235-08-Samples-t2 - UNIVERSITY OF TORONTO DEPARTMENT OF...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT 235 Y – CALCULUS II FALL-WINTER 2008-2009 SAMPLE QUESTIONS FROM PREVIOUS YEARS TERM TEST #2 1. a) Consider the function 42 2 (,) 3 2 x y fxy x y + = + . Does (,) ( 0 , 0 ) lim ( , ) xy f xy exist? If the limit exists, then evaluate it. If the limit does not exist, then explain why. b) Consider the function 8 3 8 gxy x y =+ 3 . Does (0,0) g y exist? If the partial derivative exists, then evaluate it. If it does not exist, then explain why. 2. a) Find z / x and z / y if x y sin ( y + z ) + e x z = 1 . b) Let f ( x , y ) = ( x + 3 y ) 4 , where x = u 2 v and y = 2 u v . Use the Chain Rule to evaluate f / v when u = – 1 and v = 2 . 3. a) Use the linear approximation of the function f ( x , y ) = e x – 2 y at the point ( 2 , 1 ) to approximate the value of f ( 2.01 , 1.02 ) . b) Let f ( x , y , z ) = x 2 y y 3 z + z 2 . Find the maximum possible value of the directional derivative D u f ( x , y , z ) at the point P ( 1 , 1 , 1 ) . 4. Let u = ( x 2 + y 2 ) 3 / 2 . Find the value of the constant k for which u x x + u y y = k u 1 / 3 . 5. a) Find all critical points of the function f ( x , y ) = x 3 – 15 x + 3 x y 2 + y 3 , and use the second derivative test to classify each of these critical points as a local minimum, a local maximum or a saddle point. b) Find the minimum value of the function 122 (,,) fxyz x yz = ++ , where x > 0 , y > 0 , z > 0 , subject to the constraint 4 x 2 + y 2 + 8 z 2 = 1 . 6. a) Find the volume of the solid that lies under the surface z = 1 + x + x y and above the triangular region with vertices ( 0 , 0 ) , ( 1 , 1 ) , and ( 1 , 2 ) in the xy -plane. b) Evaluate the iterated integral 2 5 11 4 0 8 1 y y dxdy x + ∫∫ . 7. Let S denote the surface given by the equation x 2 + y 2 + 2 z 2 x y + x z + y z + z = 6 . a) Find an equation of the plane that is tangent to the surface S at the point ( 1 , 1 , 1 ) . b) At what points of the surface S is the tangent plane parallel to the xy -plane ?
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Page 2 8. a) Consider the function 1 (,) u f xt e t = , where 2 2 () x b u at = , a 0 , t > 0 .
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This note was uploaded on 10/16/2009 for the course MATH MAT235 taught by Professor Recio during the Fall '08 term at University of Toronto.

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235-08-Samples-t2 - UNIVERSITY OF TORONTO DEPARTMENT OF...

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