RyanBlairUniversityofPennsylvaniaThursdaySeptember272011 RyanBlairUPenn Math103SecantsTangentsandDerivatives ThursdaySeptember272011111 Outline 1 Review 2 SecantLinesandTangentLines 3 Derivativ ID: 189821
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Math103:Secants,TangentsandDerivatives RyanBlairUniversityofPennsylvaniaThursdaySeptember27,2011 RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20111/11 Outline 1 Review 2 SecantLinesandTangentLines 3 Derivative RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20112/11 Review LimitsInvolvingIn nity 1 Limitsasxapproaches1or 1. 2 Limitsatin nitediscontinuities 3 Horizontalasymptotes 4 Verticalasymptotes RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20113/11 Review Horizontalasymptotes De nition Aliney=bisahorizontalasymptoteofthegraphofy=f(x)iflimx!1f(x)=borlimx! 1f(x)=b RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20114/11 Review Horizontalasymptotes De nition Aliney=bisahorizontalasymptoteofthegraphofy=f(x)iflimx!1f(x)=borlimx! 1f(x)=bExample:Findthehorozontalasymptotesofy=x+1+sin(x) x RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20114/11 SecantLinesandTangentLines Secantlines De nition Alineintheplaneisasecantlinetoacircleifitmeetsthecircleinexactlytwopoints. De nition Alineintheplaneisasecantlinetothegraphofy=f(x)ifitmeetsthegraphofy=f(x)inatleasttwopoints. RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20115/11 SecantLinesandTangentLines Secantlines De nition Alineintheplaneisasecantlinetoacircleifitmeetsthecircleinexactlytwopoints. De nition Alineintheplaneisasecantlinetothegraphofy=f(x)ifitmeetsthegraphofy=f(x)inatleasttwopoints.Findthesecantlinetoy=x3 2x+1betweenx=0andx=1. RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20115/11 SecantLinesandTangentLines Tangentlines De nition Alineintheplaneisatangentlinetoacircleifitmeetsthecircleinexactlyonepoint. RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20116/11 SecantLinesandTangentLines Tangentlines De nition Alineintheplaneisatangentlinetoacircleifitmeetsthecircleinexactlyonepoint. De nition Alineintheplaneisatangentlinetothegraphofy=f(x)ifitmeetsthegraphofy=f(x)inexactlyonepointlocallyandit\kisses"thegraphatthatpoint RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20116/11 SecantLinesandTangentLines Tangentlines De nition Alineintheplaneisatangentlinetoacircleifitmeetsthecircleinexactlyonepoint. De nition Alineintheplaneisatangentlinetothegraphofy=f(x)ifitmeetsthegraphofy=f(x)inexactlyonepointlocallyandit\kisses"thegraphatthatpoint\kisses"=(nearthepointofintersectionthegraphiscompletelycontaintoonesideofthetangentline) RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20116/11 SecantLinesandTangentLines Sincewedon'thavetwopointsweneedanewideato ndtheslopeofatangentline. RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20117/11 SecantLinesandTangentLines Sincewedon'thavetwopointsweneedanewideato ndtheslopeofatangentline.Findtheslopeofsecantlinestof(x)=1 1+xonthefollowingintervals: 1 [1;3] 2 [1;2] 3 [1;1:5] RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20117/11 SecantLinesandTangentLines Sincewedon'thavetwopointsweneedanewideato ndtheslopeofatangentline.Findtheslopeofsecantlinestof(x)=1 1+xonthefollowingintervals: 1 [1;3] 2 [1;2] 3 [1;1:5]http://en.wikipedia.org/wiki/File:Sec2tan.gif RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20117/11 SecantLinesandTangentLines Sincewedon'thavetwopointsweneedanewideato ndtheslopeofatangentline.Findtheslopeofsecantlinestof(x)=1 1+xonthefollowingintervals: 1 [1;3] 2 [1;2] 3 [1;1:5]http://en.wikipedia.org/wiki/File:Sec2tan.gifTangentlinesarethelimitsofsecantlines RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20117/11 SecantLinesandTangentLines TangentLines De nition Thetangentlinetoacurvey=f(x)atapoint(a;f(a))isthelinethrough(a;f(a))withtheslopelimh!0f(a+h) f(a) h RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20118/11 SecantLinesandTangentLines TangentLines De nition Thetangentlinetoacurvey=f(x)atapoint(a;f(a))isthelinethrough(a;f(a))withtheslopelimh!0f(a+h) f(a) hFindtheslopeofthelinetangenttoy=sin(x)atx=0. RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20118/11 Application:InstantaneousVelocity De nition Ifs(t)isapositionfunctionde nedintermsoftimet,thentheinstantaneousvelocityattimet=aisgivenbyv(a)=limh!0s(a+h) s(a) h RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20119/11 Application:InstantaneousVelocity De nition Ifs(t)isapositionfunctionde nedintermsoftimet,thentheinstantaneousvelocityattimet=aisgivenbyv(a)=limh!0s(a+h) s(a) hExampleSupposeapennyisdroppedfromthetopofDRLwhichis19.6metershigh.Thepositionofthepennyintermsofhightabovethestreetisgivenbys(t)=19:6 4:9t2.Atwhatspeedisthepennytravelingwhenithitstheground. RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,20119/11 Derivativeasafunction Givenanyfunctionf(x)wewantto ndanewfunctionthat,foranyx-value,outputstheslopeoff(x)atthatvalue. RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,201110/11 Derivativeasafunction Givenanyfunctionf(x)wewantto ndanewfunctionthat,foranyx-value,outputstheslopeoff(x)atthatvalue. De nition f0(x)=limh!0f(x+h) f(x) h RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,201110/11 Derivativeasafunction Givenanyfunctionf(x)wewantto ndanewfunctionthat,foranyx-value,outputstheslopeoff(x)atthatvalue. De nition f0(x)=limh!0f(x+h) f(x) hNotation.Otherwaysofwritingthederivativeofy=f(x).f0(x)=y0=dy dx=df dx=d dxf(x)=Df(x)=Dxf(x) RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,201110/11 TheSandwichTheorem Theorem Iff(x)g(x)h(x)whenxisnearcandlimx!cf(x)=limx!ch(x)=Lthenlimx!cg(x)=L RyanBlair(UPenn) Math103:Secants,TangentsandDerivatives ThursdaySeptember27,201111/11