Math 6A W2017 Practice Final - Practice Problems Math 6A...

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Practice Problems Math 6A Winter 2017 1. (a) Write down the equation to find the point on the surface ( x + y ) z 2 = 1 in the first octant, closest to the origin, using the method of Lagrange multiplies. (b) Solve these equations to find the closest point. 2. Let f ( x, y ) = x 3 - 3 x 2 - 3 y 2 + 3 xy 2 . (a) Find all the critical points of f . (b) Use the second derivative test to determine the type of all the critical points. 3. Let D be the disk: x 2 + y 2 1. Let f ( x, y ) = x 2 + 2 y 2 . (a) Find the maximum and minimum values of the function f on the boundary of D , i.e., the circle x 2 + y 2 = 1. (b) Find the global maximum and global minimum values of f on the disk D . 4. Let ~ c be the path ~ c ( t ) = (cos t, sin t, t ) , 0 t 2 π. (a) Compute the length of the curve ~ c . (b) Let f ( x, y, z ) = y sin z . Compute the path integral Z ~ c f ds of the function f along the path ~ c . 5. Let ~ F be the vector field ~ F ( x, y, z ) = ( y + z, x, x ) . (a) Compute div( ~ F ). (b) Compute curl( ~ F ). (c) Find a function f ( x, y, z ) such that ~ F = f and f (0 , 0 , 0) = 0.
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