{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

08spr-f

# 08spr-f - Math 51 Final Exam June 6 2008 Name Section...

This preview shows pages 1–5. Sign up to view the full content.

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

View Full Document
Math 51, Spring 2008 Final Exam — June 6, 2008 Page 2 of 16 1. (16 points) Let f ( x, y ) = x 2 y - 4 xy + 1 2 y 2 + 1 . (a) Calculate formulas for the gradient of f and the Hessian matrix of f at the point ( x, y ). (b) Find all critical points of f . (c) For each critical point, determine if it corresponds to a local maximum, local minimum or saddle point of f . Show your reasoning. (d) Write the quadratic approximation (that is, the degree-2 Taylor polynomial) for f at the point ( x, y ) = (0 , 0).
Math 51, Spring 2008 Final Exam — June 6, 2008 Page 3 of 16 2. (12 points) Suppose S is the surface in R 3 given by the equation xz 3 + yz 2 + x 2 y = 18. (a) Find an equation of the plane tangent to S at (1 , 2 , 2). (b) Using linear approximations, estimate the z -coordinate of the point on the surface S that has x = 1 . 1 and y = 1 . 96.

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

View Full Document
Math 51, Spring 2008 Final Exam — June 6, 2008 Page 4 of 16 3. (16 points) Let f ( x, y ) be a function on R 2 . Suppose that f (1 ,
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}