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chapter_14

# chapter_14 - 11:55 L24-ch14 Sheet number 1 Page number 603...

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January 27, 2005 11:55 L24-ch14 Sheet number 1 Page number 603 black 603 CHAPTER 14 Partial Derivatives EXERCISE SET 14.1 1. (a) f (2 , 1) = (2) 2 (1) + 1 = 5 (b) f (1 , 2) = (1) 2 (2) + 1 = 3 (c) f (0 , 0) = (0) 2 (0) + 1 = 1 (d) f (1 , 3) = (1) 2 ( 3) + 1 = 2 (e) f (3 a, a ) = (3 a ) 2 ( a ) + 1 = 9 a 3 + 1 (f) f ( ab , a b ) = ( ab ) 2 ( a b ) + 1 = a 3 b 2 a 2 b 3 + 1 2. (a) 2 t (b) 2 x (c) 2 y 2 + 2 y 3. (a) f ( x + y, x y ) = ( x + y )( x y ) + 3 = x 2 y 2 + 3 (b) f ( xy, 3 x 2 y 3 ) = ( xy ) ( 3 x 2 y 3 ) + 3 = 3 x 3 y 4 + 3 4. (a) ( x/y ) sin( x/y ) (b) xy sin( xy ) (c) ( x y ) sin( x y ) 5. F ( g ( x ) , h ( y )) = F ( x 3 , 3 y + 1 ) = x 3 e x 3 (3 y +1) 6. g ( u ( x, y ) , v ( x, y )) = g ( x 2 y 3 , πxy ) = πxy sin ( x 2 y 3 ) 2 ( πxy ) = πxy sin ( πx 5 y 7 ) 7. (a) t 2 + 3 t 10 (b) 0 (c) 3076 8. te 3 ln ( t 2 +1 ) = t ( t 2 + 1) 3 9. (a) v = 7 lies between v = 5 and v = 15, and 7 = 5 + 2 = 5 + 2 10 (15 5), so WCI 19 + 2 10 (13 19) = 19 1 . 2 = 17 . 8 F (b) v = 28 lies between v = 25 and v = 30, and 28 = 25 + 3 5 (30 25), so WCI 19 + 3 5 (25 19) = 19 + 3 . 6 = 22 . 6 F 10. (a) At T = 35 , 14 = 5 + 9 = 5 + 9 10 (15 5), so WCI 31 + 9 10 (25 31) = 25 . 6 F (b) At v = 15 , 32 = 30 + 2 5 (35 30), so WCI 19 + 2 5 (25 19) = 21 . 4 F 11. (a) The depression is 20 16 = 4, so the relative humidity is 66%. (b) The relative humidity 77 (1 / 2)7 = 73 . 5%. (c) The relative humidity 59 + (2 / 5)4 = 60 . 6%. 12. (a) 4 C (b) The relative humidity 62 (1 / 4)9 = 59 . 75%. (c) The relative humidity 77 + (1 / 5)(79 77) = 77 . 4%. 13. (a) 19 (b) 9 (c) 3 (d) a 6 + 3 (e) t 8 + 3 (f) ( a + b )( a b ) 2 b 3 + 3 14. (a) x 2 ( x + y )( x y ) + ( x + y ) = x 2 ( x 2 y 2 ) + ( x + y ) = x 4 x 2 y 2 + x + y (b) ( xz )( xy )( y/x ) + xy = xy 2 z + xy 15. F ( x 2 , y + 1 , z 2 ) = ( y + 1) e x 2 ( y +1) z 2 16. g ( x 2 z 3 , πxyz, xy/z ) = ( xy/z ) sin ( πx 3 yz 4 )

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January 27, 2005 11:55 L24-ch14 Sheet number 2 Page number 604 black 604 Chapter 14 17. (a) f ( 5 , 2 , π, 3 π ) = 80 π (b) f (1 , 1 , . . . , 1) = n k =1 k = n ( n + 1) / 2 18. (a) f ( 2 , 2 , 0 , π/ 4) = 1 (b) f (1 , 2 , . . . , n ) = n ( n + 1)(2 n + 1) / 6, see Theorem 2(b), Section 5.4 19. 1 x y 20. 2 x y 21. x y 22. x y 23. (a) all points above or on the line y = 2 (b) all points on or within the sphere x 2 + y 2 + z 2 = 25 (c) all points in 3-space 24. (a) all points on or between the vertical lines x = ± 2. (b) all points above the line y = 2 x (c) all points not on the plane x + y + z = 0 25. 3 x y z 26. (0, 3, 0) z y x
January 27, 2005 11:55 L24-ch14 Sheet number 3 Page number 605 black Exercise Set 14.1 605 27. z y x 28. z y x 29. z y x 30. (2, 0, 0) (0, 2, 0) (0, 0, 4) z y x 31. z y x (0, 0, 1) 32. (0, 1, 0) (1, 0, 0) z y x 33. z y x (0, 0, 1) (0, –1, 0) 34. z y x 35. (a) f ( x, y ) = 1 x 2 y 2 , because f = c is a circle of radius 1 c (provided c 1), and the radii in (a) decrease as c increases. (b) f ( x, y ) = x 2 + y 2 because f = c is a circle of radius c , and the radii increase uniformly. (c) f ( x, y ) = x 2 + y 2 because f = c is a circle of radius c and the radii in the plot grow like the square root function.

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January 27, 2005 11:55 L24-ch14 Sheet number 4 Page number 606 black 606 Chapter 14 36. (a) III, because the surface has 9 peaks along the edges, three peaks to each edge (b) IV, because the center is relatively ﬂat and the deep valley in the first quadrant points in the direction of the positive x -axis (c) I, because the deep valley in the first quadrant points in the direction of the positive y -axis (d) II, because the surface has four peaks 37. (a) A (b) B (c) increase (d) decrease (e) increase (f) decrease 38. (a) Medicine Hat, since the contour lines are closer together near Medicine Hat than they are near Chicago.
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