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Unformatted text preview: Math 237 Final F07
NOTE: This exam does not necessarily reﬂect which type of questions will be asked on this semester’s ﬁnal. i.e. something covered on this test may not be covered on this semester’s ﬁnal and material not covered on this test may be covered on this year’s ﬁnal. 1. Short Answer Problems  a) State the second derivative test.  b) Does the mapping F (x, y ) = (y − x2 , y + x2 ) have an inverse transformation in a neighbourhood of (1, 2)? Justify your answer  c) Find the derivative matrix of the mapping F (x, y ) = (x2 sin y, xy 2 ). 2. Let f (x, y ) = xy −. yx  a) Find the equation of the tangent plane to the surface z = f (x, y ) at (1, 1, 0).  b) Use the 2nd degree Taylor Polynomial to approximate f (1.1, 0.9). 3. Consider the function f (x, y, z ) = ln(x + eyz ).  a) Write the deﬁnition of the directional derivative.  b) In what direction(s) does f have a rate of change of −1 at the point (0, 1, 0)?  c) Is there a direction in which f has a rate of change of 2 at the point (0, 1, 0)? Justify your answer. 4. Consider the function f (x, y ) =  a) What is the domain of f ?  b) Where is f continuous on its domain?  c) Where is f diﬀerentiable on its domain?  5. Let u = x3 f yz ∂u ∂u ∂u , . Show that x +y +z = 3u. xx ∂x ∂y ∂z State any assumptions you need to make about f .
xy 2 , x2 +2y 2 0, (x, y ) = (0, 0) (x, y ) = (0, 0).  6. Use the method of Lagrange multipliers to prove that if x2 + y 2 + z 2 = 1 then x2 y 2 z 2 ≤ 313 .  7. Find the maximum and minimum points of the function f (x, y ) = xyex+2y−2 on the triangle with vertices (0, 0), (0, 1) and (2, 0).  8. Evaluate
D z dV , where D is bounded by x = 0, x = 1, z = 0, y + z = 2, and y = z .  9. Evaluate
Dxy x2 dA, where Dxy is bounded by the ellipse 5x2 + 4xy + y 2 = 1.  10. Find the volume of the region bounded by (x2 + y 2 + z 2 )2 = x. 2 ...
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This note was uploaded on 04/18/2010 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.
- Spring '08