14.6-14.7

14.6-14.7 - y 2 = 9 + xz that are closest to the origin. 5....

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MATH 231 Group Work Quiz Related 1. Find the directional derivative of f at the given point in the direction indicated by the angle θ or in the direction of the vector v . (a) f ( x,y ) = x 2 + y 2 , ( 1 2 , 1 2 ), θ = 3 π 4 . (b) f ( x,y ) = ln( x 2 + y 2 ), (2 , 1), v = h- 1 , 2 i . (c) f ( x,y ) = ( x + 3 y + 9 z 2 ) 3 / 2 , (1 , 1 , 2), v = 2 i - j + k . (d) f ( x,y ) = x - y x + y , (1 , 1), θ = π 2 . 2. Find the local maximum, minimum, and saddle points of the following functions: (a) f ( x,y ) = x 3 y + 12 x 2 - 8 y (b) f ( x,y ) = e 4 y - x 2 - y 2 (c) f ( x,y ) = xy + 1 x + 1 y (d) f ( x,y ) = y 2 - 2 y cos x 1 x 7 3. Find the maximum rate of change of f at the given point and the direction in which it occurs for the functions and points given in (1) . 4. Find the points on the surface
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Unformatted text preview: y 2 = 9 + xz that are closest to the origin. 5. Find the equations for the tangent plane and the normal line to the given surface at the given point. (a) y = x 2-z 2 at (4 , 7 , 3). (b) x-z = 4 arctan( yz ) at (1 + , 1 , 1). (c) yz = ln( x + z ) at (0 , , 1). 6. Find the point on the plane x + 2 y-z = 5 which is closest to the point (1 , 2 , 3). 7. Show that any plane that is tangent to the cone x 2 + y 2 = z 2 passes through the origin. 1...
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