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Note Week 10.pdf

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Department
Mathematics
Course
MATH 127
Professor
Fiona Dunbar
Semester
Winter

Description
10Week10101SigmaNotationYoushouldreadAppendixEofthetextbookanddoexercises125odd293133Supposewewanttoaddtogethertherst100integersWecanwritethisusingthreedotstoconveytheideaofcontinuingthepatternS1239899100AnotherwaytoexpressthesumSis100XSii1ThisiscalledsigmanotationIngeneralwhenwewanttoaddtogetherthenumbersaforitakingontheintegervaluesmthroughininclusiveweexpressthisasnXaaaaaimm1m2nimWeneedtorecognizethedescriptionofainoursumThesubscriptijustmeanswewantaformulaiintermsofiAnotherwayofsayingawouldbefii2222Supposewewanttorepresent3459usingsummationnotationWerecognizethatweareaddingtogethersquaresofnumbersbetween3and99X222222Som3n9andaiandwehave3459iii3Checkbysubstitutingvaluesforistartingat3andgoingallthewayupto9PropertiesofSummationsnnXX1cacaiiimimnnnXXX2ababiiiiimimimnnXX3ccnInparticular1ni1i191ThefollowingthreeformulaswillbeusefulinevaluatingsumsnXnn1i2i1nXnn12n12i6i1n22Xnn13i4i1WewilldemonstratehowtoobtaintherstformulaWewilldenethesumSasfollowsS123n1nWritingthisanotherwaywehaveSnn1321Addingthetwoequationstogetherwehave2Sn1n1n1n1n12Snn1nn1S2102AreasandDistancesYoushouldreadsection51ofthetextbookanddoexercises1571317191021TheAreaProblemFindtheareaoftheregionSthatliesunderthecurveyfxfromatobNotethatSisboundedbythegraphofacontinuousfunctionfx0theverticallinesxaandxbandthexaxisItiseasytondtheareaofobjectswithstraightedgesbutitismoredicultwhenashapehascurvededgesRecallthatweapproximatedtheslopeofacurvebytheslopeofthesecantlineandthentookthelimitoftheseapproximationsWewilldoasimilarprocedureforareasWewillapproximateSbyrectanglesandtakethelimitoftheseareasasweincreasethenumberofrectangles92
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