MIDTERM2B - MATH 2401, FAll 2009 Midterm II, Solutions...

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Unformatted text preview: MATH 2401, FAll 2009 Midterm II, Solutions Problem 1 .(40 pt) Calculations. (a)(10 pt) Find the directional derivative of f ( x, y, z ) = x 2 + 2 xyz- yz 2 at P (1 , 1 , 2) in the direction of 2 i + j- 3 k Solution: ∇ f = (2 x + 2 yz ) i + (2 xz- z 2 ) j + (2 xy- 2 yz ) k , ∇ f (1 , 1 , 2) = 6 i- 2 k . Normalize the vector: u = 1 √ 14 (2 i + j- 3 k ) . f u (1 , 1 , 2) = ∇ f (1 , 1 , 2) • u = 18 √ 14 . (b)(10 pt) Compute df dt for f ( x, y, z ) = x 2 + y 2 along r ( t ) = a cos (2 t ) i + b sin (2 t ) j + 2 bt k . Solution: ∇ f = 2 x i + 2 y j , ∇ f ( r ( t )) = 2 a cos (2 t ) i + 2 b sin (2 t ) j , r ( t ) =- 2 asin (2 t ) i + 2 bcos (2 t ) j + 2 b k , df dt = ∇ f ( r ( t )) • r ( t ) = 4( b 2- a 2 ) sin (2 t ) cos (2 t ) . (c)(10 pt) Find dy dx if x 2 / 3 + y 2 / 3 = 1 at the point (1 , 0). Solution: Set u ( x, y ) = x 2 / 3 + y 2 / 3- 1 = 0. ∂u ∂x = 2 3 x- 1 / 3 ....
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This note was uploaded on 11/07/2009 for the course MATH 2401 taught by Professor Morley during the Spring '08 term at Georgia Tech.

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MIDTERM2B - MATH 2401, FAll 2009 Midterm II, Solutions...

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