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Calculus

# Calculus - Chapter 12 Functions of Several Variables and...

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Chapter 12 Functions of Several Variables and Partial Differentiation 372 Chapter 12 Section 12.1 5. The domain is the set of all values ( x , y ) except the line y = – x . {( x , y ) x y } 0 y x 10 10 7. The domain is the half-plane given by y > – x – 2. { x , y x + y + 2 > 0 or y > − ( x + 2)} y x 10 10 –2 9. Range: { f ( x , y ) f ( x , y ) 0} 11. Range: { f ( x , y ) 1 f ( x , y ) 1} 13. Range: { f ( x , y ) f ( x , y ) ≥ − 1} 15. f (1, 2) = 3, f (0, 3) = 3 17. (150,1000) 312 (150, 2000) 333 (150, 3000) 350 About 19 feet. R R R = = = a. b. c. d. 19. f ( x , y ) = x 2 + y 2 –4 –2 0 2 4 –4 –2 0 2 4 0 5 10 x y z The traces give a graph that is representative of the function. 21. f ( x , y ) = x 2 + y 2 –2 0 2 –2 0 2 0 2 4 x y z The traces give a graph that is representative of the function. 23. –2 –1 0 1 2 –2 –1 0 1 2 –5 0 5 x y z 25. –4 –2 0 2 4 –4 –2 0 2 4 –10 –5 0 5 10 x y z 27. –2 –1 0 1 2 –2 –1 0 1 2 –5 0 5 x y z 29. –5 0 5 –5 0 5 –4 –2 0 2 4 x y z

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Chapter 12 Functions of Several Variables and Partial Differentiation 373 31. –2 –1 0 1 2 –2 –1 0 1 2 –5 –2.5 0 2.5 5 y y z x 33. –2 0 2 –2 0 2 –0.2 0 0.2 x y z 35. –4 –2 0 2 4 –4 –2 0 2 4 –2 0 2 4 –4 37. –2 –1 0 1 2 0 2 4 –10 0 10 x y z 39. –2 2 –2 2 x y 41. - 4 - 2 2 4 - 4 - 2 2 4 x y 43. –2 2 –2 2 x y 45. –2 2 –2 2 x y 47. –2 –1 0 1 2 –2 –1 0 1 2
Chapter 12 Functions of Several Variables and Partial Differentiation 374 49. –6 –4 –2 0 2 4 6 –6 –4 –2 0 2 4 6 51. a. Surface B b. Surface D c. Surface A d. Surface F e. Surface C f. Surface E 53. a. Contour A b. Contour D c. Contour C d. Contour B 55. –2 0 2 –2 0 2 –2.5 0 2.5 5 7.5 x y z 57. a. Looking from the positive x -axis you observe traces of y 2 curves: Surface B b. Looking from the positive y -axis you observe traces of x 2 curves: Surface A 59. Since deformation of the square grid depends on changes in height, no deformation is observed from the z -axis. 61. The wave travels parallel to the line y = x . From (100, 100, 0) the waves fill in the vertical region between z = 1 and z = 1, so the top and bottom of the shape are straight lines. What the ends look like depends on how the plotting window is set. 63. For example: (100,100 3, 0) 65. The stadium would be located in the upper left hand corner in the concentric ellipses. The line could be a highly used roadway and the other circular level curves could represent businesses along the roadway. 67. The point of maximum power would be at the smallest of the concentric ellipses. Power would increase away from the frame, and players would want to hit the ball at this spot to obtain maximum power. 69. Using a maximum of 4 for HS, 800 for SATV and S ATM , we get PGA = 3.942 so that it is not possible to give a PGA of 4. Using a minimum of 0 for HS , 200 for SATV and SATM , we get PGA = –.57 so that it is possible to have a PGA that is negative. The HS variable is most important because of the weight carried by the term .708 HS . This term has a range from 0 to 2.832, a much larger range than the other terms.

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