fitch2 - given by f at this point. 4. (14 points) Find lim...

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NAME__________________________________ Math 22 10:30 Test 2 Fall 2000 You have 50 minutes to complete this test. You must show all work to receive full credit. Work 7 out of the 8 problems, and clearly cross out the one you don’t want me to grade. If you do not, I will grade only the first 7 whether you worked them or not. Each problem is worth 14 points, and you get two points for free (for a total of 100). Be sure to do the easier ones first! If you have any questions, please come to the front and ask. 1. (14 points) If u = x + y y + z , x = p + r + t , y = p - r - t , and z = p + r - t , find u t at the point where p = 1, r = 1, and t = 2. 2. (14 points) Find the rate of change of f x , y , z ( 29 = x 2 + y 2 + z 2 at the point 1,2, - 2 ( 29 in the direction towards the point 2,3, - 3 ( 29 . 3. (14 points) Sketch a contour map for the surface given by f x , y ( 29 = x 2 + y 2 - 1 , showing at least 3 level curves, including the one containing the point where x = 1 and y = 2. Be sure to label your level curves. On your graph, also sketch (approximately) f 1,2 ( 29 . Then find the equation of the plane tangent to the surface
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Unformatted text preview: given by f at this point. 4. (14 points) Find lim x , y ( 29 0,0 ( 29 xy 3 x 2 + y 6 , or explain in detail why this limit does not exist. 5. (14 points) Find and classify all critical points of f x , y ( 29 = -x 4 + 4 xy 3-2 y 2 + 1. 6. (14 points) Use differentials to approximate the length of the hypotenuse of a right triangle whose other two sides are lengths 4.2 and 2.9. 7. (14 points) Suppose the sum of the lengths of all the edges of a rectangular box is given to be a constant, say p . Use Lagrange multipliers to prove (not just guess) that in order to have the largest possible volume, this box must actually be a cube. What is its maximum volume? 8. (14 points) Find the absolute extrema of f x , y ( 29 = 2 x 3 + y 4 over the square region whose boundary connects the points 1,1 ( 29 , 1,-1 ( 29 ,-1,-1 ( 29 ,and-1,1 ( 29 in the xy-plane....
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fitch2 - given by f at this point. 4. (14 points) Find lim...

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