanx(lncosx)2dx(e)Zp320arctan2tdt(f)Z30dxx1(g)Zx(x3)2dx(h)Zln(ln3)ln(ln2)eexexdx(i)Z1+4xp1+x+2x2dx(j)4Z01x24x+8dx(k)Z10px1+pxdx(l)Z8011+3pxdx(m)Z20sin2x2+cosxdx(n)Z331x(x+1)dx(o)Z41dxx24(p)Zsin5xcos7xdx(q)Zsin(lnx)dx(r)Z10x21+x2dx(s)Z81x13x43+1dx(t)Zxtan22xdx(u)Z111ex+exdx(v)Z411x25x+6dx(w)Zdxx(lnx)2(x)2Z0sinx1+cosxdx(y)Zsin(e2x)e2xdx(z)4Z1epxpx(1+epx)dxMath222|Exam1Review22.Determinewhetherthefollowingintegralsconvergeordiverge.(a)Z11dx5+x5(b)Z1eeln(lnx)dx(c)Z11ex2dx3.(a)Whathappenswhenyoutrytotest1R11x22dxforconvergencebyusingtheDirectComparisonTestwithg(x)=1x2?(b)Determinetheconvergenceof1R11x22dxusingtheLimitComparisonTestwithg(x)=1x2.4.Solvethedierentialequationorinitialvalue(orboundaryvalue)problem.(a)y0+2xy=xex2(b)dydx=y2x(x2+2)4;y(0)=1(c)y0=xxy;y(0)=6(d)2y00+5y0+3y=0;y(0)=3;y0(0)=4(e)y004y0+13y=0;y(0)=1;y0(0)=55.Achemicalmanufacturingcompanyhasa1000-gallonholdingtankwhichitusestocontrolthereleaseofpollutantsintoasewagesystem.Initiallythetankhas360gallonsofuidcontaining2poundsofpollutantpergallon.Fluidcontaining3poundsofpollutantpergallonentersthetankattherateof80gallonsperhourandisuniformlymixedwiththeuidalreadyinthetank.Simultaneously,uidisreleasedfromthetankattherateof40gallonsperhour.Determinehowmanypoundsofpollutantispresentinthetankafteronehour.Math222|Exam1Review36.AnadvertisingcompanyistryingtoexposeanewproducttothegreaterMilwaukeemetropolitanareabyadvertisingontelevision.SupposethattheexposuretonewpeopleisproportionaltothenumberofpeoplewhohavenotseentheproductoutofatotalpopulationofMviewers.LetP(t)denotethenumberofviewerswhohavebeenexposedtotheproductattimet.Thecompanyhasdeterminedthatnoonewasawareoftheproductatthestartofthecampaign[P(0)=0]andthat30%oftheviewerswereawareoftheproductafter10days.(a)Determinethedierentialequationthatdescribesthenumberofviewerswhoareawareoftheproductattimet.(b)Determinethesolutionofthedierentialequationfrompart(a)thatsatisestheinitialconditionP(0)=0.(c)Howlongwillittakefor90%oftheviewingpopulationtobeawareoftheproduct?SomePossiblyUsefulTrigIdentitiessin2u=2sinucosucos2u=cos2usin2u=12sin2u=2cos2u1sin2u=1cos2u2cos2u=1+cos2u2sinmxcosnx=12[sin(m+n)x+sin(mn)x]sinmxsinnx=12[cos(m+n)xcos(mn)x]cosmxcosnx=12[cos(m+n)x+cos(mn)x]gomez/m222/s09/Handouts/review1.pdf