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20cMidterm2 - (3 All parts of this problem refer to the...

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Math 20C Test 2B 100 points November 24, 2008 --------------------------------------------------------------------------------------------------- Directions: Justify your answers and show all work. The four parts of Problem (3) are worth 13 pts each, and the other three problems are worth 16 pts each. --------------------------------------------------------------------------------------------------- (1) Let R denote that region inside the circle x 2 + y 2 = 2 which lies above the x-axis and below the line y = x. Find the x-coordinate of R’s center of gravity. Hint : First explain why outer integral goes from y = 0 to y = 1. ---------------------------------------------------------------------------------------------------------- (2) The height of a box, added to the perimeter of its base, gives a total of 12 feet. Find the largest possible volume of such a box. ---------------------------------------------------------------------------------------------------------
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Unformatted text preview: (3) All parts of this problem refer to the point P = (1, 3, -1) on the mountainous surface z = f(x,y) defined (implicitly) by 8x 2 + zy 2 + z 4 = 0 . (A) Write down the equation of the tangent plane to the surface at P. (B) Explain why the gradient of f at the point P is the vector (-16/5, 6/5) . (C) Find the slope (directional derivative) of f at the point P in the southeast direction. (D) Find the absolute value of the slope at P of the path of a creek that flows freely on the mountainous surface through the point P. Explain briefly.------------------------------------------------------------------------------------------------(4) Consider the point G = (1, 5, -1) on the plane x + 2y + 3z = 8. Find another point E on this plane such that the line EG is steepest possible (i.e., the line EG is closest to vertical)....
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