Calculus: Early Transcendentals, by Anton, 7th Edition,ch14

# Calculus - Early Transcendentals

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569 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) At T = 25 there is a drop in temperature of 12 degrees when v changes from 5 to 10, thus WCI (2 / 5)( 12) + 22 = 22 24 / 5 = 17 . 2 F. (b) At v = 5 there is an increase in temperature of 5 degrees as T changes from 25 to 30 degrees, thus WCI (3 / 5)5 + 22 = 25 F. 10. (a) T (4 / 5)( 7) + 22 = 22 5 . 6 = 16 . 4 F (b) T (2 / 5)6 + 16 = 16 + 2 . 4 = 18 . 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 ) 17. (a) f ( 5 , 2 , π, 3 π ) = 80 π (b) f (1 , 1 , . . . , 1) = n k =1 k = n ( n + 1) / 2

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570 Chapter 14 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 27. z y x 28. z y x
Exercise Set 14.1 571 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. 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

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572 Chapter 14 38. (a) Medicine Hat, since the contour lines are closer together near Medicine Hat than they are near Chicago.
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