# B find equation of the tangent plane to the surface x

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(b) Find equation of the tangent plane to the surface x 2 + 2 y - xz 2 = 4 at the point (1 , 2 , 1).

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4 FALL 2012 — MA 227 — FINAL EXAM SATURDAY, DECEMBER 08, 2012 4. Find the local maximum, minimum and saddle points (if any) of the function f ( x, y ) = x 2 - 4 xy + y 2 - 2 y + 2 .
FALL 2012 — MA 227 — FINAL EXAM SATURDAY, DECEMBER 08, 2012 5 5. (a) Find the linear approximation for the function f ( x, y ) = ye x - 2 - x 2 y 3 near the point (2 , 1). (b) Let f ( x, y ) = x 2 y - e y and x = s - t, y = s 2 t . Find the partial derivatives ∂f/∂s and ∂f/∂t . You don’t need to simplify your answer!

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6 FALL 2012 — MA 227 — FINAL EXAM SATURDAY, DECEMBER 08, 2012 6. Find the absolute maximum and absolute minimum points of the function f ( x, y ) = x 2 - y 2 + y on the region 0 x 1 , - 1 y 1. Be sure to provide the coordinates of the points and the values of absolute maximum and minimum.
FALL 2012 — MA 227 — FINAL EXAM SATURDAY, DECEMBER 08, 2012 7 7. Evaluate the integral ZZ D ( x - y ) 3 sin( x + 2 y ) dA where D is the parallelogram enclosed by the lines x - y = 0, x - y = 1, x + 2 y = 0, and x + 2 y = π 4 . Use the change of the variables u = x - y , v = x + 2 y .

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