Calculus(2)Ch14 - Chapter 14 Partial Derivatives 1 � 14 1...

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Chapter 14. Partial Derivatives1§14.1Functions of Several VariablesFunctions of two variablesz=f(x, y),(x, y)∈ D.The setDia called thedomainoffand{f(x, y)|(x, y)∈ D }israngeoff.x, y:independent variablesz:dependent variableEx1. FindDand evaluatef(3,3).
(b)f(x, y) =xln(y(sol)62
4. Find the domain and range ofg(x, y) =p9-x2-y2.(sol)
Yonsei UniversityD.H. Kim
Chapter 14. Partial Derivatives2Ex5. Sketch the graph off(x, y) = 6-3x-2y.(sol)x-intercept:(2,0,0),y-intercept:(0,3,0),z-intercept:(0,0,6)
8. Find the domain and range and sketch the graph off(x, y) = 4x2+y2.(sol)
Yonsei UniversityD.H. Kim
Chapter 14. Partial Derivatives3Level curvesFor a constantk,f(x, y) =kislevel curvesoff(traces of the graph offin the planez=kprojected down to thexy-plane).Ex9.f(1,3) =?f(4,5) =?(sol)f(1,3)73,f(4,5)56Yonsei UniversityD.H. Kim
Chapter 14. Partial Derivatives4Ex10. Sketch the level curves off(x, y) = 6-3x-2yfork=-6,0,6,12.(sol)
11. Sketch the level curves ofg(x, y) =p9-x2-y2fork= 0,1,2,3.(sol)
14.f(x, y, z) = ln(z-y) +xysinzD={(x, y, z)R3|z > y}Ex15. Find thelevel surfacesoff(x, y, z) =x2+y2+z2.(sol)
Chapter 14. Partial Derivatives5§14.2Limits and ContinuityDef.f:D ⊂R2Ris a function.lim(x,y)(a,b)f(x, y) =L² >0,δ >0such that0<|(x, y)-(a, b)|< δ⇒ |f(x, y)-L|< ²Note.lim(x,y)(a,b)f(x, y) =Llimxaybf(x, y) =Lf(x, y)Las(x, y)(a, b)Ex1. Show thatlim(x,y)(0,0)x2-y2x2+y2does not exist.(sol)Letf(x, y) =x2-y2x2+y2(i)(x, y)(0,0)alongx-axis :y= 0f(x,0) =x2x2= 1,x6
Ex2.f(x, y) =xyx2+y2lim(x,y)(0,0)f(x, y)?(sol)(i)(x, y)(0,0)alongx-axis :y= 0f(x,0) =0x2= 0,x6= 0f(x, y)0as(x, y)(0,0)alongx-axis(ii)(x, y)(0,0)alongy-axis :x= 0f(0, y) =0y2= 0,y6= 0f(x, y)0as(x, y)(0,0)alongy-axis
Yonsei UniversityD.H. Kim
Chapter 14. Partial Derivatives6Ex3.f(x, y) =xy2x2+y4lim(x,y)(0,0)f(x, y)?(sol)(i)(x, y)(0,0)alongy=mx:f(x, mx) =m2x3x2+m4x4=m2x1 +m2x20as(x, y)(0,0)alongy=mx(ii)(x, y)(0,0)alongx=y2:f(y2, y) =y4y4+y4=12f(x, y)12as(x, y)(0,0)alongx=y2Ex4. Findlim(x,y)(0,0)3x2yx2+y2if it exists.(sol)Let² >0. We want to findδ >0such that0<px2+y2< δflflflfl3x2yx2+y2-0flflflfl< ²Note thatflflflfl3x2yx2+y2flflflfl=3x2|y|x2+y23|y| ≤3px2+y2<3δTakeδ=²3lim(x,y)(0,0)3x2yx2+y2= 0ContinuityDef.lim(x,y)(a,b)f(x, y) =f(a, b)fiscontinuousat(a, b)We sayfiscontinuous onDiffis continuous at every point(a, b)∈ DYonsei UniversityD.H. Kim
Chapter 14. Partial Derivatives7Note.1 All polynomials are continuous onR2.

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