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1、青岛格兰德中学2012——2013学年度第二学期高一年级中英班数学学科模块检测1.(a)Simplify.(b)Hence,orotherwise,express–asasinglefractioninitssimplestform.2.Express–asasinglefractioninitssimplestform.73.Thefunctionf(x)isdefinebya.Sketchthegraphoff(x)b.Writedowntherangeoffc.Findthevaluesofxfor
2、whichf(x)<24..Aheatedmetalballisdroppedintoaliquid.Astheballcools,itstemperature,T°C,tminutesafteritenterstheliquid,isgivenbyT=400e–0.05t+25,t³0.(a)Findthetemperatureoftheballasitenterstheliquid.(b)FindthevalueoftforwhichT=300,givingyouranswerto3significa
3、ntfigures.(c)Findtherateatwhichthetemperatureoftheballisdecreasingattheinstantwhent=50.Giveyouranswerin°Cperminuteto3significantfigures.(d)FromtheequationfortemperatureTintermsoft,givenabove,explainwhythetemperatureoftheballcanneverfallto20°C.75.f(x)=2x3–
4、x–4.(a)Showthattheequationf(x)=0canbewrittenasx=.Theequation2x3–x–4=0hasarootbetween1.35and1.4.(b)Usetheiterationformulaxn+1=,withx0=1.35,tofind,to2decimalplaces,thevalueofx1,x2andx3.Theonlyrealrootoff(x)=0isa.(c)Bychoosingasuitableinterval,provethata=1.3
5、92,to3decimalplaces.77学校班级姓名学号Thefunctionsfandgaredefinedbyf:x2x+ln2,xÎℝ,g:xe2x,xÎℝ.(a)Provethatthecompositefunctiongfisgf:x4e4x,xÎℝ.(b)Sketchthecurvewithequationy=gf(x),andshowthecoordinatesofthepointwherethecurvecutsthey-axis.(c)Writedowntherangeofgf.7.
6、Thefunctionfisdefinedbyf:x–,x>1.7(a)Showthatf(x)=,x>1.(b)Findf–1(x).Thefunctiongisdefinedbyg:xx2+5,xÎℝ.(c)Solvefg(x)=.8.Aparticularspeciesoforchidisbeingstudied.Thepopulationpattimetyearsafterthestudystartedisassumedtobep=,whereaisaconstant.Giventhatthere
7、were300orchidswhenthestudystarted,7(a)showthata=0.12,(b)usetheequationwitha=0.12topredictthenumberofyearsbeforethepopulationoforchidsreaches1850.(c)Showthatp=.(d)Henceshowthatthepopulationcannotexceed2800.7
