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s07wk06 - show that the limit doesn’t exist Problem...

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Math 23 B. Dodson Week 6 Homework: 14.2 limits 14.3 partial derivatives, 2nd order deriv Week 6 Homework: 14.2 limits Problem 14.2.6: Find the limit lim ( x,y ) (6 , 3) ( xy cos ( x - 2 y )) . Solution: We see that the function f ( x, y ) = xy cos ( x - 2 y ) is continuous. (why?) Since xy and x - 2 y are polynomials; cos x - 2 y is a composite of continuous functions; and then f is a product of continuous functions (that is, of xy and cos x - 2 y ). So the limit is f (6 , 3) = 6 · 3 cos 6 - 6 = 18 · 1 = 18 . Problem 14.2.13: Show that the limit lim ( x,y ) (0 , 0) 2 x 2 y x 4 + y 2 does not exist.
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2 . Solution: Along the x -axis ( x, 0) the limit is lim ( x, 0) (0 , 0) 2 x 2 y x 4 + y 2 = lim x 0 0 x 4 = 0 , recalling that x = 0 while x 0 . But along the parabola y = x 2 , so ( x, x 2 ), lim ( x,x 2 ) (0 , 0) 2 x 2 y x 4 + y 2 = lim x 0 2 x 4 2 x 4 = 1 , again using that x = 0 while x 0 . Since we have two different limits among points with ( x, y ) (0 , 0) , the limit does not exist. Note that the limit along lines (0 , y ) (the y -axis) gives 0 y 2 0 and ( x, mx ) (with y = mx, m = 0) gives 2 mx 3 x 4 + m 2 x 2 2 mx x 2 + m 2 0 m 2 = 0 , (since m = 0) . So the limits along all lines approach 0, and we needed the parabola to get a non-zero value and
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Unformatted text preview: show that the limit doesn’t exist. Problem 14.3.15: Find the partial derivatives of the function z = f ( x, y ) = xe 3 y . Find f x (2 , 1) . 3 Solution: For f x we (temporarily) hold y constant, so e 3 y constant, giving f as constant · x . We then take the derivative in x , with (constant · x )’ = constant, or f x = e 3 y . Likewise, with x constant, f y = x ∂ ∂y ( e 3 y ) = x d dy ( e 3 y ) = 3 xe 3 y . Finally, f x (2 , 1) = e 3 gives the rate of change of f at (2 , 1) with respect to x ; which we may constrast with f y (2 , 1) = 6 e 3 , the rate of change of f at (2 , 1) with respect to y . For example, f is growing six times more rapidly in y than in x . We also solved #49, 14.3, and in particular verified that the second partials z xy = z yx ....
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