PracticeExam1-soln

PracticeExam1-soln - Solutions to practice final1 1(a)....

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Unformatted text preview: Solutions to practice final1 1(a). Answer : Apply two path test. Let y = mx 2 , m = 1 and m 0. Then lim ( x, y ) (0 , 0) xy 1 / 2 x 2 y = lim ( x, y ) (0 , 0) x mx x 2 mx 2 = m 1 m . If m = 4, Limit = 2 3 . If m = 9, Limit = 3 8 . So by two path test, the limit does not exist . 1(b). Answer : F ( x ) = f ( e x 2 )( e x 2 ) = f ( e x 2 ) e x 2 2 x. Then F (1) = f ( e 1 ) e 2 = 4 e 2 = 8 e . 1(c). Answer : (A) describes an elliptic paraboloid. For 0, the equation of level curve is x 2 4 + y 2 9 = 0, it is just the point (0 , 0). For 1, the equation of level curve is x 2 4 + y 2 9 = 1, it is a ellipse. There is no level curve corresponding 1. 1(d). Answer : Let u = x 2 , then du = 2 x dx . Then xe x 2 dx = 1 2 e u du = 1 2 e u + C = 1 2 e x 2 + C . Thus xe x 2 dx = lim b b xe x 2 dx = lim b 1 2 e x 2 | b = lim b 1 2 [ e b 2 e ] = 1 2 ....
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PracticeExam1-soln - Solutions to practice final1 1(a)....

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