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Midterm

# 2002 Test 2 solution

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Department
Mathematics
Course
MAT137Y1
Professor
N/ A
Semester
Winter

Description
MAT137Y20022003SolutionstoTermTest21Evaluatethefollowingexpressionsdcosx5idx221sin1tanx2221tanxcos1tanx2tanxsecx2sin221sin1tanxsinxcosx2221sin1tanxdy22dx1sin1tanx112x1x125iilim11x15x1x15LetLdenotethelimitApplyingLHospitalsRuletwice1111232xx12525224Llimlim1166511511xx4624xx5525pt1sin25iiilim1t1cos1t2pppp2costsintp2242ApplyingLHospitalsRuletwiceLlimlim11ttsin1tcos1t4ptanh25ivlim0hsechThenumeratorgoesto0andthedenominatorgoesto1Thelimitis05vfxwherefxxusingonlythedenitionofderivativefxhfxxhxxhxxhxfxlimlimlimhhh000hhxhxhxhx11limh02xxhx22Letfx1xx5aFindalllocalextremaoffdoesnotexistat0WeconsidercasesonxTondcriticalpointsnotethatf21xxx02x1x0 fxfx21xxx02x1x0Noticethatfx0forallx0andfx0forallx0Thereforetheonlycriticalpointisx01Thisisnotamistakeeventhough2x10impliesxthepointisnotintheintervalx0 2Sincefisincreasingforx0anddecreasingforx0x0mustbealocalandabsoluteminimum 103bFindtheabsoluteextremaoffon10themaximummustoccuratx1andtheminimummustoccuratSincefisdecreasingonx01
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