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MTH 254

# MTH 254 - 882 Chapter Twelve SOLUTIONS II II(c Shi the...

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Unformatted text preview: 882 Chapter Twelve SOLUTIONS II ! II :/ (c) Shi the diagram 2 units to the right and 2 units up. See Figue 12.88. y y 0--2-1-0--1--2 x Figure 12.88: f(x - 2, y - 2) Figure 12.89: f( -x, y) (d) Reflect the diagram about the y-axs. See Figure 12.89. 30. (a) See Figure 12.90. y \ b i 1 1 y b i 1 Figure 12.90 Figure 12.91 (b) See Figue 12.91. 31. (a) (i) y (ii) Y x (b) The function f(x, y) = g(y - x) is constant on lies y - x = k. Thus a1lines paralel to y = x are level cures of f. 32. Since f(x,y) = x2 - y2 = (x - y)(x + y) = 0 gives x - y = 0 or x + y = 0, the contours f(x,y) = 0 are the lies y = x or y = -x. In the regions between them, f(x, y) )0 0 or f(x, y) 0( 0 as shown in Figure 12.92. The surace z = f(x, y) is above the xy-plane where f )0 0 (that is on the shaded regions contag the x-axs) and is below the xy-plane where f 0( O. Ths means that a person could sit on the surace facing along the positive or negative x-axs, and with hislher legs hangig down the sides below the y-axs. Thus, the graph of the function is saddle-shaped at the origi. ...;r. &amp;quot; 12.4 SOLUTIONS 88 y 1 =0 x g=O y=x 1=0 y= -x x y=x y= -x Figure 12.92 Figure 12.93 33. We need thee lines with g(x, y) = 0, so that the xy-plane is divided into six regions. For example g(x, y) = y(x - y)(x + y) has the contour map in Figue 12.93. (Many other answers to ths question are possible.) Solutions for Section 12.4 Exercises 1. (a) Yes. (b) The coeffcient of m is 15 dollars per month. It represents the monthy charge to use this service. The coeffcient of t is 0.05 dollars per miute. Each miute the customer is on-line costs 5 cents. (c) The intercept represents the base charge. It costs \$35 just to get hooked up to ths servce. (d) We have f(3, 800) = 120. A customer who uses ths servce for thee months and is on-line for a tota of 800 miutes is charged \$120. 2. (a) Since z is a liear function of x and y with slope 2 in the x-diection, and slope 3 in the y-diection, we have: z = 2x + 3y + c We can write an equation for changes in z in terms of changes in x and y: L\z = (2(x + L\x) + 3(y + L\y) + c) - (2x + 3y + c) = 2L\x + 3L\y Since L\x = 0.5 and L\y = -0.2, we have L\z = 2(0.5) + 3( -0.2) = 0.4 So a 0.5 change in x and a -0.2 change in y produces a 0.4 change in z. (b) As we know that z = 2 when x = 5 and y = 7, the value of z when x = 4.9 and y = 7.2 wil be z = 2 + L\z = 2 + 2L\x + 3L\y where L\z is the change in z when x changes from 4.9 to 5 and y changes from 7.2 to 7. We have L\x = 4.9 - 5 =-0.1 and L\y = 7.2 - 7 = 0.2. Therefore, when x = 4.9 and y = 7.2, we have z =2 + 2 . (-0.1) + 5 . 0.2 = 2.4 884 Chapter Twelve /SOLUTIONS 3. (a) Substitutig in the values for the slopes, we see that the formula for the plane is z = c + 5x - 3y for some value of c. Substituting the point (4, 3, - 2) gives c = -13. The formula for the plane is z = -13 + 5x - 3y....
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MTH 254 - 882 Chapter Twelve SOLUTIONS II II(c Shi the...

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