15.1%20Ex%2044%20-%2053

15.1%20Ex%2044%20-%2053 - 620 C H A P T E R 15 D I F F E R...

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620 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES (ET CHAPTER 14) i B D E C P A ii iii 400 450 0 1 2 km Contour interval = 10 meters 470 iv 44. Match the contour maps (A) and (B) in Figure 23 with the two functions f ( x , y ) = x 2 y and g ( x , y ) = 2 x y . c = 2 y xx c = 2 c = 0 c = 2 c = 0 c = 2 2 2 1 2 1 y 2 2 2 (A) (B) 1 1 2 FIGURE 23 SOLUTION The level curves of the function f ( x , y ) = x 2 y are the lines x 2 y = c or y = x 2 c 2 .Theleve l curves of g ( x , y ) = 2 x y are the lines 2 x y = c or y = 2 x c . The slope of the lines in the contour map of g is greater than the slope in the contour map of f . Therefore (A) is a contour map of f and (B) is a contour map of g . 45. Which linear function has the contour map shown in Figure 24 (with level curve c = 0 as indicated), assuming that the contour interval is m = 6? What if m = 3? c = 0 6 3 6 3 1 2 2 1 x y FIGURE 24 We denote the linear function by f ( x , y ) = α x + β y + γ (1) The level curves of f are x + y + = c (2) By the given information, the level curve for c = 0 is the line passing through the points ( 0 , 1 ) and ( 3 , 0 ) .We±nd the equation of this line: y = 0 ( 1 ) 3 0 ( x + 3 ) y =− 1 3 ( x + 3 ) x + 3 y + 3 = 0 Setting c = 0in(2)gives x + y + = 0. Hence, 1 = 3 = 3 = 3 , = 3 Substituting into (2) gives x + 3 y + 3 = c or x + 3 y + 3 = c (3)
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SECTION 15.1 Functions of Two or More Variables (ET Section 14.1) 621 Case 1: m = 6. In this case, the closest line above c = 0 corresponds to c = 6. This line is parallel to x + 3 y + 3 = 0 and passes through the origin, hence its equation is x + 3 y = 0. Setting c = 6 in (3) gives
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This homework help was uploaded on 04/22/2008 for the course MATH 32A taught by Professor Gangliu during the Winter '08 term at UCLA.

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15.1%20Ex%2044%20-%2053 - 620 C H A P T E R 15 D I F F E R...

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