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2220pracprelim1

# 2220pracprelim1 - x 2 9 y 2 4 z 2 = 9 where the edges of...

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Math 222 Prelim I Spring 2008 Calculators are not permitted. Show all your work. 1. Let S be the level surface f ( x, y, z ) = x 2 + 3 y 2 + z 2 + xz = 6. (5 pts) (a) Find the tangent plane to S at the point (1 , 1 , 1). (5 pts) (b) Find all points on S where the tangent plane is horizontal (parallel to the xy -plane). (5 pts) (c) Show that the points on S where the tangent plane is vertical (parallel to a plane containing the z -axis) form an ellipse in some sloping plane. 2. Let f ( x, y ) = x 2 + y 2 2 x . (5 pts) (a) Show that the level curves of f are circles passing through the origin. (5 pts) (b) Draw a sketch showing the level curves that pass through the six points ( ± 1 , 0), ( ± 2 , 0), ( ± 3 , 0). (5 pts) (c) Using the level curves of f , determine whether lim ( x,y ) (0 , 0) f ( x, y ) exists, and give a reason for your answer. (20 pts) 3. Find all critical points of f ( x, y ) = sin x cos y in the range π < x < π and π < y < π (note that these are strict inequalities) and determine whether each critical point is a local minimum, local maximum, or saddle. (20 pts) 4. Find the maximal volume of a rectangular box that is contained inside the ellipsoid
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Unformatted text preview: x 2 +9 y 2 +4 z 2 = 9, where the edges of the box are assumed to be parallel to the coordinate axes. (You can assume that a box of maximum volume exists.) 5. Suppose that for some di±erentiable function f ( x, y, z ) we know that the maximum value of the directional derivatives D u at the point (1 , 1 , 1) is 2, and this maximum occurs in the direction of the vector (1 , 2 , 2). (8 pts) (a) From this information, compute ∇ f (1 , 1 , 1). (7 pts) (b) Compute D u (1 , 1 , 1) in the directions of the vectors (2 , 1 , − 2) and (1 , 1 , 0). (15 pts) 6. Suppose that a di±erentiable function f ( x, y ) has ∂f ∂x (5 , 3) = 4 and ∂f ∂y (5 , 3) = 6. Suppose also that x and y are related to variables u and v by x = u 2 + v 2 and y = u 2 − v 2 . Compute ∂f ∂u and ∂f ∂v at ( u, v ) = (2 , 1)....
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