# 5 let f x y z x 2 3 xy 4 y 2 e z 9 a find a point x y

This preview shows pages 2–3. Sign up to view the full content.

5. Let f ( x, y, z ) = x 2 + 3 xy + 4 y 2 + e z - 9. a ) Find a point ( x 0 , y 0 , z 0 ) satisfying f ( x 0 , y 0 , z 0 ) = 0 . b ) Can x be expressed as a function g ( y, z ) in some neighborhood of ( x 0 , y 0 , z 0 )? c ) Compute dg ( y 0 , z 0 ).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
MATHEMATICAL ECONOMICS MIDTERM #2, NOVEMBER 12, 2002 Page 3 Answer: a ) The point ( x 0 , y 0 , z 0 ) = (1 , 1 , 0) satisfies the equation. b ) We compute ∂f/∂x = 2 x + 3 y . Plugging in x = 1 and y = 1, we obtain 5. The derivative is invertible (not zero). The Implicit Function Theorem then yields the desired function g . In fact, it is possible to compute g ( y, z ) = [ - 3 y + p 36 - 7 y 2 - 4 e z ] / 2. c ) The derivative is given by dg (1 , 1 , 0) = - 1 5 ∂f ∂y , ∂f ∂z (1 , 1 , 0) = - 1 5 (3 x + 8 y, e z ) (1 , 1 , 0) = - 11 5 , - 1 5 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern