math16B_review_exer1_ans

# math16B_review_exer1_ans - f rather than minimizes f under...

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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 ﬁnd the critical points of F ( x,y,λ ) = 3 x 3 + 4 y 3 + λ (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....
View Full Document

## This note was uploaded on 04/16/2011 for the course MATH 16B taught by Professor Sarason during the Spring '06 term at Berkeley.

Ask a homework question - tutors are online