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12DIFFERENTIATIONIN SEVERAL VARIABLES12.1 Functions of Two or More VariablesPreliminary Questions1.What is the difference between a horizontal trace and a level curve? How are they related?2.Describe the trace off .x;y/Dx2sin.x3y/in thexz-plane.3.Is it possible for two different level curves of a function to intersect? Explain.4.Describe the contour map off .x; y/Dxwith contour interval 1.5.How will the contour maps off .x;y/Dxandg.x;y/D2xwith contour interval 1 look different?SOLUTIONThe level curves off .x;y/Dxare the vertical linesxDc, and the level curves ofg.x; y/D2xare the verticallines2xDcorxDc2. Therefore, the contour map off .x; y/Dxwith contour interval1consists of vertical lines with distanceone unit between adjacent lines, whereas in the contour map ofg.x; y/D2x(with contour interval1) the distance between twoadjacent vertical lines is12.ExercisesIn Exercises 1–4, evaluate the function at the specified points.1.f .x;y/DxCyx3,.2; 2/,.1; 4/SOLUTIONWe substitute the values forxandyinf .x;y/and compute the values offat the given points. This givesf .2; 2/D2C223D18f .1; 4/D1C4.1/3D52.g.x;y/Dyx2Cy2,.1; 3/,.3;2/SOLUTIONWe substitute.x; y/D.1; 3/and.x;y/D.3;2/in the function to obtaing.1; 3/D312C32D310Ig.3;2/D232C.2/2D2133.h.x; y; z/Dxyz2,.3; 8; 2/,.3;2;6/SOLUTIONSubstituting.x; y; z/D.3; 8; 2/and.x; y; z/D.3;2;6/in the function, we obtainh.3; 8; 2/D3822D3814D6h.3;2;6/D3.2/.6/2D6136D16
1506C H A P T E R12DIFFERENTIATION IN SEVERAL VARIABLES4.Q.y; z/Dy2Cysinz,.y; z/D2;2;2;6SOLUTIONWe haveQ2;2D22C2sin2D4C21D6Q2;6D.2/22sin6D4212D3In Exercises 5–12, sketch the domain of the function.