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# Chapter 12 - INSTRUCTORS SOLUTIONS MANUAL SECTION 12.1(PAGE...

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INSTRUCTOR’S SOLUTIONS MANUAL SECTION 12.1 (PAGE 645) CHAPTER 12. PARTIAL DIFFERENTIA- TION Section 12.1 Functions of Several Variables (page 645) 1. f ( x , y ) = x + y x y . The domain consists of all points in the xy -plane not on the line x = y . 2. f ( x , y ) = . Domain is the set of points ( x , y ) for which 0, that is, points on the coordinate axes and in the ﬁrst and third quadrants. 3. f ( x , y ) = x x 2 + y 2 . The domain is the set of all points in the -plane except the origin. 4. f ( x , y ) = x 2 y 2 . The domain consists of all points not on the lines x y . 5. f ( x , y ) = ± 4 x 2 + 9 y 2 36. The domain consists of all points ( x , y ) lying on or out- side the ellipse 4 x 2 + 9 y 2 = 36. 6. f ( x , y ) = 1 / ± x 2 y 2 . The domain consists of all points in the part of the plane where | x | > | y | . 7. f ( x , y ) = ln ( 1 + ) . The domain consists of all points satisfying > 1, that is, points lying between the two branches of the hy- perbola =− 1. 8. f ( x , y ) = sin 1 ( x + y ) . The domain consists of all points in the strip 1 x + y 1. 9. f ( x , y , z ) = xyz x 2 + y 2 + z 2 . The domain consists of all points in 3-dimensional space except the origin. 10. f ( x , y , z ) = e . The domain consists of all points ( x , y , z ) where > 0, that is, all points in the four octants x > 0, y > 0, z > 0; x > 0, y < 0, z < 0; x < 0, y > 0, z < 0; and x < 0, y < 0, z > 0. 11. z = f ( x , y ) = x x y z ( 2 , 0 , 2 ) ( 2 , 3 , 2 ) z = x 3 Fig. 12.1.11 12. f ( x , y ) = sin x , 0 x 2 π, 0 y 1 x y z z = sin x 2 π 1 Fig. 12.1.12 13. z = f ( x , y ) = y 2 x y z z = y 2 Fig. 12.1.13 14. f ( x , y ) = 4 x 2 y 2 ,( x 2 + y 2 4 , x 0 , y 0 ) 443

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SECTION 12.1 (PAGE 645) R. A. ADAMS: CALCULUS x y z z = 4 x 2 y 2 2 2 4 Fig. 12.1.14 15. z = f ( x , y ) = ± x 2 + y 2 x y z z = ± x 2 + y 2 Fig. 12.1.15 16. f ( x , y ) = 4 x 2 x y z z = 4 x 2 Fig. 12.1.16 17. z = f ( x , y ) =| x |+| y | x y z Fig. 12.1.17 18. f ( x , y ) = 6 x 2 y x y z 6 z = 6 x 2 y 3 6 Fig. 12.1.18 19. f ( x , y ) = x y = C , a family of straight lines of slope 1. y x x y = c c =− 3 c =− 2 c =− 1 c = 0 c = 1 c = 2 c = 3 Fig. 12.1.19 20. f ( x , y ) = x 2 + 2 y 2 = C , a family of similar ellipses centred at the origin. 444
INSTRUCTOR’S SOLUTIONS MANUAL SECTION 12.1 (PAGE 645) y x x 2 + 2 y 2 = c c = 1 c = 4 c = 9 c = 16 Fig. 12.1.20 21. f ( x , y ) = xy = C , a family of rectangular hyperbolas with the coordinate axes as asymptotes. y x = c c = 1 c = 4 c = 9 c =− 1 c =− 4 c =− 9 c = 0 Fig. 12.1.21 22. f ( x , y ) = x 2 y = C , a family of parabolas, y = x 2 / C , with vertices at the origin and vertical axes. y x x 2 y = c c = 0 . 5 c = 1 c = 2 c =− 2 c =− 1 c =− 0 . 5 Fig. 12.1.22 23. f ( x , y ) = x y x + y = C , a family of straight lines through the origin, but not including the origin. y x x y x + y = c c =− 2 c = 2 c =− . 5 c = 0 c = . 5 c = 1 c =− 1 Fig. 12.1.23 24. f ( x , y ) = y x 2 + y 2 = C . This is the family x 2 + ( y 1 2 C ) 2 = 1 4 C 2 of circles passing through the origin and having centres on the y -axis. The origin itself is, however, not on any of the level curves. y x c = 1 c = 2 c = 3 c =− 3 c =− 2 c =− 1 y x 2 + y 2 = c Fig. 12.1.24 25. f ( x , y ) = xe y = C . This is the family of curves y = ln x C .

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Chapter 12 - INSTRUCTORS SOLUTIONS MANUAL SECTION 12.1(PAGE...

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