math16B_review_exer1_ans

math16B_review_exer1_ans - f , rather than minimizes f ,...

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Math 16B Spring 2011 Sarason REVIEW EXERCISES 1 - ANSWERS AND COMMENTS 1. ∂x (tan( x 2 y )) = sec 2 ( x 2 y ) ∂x ( x 2 y ) = 2 xy sec 2 ( x 2 y ) ∂y (tan( x 2 y )) = sec 2 ( x 2 y ) ∂y ( x 2 y ) = x 2 sec 2 ( x 2 y ) ∂x (ln(tan( x 2 y )) = 1 tan( x 2 y ) ∂x (tan( x 2 y )) = 2 xy sec 2 ( x 2 y ) tan( x 2 y ) (from above). The last function can be re-expressed as 2 xy/ sin( x 2 y ) cos( x 2 y ), and as 4 xy/ sin(2 x 2 y ). ∂x (ln(tan( x 2 y )) = 1 tan( x 2 y ) ∂x (tan( x 2 y )) = x 2 sec 2 ( x 2 y ) tan( x 2 y ) (from above). The last function can be re-expressed as x 2 / sin( x 2 y ) cos( x 2 y ), and as 2 x 2 / sin(2 x 2 y ). 2. I 1 = 1, I 2 = π 2 - 1. For both integrals, it is easiest to integrate first with respect to y . 3. I = 1 5 . Integration first with respect to x , then with respect to y , is simpler than the other way around. 4. (1 , 1) is a relative minimum; ( - 1 , - 1) is a relative maximum; (1 , - 1) and ( - 1 , 1) are saddle points. 5. (2 , 2) is a relative minimum; (0 , 0) is a saddle point. 6. (a) (1 , 1); (b) (1 , 1). If you interpret f as a production function and g as a cost function, it becomes intuitively clear that the point found in (a) maximizes
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Unformatted text preview: f , rather than minimizes f , under the given constraint, while in (b) the point found minimizes g , rather than maximizes g , under the given constraint. The same conclusions follow if one notes that f gets arbitrarily close to 0 in the region x > 0, y > 0, under the constraint on g , while g is unbounded in the region x > 0, y > 0, under the constraint on f . 7. The maximum is 4, attained at (0 , 1); the minimum is-4, attained at (0 ,-1). This problem is tricky because, when you nd the critical points of F ( x,y, ) = 3 x 3 + 4 y 3 + (1-x 2-y 2 ) , you end up with 6 candidates for the maximizing and minimizing points: (0 , 1), (0 ,-1), (1 , 0), (-1 , 0), ( 4 5 , 3 5 ), (-4 5 ,-3 5 ). Check your algebra if you did not come up with all 6 points....
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