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Unformatted text preview: MAT 397 Final exam August 10, 2001 Show all work! Incomplete or incorrect answers may receive no credit. I. Find the equation of the plane passing through the points (1,0,3), (2,1,0) and (0,2,1). 2. Let f(x,_v,z) = xzy + vz + x23.
(a) Find the gradient of fat the point (2,3,4). (b) What is the maximum value over all possible directions of the directional derivative of f at the
point (23,!) ? (c) Find the equation of the tangent plane to the level surface f(x,_y,z) = 7 at the point (2,3,4). 3. Find the linearization of f (x, y) = 1le  y2 at (5,4) and use it to obtain an approximate value for
«5.012 — 3.982 . No credit forjust plugging in the numbers in a calculator. 4. Find and classify (local min, max or saddle) the critical points of f(x, y) = 8xy + l — 81.
y x 5. Use Lagrange multipliers to find the point on the circle x2 + y2 = 4which is closest to the point
(3,4). Hint: It is easier to work with the square of the distance. 3 {3:
6. Reverse the order of integration but do not evaluate I I dxdy. Hint: sketch the region of 4i:
2 integration 7. Find the area of the portion of the surface 2 = w/x2 + y2 that lies above the annular region in
the xy plane between the circles x2 + y2 =1 and x2 + y2 = 4. 8. Find the volume of the solid bounded by the cylinders x2 + y2 = l, x2 + y2 = 4, the plane z = 0, and the cone z = 11x2 + yz. 9. Let R be the square in the xy plane with vertices at (0,2), (1,1), (2,2) and (1,3) and let F be the
transformation given by u = x — y,v = x + y. Sketch the region F (R) in the uvplane and use the change of variables to evaluate the integral “RU — y)/(x + y)dA. ...
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 Fall '07
 Griffin

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