12.4Chain

# 12.4Chain - 12.4ChainRule...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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? ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online