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# EVEN12 - CHAPTER 12 Functions of Several Variables Section...

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C H A P T E R 1 2 Functions of Several Variables Section 12.1 Introduction to Functions of Several Variables . . . . . . . 308 Section 12.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . 312 Section 12.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . 315 Section 12.4 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . 321 Section 12.5 Chain Rules for Functions of Several Variables . . . . . . 325 Section 12.6 Directional Derivatives and Gradients . . . . . . . . . . . 330 Section 12.7 Tangent Planes and Normal Lines . . . . . . . . . . . . . 334 Section 12.8 Extrema of Functions of Two Variables . . . . . . . . . . 340 Section 12.9 Applications of Extrema of Functions of Two Variables . 345 Section 12.10 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . 350 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

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C H A P T E R 1 2 Functions of Several Variables Section 12.1 Introduction to Functions of Several Variables Solutions to Even-Numbered Exercises 308 2. No, z is not a function of x and y . For example, corresponds to both z ± 2 x , y 1, 0 xz 2 2 xy y 2 4 4. Yes, z is a function of x and y . z 8 x ln y z x ln y 8 0 6. (a) (b) (c) (d) (e) (f) f t , 1 4 t 2 4 t 2 f x , 0 4 x 2 0 4 x 2 f 1, y 4 1 4 y 2 3 4 y 2 f 2, 3 4 4 36 36 f 0, 1 4 0 4 0 f 0, 0 4 f x , y 4 x 2 4 y 2 8. (a) (b) (c) (d) (e) (f) ln 2 ln e ln 2 1 g e , e ln e e ln 2 e g 2, 3 ln 2 3 ln 1 0 g 0, 1 ln 0 1 0 g e , 0 ln e 0 1 g 5, 6 ln 5 6 ln 11 g 2, 3 ln 2 3 ln 5 g x , y ln x y 10. (a) (b) f 6, 8, 3 6 8 3 11 f 0, 5, 4 0 5 4 3 f x , y , z x y z 12. (a) (b) V 5, 2 5 2 2 50 V 3, 10 3 2 10 90 V r , h r 2 h 14. (a) g 4, 1 1 4 1 t dt ln t 1 4 ln 4 g x , y y x 1 t dt (b) g 6, 3 3 6 1 t dt ln t 3 6 ln 3 ln 6 ln 1 2 16. (a) (b) y 3 x 2 y y y 3 x 2 y y , y 0 3 xy 3 x y y 2 2 y y y 2 3 xy y 2 y f x , y y f x , y y 3 x y y y y 2 3 xy y 2 y 3 xy 3 x y y 2 3 xy y 2 x 3 x y x 3 y , x 0 f x x , y f x , y x 3 x x y y 2 3 xy y 2 x f x , y 3 xy y 2
Section 12.1 Introduction to Functions of Several Variables 309 18. Domain: Range: 0 z 2 x , y : x 2 4 y 2 1 1 x 2 4 y 2 1 1 x 2 4 y 2 4 4 x 2 4 y 2 0 f x , y 4 x 2 4 y 2 20. Domain: Range: 0 z x , y : 1 y x 1 f x , y arccos y x 22. Domain: Range: all real numbers x , y : xy > 6 xy > 6 xy 6 > 0 f x , y ln xy 6 30. (a) Domain: is any real number, is any real number Range: (b) when which represents points on the y -axis. (c) No. When x is positive, z is negative. When x is negative, z is positive. The surface does not pass through the first octant, the octant where y is negative and x and z are positive, the octant where y is positive and x and z are negative, and the octant where x , y and z are all negative. x 0 z 0 2 z 2 y x , y : x 32. Plane Domain: entire xy -plane Range: x y 2 3 3 4 4 6 z < z < f x , y 6 2 x 3 y 34. Plane: x y 4 4 4 3 2 2 3 4 4 z z 1 2 x g x , y 1 2 x 36. Cone Domain of f : entire xy -plane Range: x y 1 1 2 2 3 3 2 z z 0 z 1 2 x 2 y 2 38. Domain of f : entire xy -plane Range: x y 5 25 20 15 10 5 z z 0 f x , y xy , 0, x 0, y 0 elsewhere 40. Semi-ellipsoid Domain: set of all points lying on or inside the ellipse Range: 4 4 4 2 4 y x z 0 z 1 x 2 9 y 2 16 1 f x , y 1 12 144 16 x 2 9 y 2 24. Domain: Range: all real numbers x , y : x y z xy x y 26.

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