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D12 jackson mi-0 4xydx = 2y if 0 ≤ y ≤ 1 0 otherwise (d) YES, X and Y are independent, since fX(x)fY (y) = ˆ 2x·2y = 4xy if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 0otherwise is exactly the same as f(x,y), the joint density, for all x and y Example 4 X and Y are independent continuous random variables, each with pdf g(w) = ˆ 2w if 0 ≤ w ≤ 1 0, otherwiseHave students predict what they think the graphs of y = 2 sin(x) and y = ½ sin(x) will look like The purple curve is the sine graph The red curve is the transformation y = 2 sin(x) y = ½ sin(x) Notice that the minimum and maximum values of the function have increased from 1 and 1 to 2 and 2 Notice that the minimum and maximum values of
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D (xy)dA, where D is the triangular region with vertices (0,0), (−1,1), (2,1) Solution ZZ D (x y)dA = Z1 0 Z2y −y (xy)dxdy = Z1 0 (x2 2 xy) x=2y x=−y = Z1 0 9y2 2 dy = 3y3 2 y=1 y=0 = 3 2 Problem 2 Evaluate the iterated integral Z2 0 Z4 x2 xsin(y2)dydx by reversing the order of integration Solution Z2 0 Z4 x2 xsin(y 2)dydx = Z4The additive inverse of ½ is ½ 58 If x/y is the additive inverse of c/d, then (x/y) (c/d) = 0 SolutionTrue Let x/y = ½ and its additive inverse c/d = 1/2 Then, (x/y) (c/d) = ½ (½ ) = ½ – ½ = 0 59 For every rational number x, x 1 = x SolutionFalse Let x = 3 Then, 3 1 = 4 3 ≠ 4 So, it is clear that x 1æ ú ú e G q $ T ¨ ñ 27 2 &) ² O Í *!
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