12.4Chain - 12.4ChainRule...

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Unformatted text preview: 12.4ChainRule Supposez=f(x,y),x=g(t),andy=h(t).Ultimatelyzdepends dz ont.Wewanttofind dt . df Ex.f(x,y)=x2y,x=3t+1,y=6t2:Find dt . Ex.f(x,y)=ln(xy),x=et,y=et:Find Ex.fx=cost,fy=sint,x=t24t,y=lnt: df dt . df Find dt . Supposez=f(x,y),x=g(s,t),andy=h(s,t).Thenzultimately dependsonsandt. u z = x e , x = , y = v ln u : Ex. v 2 y z Find v Evaluatethepartialderivativeat(u,v)=(1,2). Ex.Supposez=f(x,y),x=g(s,t),andy=h(s,t) fx(2,1)=4,fy(2,1)=6,gs(1,3)=3,hs(1,3)=5, g(1,3)=2,andh(1,3)=1 Findfs(2,1). Ex.Supposew=f(r,s,t),r=g(x,y),s=h(x,y),andt=k(x,y) dz dz and Ex.Supposeyz=ln(x+z).Find dx dy . dz F =- x, dx Fz Fy dz =- dy Fz u dz 2 y z = x e , x = , y = v ln u . Do:1.Find du for v Evaluateitfor(u,v)=(1,2). z z 2.Ifz=f(x,y)wherex=s+tandy=st,find s t Applications Ex.Theradiusofarightcircularcylinderisincreasingatarate of2in/minandtheheightisincreasingat3in/min.Howfast isthelateralsurfaceareachangingwhentheradiusis10 inchesandtheheightis12inches? Ex.Thelengthsx,y,zoftheedgesofarectangularboxare changingwithtime.Attheinstantinquestion,V=6m3, dx dy dV = = 1 m /s and = 9 m 3 /s. x=1m,y=2m, dt dt dt Howfastiszchangingatthatinstant? ...
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