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### 06 Absolute Value Functions

Course: MATH MAT1300, Spring 2010
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ABSOLUTEVALUEFUNCTIONS 1. 6. Findalllocalextremaandallglobalextremaoff(x)=2+ 27x+12 x ontheinterval[2,5] __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________...

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ABSOLUTEVALUEFUNCTIONS 1. 6. Findalllocalextremaandallglobalextremaoff(x)=2+ 27x+12 x ontheinterval[2,5] __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ x VETCalculus 36 6.AbsoluteValueFunctions 2. Findthevaluesofallrealnumbersxwhichsatisfytheinequality x+6 26x. 2 __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ x VETCalculus 37 6.AbsoluteValueFunctions
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7.LIMITS1. Determinethefollowinglimitsshowingyourreasoning. (a) [2marks]lim e x + 4 x 2e x_ _ _(b) [2marks]lim x 3 + x 2 + 5 x 1 x 3 + 3_ _ _(c) [3marks]lim tan 2 (3t ) t0 t_ _ _VETCalculus387.Limits2.
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8. CONTINUITYANDDIFFERENTIABILITY1. Thefunctionsfandgaregivenby f ( x) = 3 x andg ( x ) = (3 x ) 2(a) [5marks] Showthatthefunctiondefinedbyy=f(g(x)isdefinedforallrealvaluesofxand sketchitsgraph. _ _ _ _ _ _ _ _f(g(x)xVETCalculus408.ContinuityandD
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9.TECHNIQUESOFDIFFERENTIATION1. Finddy ineachofthefollowingcases. dxy=excosx(a) [3marks]_ _ _ _(b) [4marks]y2(xy6)=x3_ _ _ _ _(c) [4marks]y=xxesin( t )dt_
Acadia - MATH - MAT1300
10.CURVESKETCHING1. Letfbethefunctiondefinedby f ( x) = (a) [2marks] 4x + 5 ( x + 2)(5 x)Statethepolesofthefunction_ _(b) [1mark]Evaluatelim x +f ( x)_ _(c) [1mark]Evaluatelim x f ( x)_ _(d) [1mark]Evaluatelim x4f ( x)_VETCalculus60
Acadia - MATH - MAT1300
11. APPROXIMATIONS1. Letfbethefunctiondefinedby f ( x) = (a) [1mark]Statethepoleofthefunction.2x 2 x x3 1_(b) [1mark]Evaluatelim f ( x) x_ _(c) [1mark]Evaluatelim f ( x) x _ _(d) [3marks]Statethexandyintercepts._ _ _ _
Acadia - MATH - MAT1300
12. OPTIMISATION1.2 Fora&gt;0theline a (a + 1) y = a x formsatriangleinthefirstquadrantwiththexandyaxes.(a) [2marks]Findthexandyinterceptsofthelineintermsofa._ _ _ _ (b) [2marks] Findtheareaofthetriangleasafunctionofa._ _ _ _ _
Acadia - MATH - MAT1300
13. RELATEDRATES1.2 2 Thevariablesxandyarerelatedbytheequationofacircle x + y + 4 x + 8 y = 14.(a) [4marks]Findthevalueofdy atthepoint(1,1). dx_ _ _ _ _(b) [2marks] Giventhatxisincreasingattherateof0.2unitspersecondatthepoint(1,1),findthe correspon
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14. TECHNIQUESOFINTEGRATION1. Evaluatethefollowingintegrals. (a) [5marks] /60sin 3 2 x cos 2 x dx ._ _ _ _ _ _ _ _(b) [5marks]40x 1 dx ._ _ _
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15. GROWTHANDDECAY1. Apotatoisputintoanovenwhosetemperatureremainsconstantat2000C.ThetemperatureHofthepotato changesaccordingtotimetismeasuredinminutes.Thetemperatureofthepotatois200Cwhenitisplacedintheoven.After30 minutesthetemperatureofthepotatois1200
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16. RECTILINEARMOTION1. AparticleP1movesalongthexaxiswithconstantaccelerationof2metrespersecondpersecond.Attimet =0,P1hasposition8metresandvelocity2metrespersecond.AnotherparticleP2movesalongthexaxis accordingtotheequationx2=t2+4t+2wherex2isthepositionof
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17. SIMPLEHARMONICMOTION1. Aparticleismovingalongthexaxiswithvelocitygivenby v (t ) = 3 sin (a) [2marks]t where t 3 . 2 3Determinethechangeindisplacementoftheparticleduringthegiventimeinterval._ _ _ _ _(b) [4marks]Determinethetotaldistancetravelledb
Acadia - MATH - MAT1300
18. AREA1. Thediagramshowsthegraphofy2=x2(1x).Thecurvecutsthexaxisatx=0and x=1.Findtheareaoftheloop._ _ _ _ _2. Findtheareaoftheregionboundedbythecurvesy=3+2xx2andy=5. _ _ _ _ _ _ _
Acadia - MATH - MAT1300
19. VOLUME1. LetRbetheregioninthexyplaneboundedbythexaxis,thelines x = y=cosxsinx.FindthevolumeofthesolidgeneratedwhenRisrotatedaboutthexaxis.3 , x = andthecurve 4 4_ _ _ _ _ _ _ _ _
Acadia - MATH - MAT1300
20. VECTORCALCULUS1. Agolferstrikesaballwithinitialvelocity18i+20jmetrespersecondwhereiishorizontalandjisvertical. Throughouttheflightoftheballtheaccelerationduetogravityis10j.Thepointwheretheballlandsisat thesameheightasthepointfromwhichitwasstruck.Find
Acadia - MATH - MAT1300
21.SOLUTIONS1.CartesianCoordinates1. 5,2i. 2. 3or2. 3. Letz=a+bi.Then:a (a 1) abi + (a 1)(b 1)i + b(b 1) a + (b 1)i (a 1) bi . = (a 1) 2 + b 2 (a 1) + bi (a 1) bi a (a 1) + b(b 1) = 0 Theresultispurelyimaginarywhen (a 1) 2 + b 2i.e. a 2 a + b 2 b = 0
Acadia - MATH - MAT1300
6.AbsoluteValuey1. Localmaximumof2.25atx=3.5. Globalmaximumof 32atx=2. Localandglobalminimumof2atx=3andx=4. 2. x05.4 3 2 1 1 2 3 4 5 6x 2 1 1 27.Limits 1. 2. (a) 0.5(b)2.5(c)0 (a)0(b)21 2a33.lim 2 x 3 x + 4 x 8 = lim x x 6x2 5x + 82 3x +48 x x
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12.Optimisationy 10 9 8 7 6 5 4 3 2 1 O -1 -2 -3 -4 -5 -6 -7 -8 -91. (a) yinterceptis2. (a)graph (b)4p2p41 .xinterceptisa a +1 1 1 1 (b)area= a 2 .(c)ata=1area= . 2 a +1 42A 1 2g(x) 3 f(x) B x-3-2-1(c) p = 23. (b)10.(a)dV = 30 cos(60t ).60 =
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