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Midterm

2009 Test 1 solution

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Department
Mathematics
Course
MAT137Y1
Professor
N/ A
Semester
Winter

Description
MAT137Y20092010WinterSessionSolutionstoTermTest11EvaluatethefollowinglimitsDonotprovethemusingtheformaldenitionoflimit11x410ilim2x4xx12Obtainingacommondenominatorandfactoringthedenominatorweget4x114xlimlimx4x4x4x34xx3112p2xx5x5p10iilimx1x1Wemultiplytopandbottombybothconjugatesgivinguspppp2222xxx5x5xx5x5x5x5x1x1pppplimlim22x1x1x1x1xx5x5x1xx5x5p5x1x1plim2x1x1xx5x5p5x1plim52x1xx5x521Findtheexactvalueofsin2q10iSupposeqisintherstquadrantandcosq522WecandothisbydrawingatriangleAlternativelysincesinqcosq1wehavepp262412422sinq1cosq1Hencesinqsincesinq0asweare252555intherstquadrantpp26461Thereforesin2q2sinqcosq2552510iiConsiderthefunctionfxjx3jjxjExpressfxasapiecewisefunctionbyeliminatingtheabsolutevaluesandwithoutplottinganypointssketchthegraphoffToexpressfasapiecewisefunctionweeliminatetheabsolutevaluerstx3x3xx0jxjjx3j3xx3xx0Thereforeifx0jx3jjxj3xx32xif0x3jx3jjxj3xx3ifx3jx3jjxjx3x2x31
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