Calculus Multivariable McCallum, Hughes-Hallett, Gleason 4th Ed ch12 solutions

Calculus: Multivariable

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12.1 SOLUTIONS 861 CHAPTER TWELVE Solutions for Section 12.1 Exercises 1. The distance of a point P = ( x, y, z ) from the yz -plane is | x | , from the xz -plane is | y | , and from the xy -plane is | z | . So, B is closest to the yz -plane, since it has the smallest x -coordinate in absolute value. B lies on the xz -plane, since its y -coordinate is 0 . B is farthest from the xy -plane, since it has the largest z -coordinate in absolute value. 2. The distance of a point P = ( x, y, z ) from the yz -plane is | x | , from the xz -plane is | y | , and from the xy -plane is | z | . So A is closest to the yz -plane, since it has the smallest x -coordinate in absolute value. B lies on the xz -plane, since its y -coordinate is 0 . C is farthest from the xy -plane, since it has the largest z -coordinate in absolute value. 3. Your Fnal position is (1 , - 1 , - 3) . Therefore, you are in front of the yz -plane, to the left of the xz -plane, and below the xy -plane. 4. Your Fnal position is (1 , - 1 , 1) . This places you in front of the yz -plane, to the left of the xz -plane, and above the xy -plane. 5. The point P is 1 2 + 2 2 + 1 2 = 6 = 2 . 45 units from the origin, and Q is 2 2 + 0 2 + 0 2 = 2 units from the origin. Since 2 < 6 , the point Q is closer. 6. The distance formula: d = p ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 gives us the distance between any pair of points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) . Thus, we Fnd Distance from P 1 to P 2 = 2 2 Distance from P 2 to P 3 = 6 Distance from P 1 to P 3 = 10 So P 2 and P 3 are closest to each other. 7. The equation is x 2 + y 2 + z 2 = 25 8. The equation is ( x - 1) 2 + ( y - 2) 2 + ( z - 3) 2 = 25 9. The graph is a plane parallel to the yz -plane, and passing through the point ( - 3 , 0 , 0) . See ±igure 12.1. 31 x y z x = - 3 - 3 Figure 12.1 x y z y = 1 1 Figure 12.2 10. The graph is a plane parallel to the xz -plane, and passing through the point (0 , 1 , 0) . See ±igure 12.2. 11. The graph is all points with y = 4 and z = 2 , i.e., a line parallel to the x -axis and passing through the points (0 , 4 , 2); (2 , 4 , 2); (4 , 4 , 2) etc. See ±igure 12.3.
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862 Chapter Twelve /SOLUTIONS z x y 2 4 (0 , 4 , 2) (2 , 4 , 2) (4 , 4 , 2) Figure 12.3 12. (a) 80 - 90 F (b) 60 - 72 F (c) 60 - 100 F 13. - north ¾ south 100 90 80 70 Topeka distance from Topeka predicted high temperature Figure 12.4 14. North South 60 80 100 ª Boise Figure 12.5 West East 60 80 100 ª Boise Figure 12.6 15. The amount of money spent on beef equals the product of the unit price p and the quantity C of beef consumed: M = pC = pf ( I, p ) . Thus, we multiply each entry in Table 12.1 on page 605 of the text by the price at the top of the column. This yields Table 12.1.
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Calculus Multivariable McCallum, Hughes-Hallett, Gleason 4th Ed ch12 solutions

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