test1 - SOLUTIONS FOR TEST #1. 1(a) Suppose l is the...

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Unformatted text preview: SOLUTIONS FOR TEST #1. 1(a) Suppose l is the tangent line to the curve C : g ( t ) = ( t,t 2 ,t 3 ) at (1 , 1 , 1) , and is the plane tangent to the surface xy 2 + 2 x 2 yz + z 3 =- 1 at (- 1 , ,- 1) . Find the equation of the line through (1 , 1 , 1) that is parallel to the plane and orthogonal to the line l. g ( t ) = (1 , 2 t, 3 t 2 ); The point (1 , 1 , 1) corresponds to t = 1 , so the vector g (1) = (1 , 2 , 3) is tangent to the curve g ( t ) at the point (1 , 1 , 1) . O f = ( y 2 + 4 xyz, 2 xy + 2 x 2 z, 2 x 2 y + 3 z 2 ) , where f = xy 2 +2 x 2 yz + z 3 +1 is the function that de nes the surface. A line is parallel to the tangent plane if and only if it is orthogonal to the gradient. Plugging the point (- 1 , ,- 1) we get the vector O f | (- 1 , ,- 1) = (0 ,- 2 , 3) . Suppose that our line is given by (1 + v 1 t, 1 + v 2 t, 1 + v 3 t ) . Then the vector v = ( v 1 ,v 2 ,v 3 ) should be orthogonal to the vectors (1 , 2 , 3) and (0 ,- 2 , 3) . Since the cross-product is orthogonal to the factors, it is enough to take v = (1 , 2 , 3) (0 ,- 2 , 3) = fl fl fl fl fl fl i j k- 2 3 1 2 3 fl fl fl fl fl fl = (- 12 , 3 , 2) . Therefore the line is given by (1- 12 t, 1 + 3 t, 1 + 2 t ) . (b) Find a real valued function F ( x,y,z ) whose 0-level set S is the image of the map f : R 2 R 3 de ned by f ( u,v ) = ( u- v,u + v, 1- 2 u ) . The map f is a parametrization of the plane: x = u- v, y = u + v, z = 1- 2 u. Adding all three equations we get x + y + z = 1 . Therefore, the image of f is 0-level set of the function F ( x,y,z ) = 1- x- y- z. (So, the surface is given implicitly by the equation F ( x,y,z ) = 0 , or 1- x- y- z = 0 . ) 1 2(a) De ne the function f : R 2 R by f ( x,y ) = y ln( x 2 + y 2 ) for ( x,y ) 6 = (0 , 0) for ( x,y ) = (0 , 0) . Let u = ( a,b ) be a unit vector. Write, using the de nition (as a limit!), the directional derivative of f in the direction of u at the point (0 , 0) . In what direction(s), if any, does u f exists? Is f of class C 1 on R 2 ? Justify your answer with virtually no calculations....
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test1 - SOLUTIONS FOR TEST #1. 1(a) Suppose l is the...

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