14.6-14.7solns

14.6-14.7solns - (a) y = x 2-z 2 at (4 , 7 , 3). (b) x-z =...

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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 . Answers: (a) 0 (b) 0 (c) 35 15 (d) - 1 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 Answers: (a) (2 , - 4) is a saddle point (b) (0 , 2) is a max. (c) (1 , 1) is a min. (d) (0 , 1) , ( π, 1) , (2 π, 1) are mins and ( π/ 2 , 0) , (3 π/ 2 , 0) are saddle points. 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) . Answers: (a) h 1 , 1 i , 2. (b) h 4 / 5 , 2 / 5 i , 2 5 (c) h 1 , 3 , 36 i , 3420 (d) h 1 / 2 , - 1 / 2 i , 1 2 4. Find the points on the surface y 2 = 9 + xz that are closest to the origin. Answer: (0 , ± 3 , 0). 5. Find the equations for the tangent plane and the normal line to the given surface at the given point.
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Unformatted text preview: (a) y = x 2-z 2 at (4 , 7 , 3). (b) x-z = 4 arctan( yz ) at (1 + π, 1 , 1). 1 (c) yz = ln( x + z ) at (0 , , 1). Answers: (a) Tangent plane: 8 x-y-6 z = 7, normal line: x-4 8 = y-7-1 = z-3-6 . (b) Tangent plane: x-2 y-3 z =-4 + π , normal line: x-1-π = y-1-2 = z-1-3 . (c) Tangent plane: x-y + z = 1, normal line:-x = y = 1-z . 6. Find the point on the plane x + 2 y-z = 5 which is closest to the point (1 , 2 , 3). Answer: (3 / 2 , 3 , 5 / 2). 7. Show that any plane that is tangent to the cone x 2 + y 2 = z 2 passes through the origin. Answer: Has normal vector h 2 x, 2 y, 2 z i . Work from there. 2...
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This homework help was uploaded on 04/17/2008 for the course MATH 231 taught by Professor Bociu during the Spring '08 term at UVA.

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14.6-14.7solns - (a) y = x 2-z 2 at (4 , 7 , 3). (b) x-z =...

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