# Calculus: One and Several Variables

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-15 JWDD027-Salas-v1 December 5, 2006 16:40 758 SECTION 15.1 CHAPTER 15 SECTION 15.1 1. dom ( f ) = the first and third quadrants, including the axes; range ( f ) = [0 , ) 2. dom ( f ) = the set of all points ( x, y ) with xy 1; the two branches of the hyperbola xy = 1 and all points in between; range ( f ) = [0 , ) 3. dom ( f ) = the set of all points ( x, y ) except those on the line y = x ; range ( f ) = ( −∞ , 0) (0 , ) 4. dom ( f ) = the set of all points ( x, y ) other than the origin; range ( f ) = (0 , ) 5. dom ( f ) = the entire plane; range ( f ) = ( 1 , 1) since e x e y e x + e y = e x + e y 2 e y e x + e y = 1 2 e x y + 1 and the last quotient takes on all values between 0 and 2. 6. dom ( f ) = the set of all points ( x, y ) other than the origin; range ( f ) = [0 , 1] 7. dom ( f ) = the first and third quadrants, excluding the axes; range ( f ) = ( −∞ , ) 8. dom ( f ) =the set of all points ( x, y ) between the branches of the hyperbola xy = 1; range ( f ) = ( −∞ , ) 9. dom ( f ) = the set of all points ( x, y ) with x 2 < y —in other words, the set of all points of the plane above the parabola y = x 2 ; range ( f ) = (0 , ) 10. dom ( f ) = the set of all points ( x, y ) with 3 x 3 , 1 y 1 (a rectangle); range ( f ) = [0 , 3] 11. dom ( f ) = the set of all points ( x, y ) with 3 x 3 , 2 y 2 (a rectangle); range ( f ) = [ 2 , 3] 12. dom ( f ) = all of space; range ( f ) = [ 3 , 3] 13. dom ( f ) = the set of all points ( x, y, z ) not on the plane x + y + z = 0; range ( f ) = {− 1 , 1 } 14. dom ( f ) = the set of all points ( x, y, z ) with x 2 = y 2 —that is, all points of space except for those which lie on the plane x y = 0 or on the plane x + y = 0; range ( f ) = ( −∞ , ) 15. dom ( f ) = the set of all points ( x, y, z ) with | y | < | x | ; range ( f ) = ( −∞ , 0] 16. dom ( f ) =the set of all points ( x, y, z ) not on the plane x y = 0; range ( f ) = ( −∞ , ) 17. dom ( f ) = the set of all points ( x, y ) with x 2 + y 2 < 9 —in other words, the set of all points of the plane inside the circle x 2 + y 2 = 9; range ( f ) = [2 / 3 , ) 18. dom ( f ) = all of space; range ( f ) = [0 , ) 19. dom ( f ) = the set of all points ( x, y, z ) with x + 2 y + 3 z > 0 — in other words, the set of all points in space that lie on the same side of the plane x + 2 y + 3 z = 0 as the point (1 , 1 , 1); range ( f ) = ( −∞ , ) 20. dom ( f ) = the set of all points ( x, y, z ) with x 2 + y 2 + z 2 4 — in other words, the set of all points inside and on the sphere x 2 + y 2 + z 2 = 4; range ( f ) = [1 , e 2 ] 21. dom ( f ) = all of space; range ( f ) = (0 , 1]

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-15 JWDD027-Salas-v1 December 5, 2006 16:40 SECTION 15.1 759 22. dom ( f ) = the set of all points ( x, y, z ) with 1 x 1 , 2 y 2 , 3 z 3 (a rectangular solid); range ( f ) = [0 , 3] 23. dom( f ) = { x : x 0 } ; range( f ) = [0 , ) dom( g ) = { ( x, y ) : x 0 , y real } ; range( g ) = [0 , ) dom( h ) = { ( x, y, z ) : x 0 , y, z real } ; range( h ) = [0 , ) 24. dom ( f ) = the entire plane, range ( f ) = [ 1 , 1] dom ( g ) = all of space, range ( g ) = [ 1 , 1] 25. lim h 0 f ( x + h, y ) f ( x, y ) h = lim h 0 2( x + h ) 2 y (2 x 2 y ) h = lim h 0 4 xh + 2 h 2 h = 4 x lim h 0 f ( x, y + h ) f ( x, y ) h = lim h 0 2 x 2 ( y + h ) (2 x 2 y ) h = 1 26. lim h 0 f ( x + h, y ) f ( x, y ) h = lim h 0 xy + hy + 2 y ( xy + 2 y ) h = lim h 0 y = y .
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