Tutorial4_sol

Tutorial4_sol - MATH1211/10-11(1)/Tu4sol s Department of...

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Unformatted text preview: MATH1211/10-11(1)/Tu4sol s Department of Mathematics, The University of Hong Kong MATH1211 Multivariable Calculus (2010–2011, First semester) Brief Solution to Tutorial 4 1. (a) f ( x,y ) = xe 2 y + x 3 , which is differentiable. ∇ f = ( e 2 y +3 x 2 , 2 xe 2 y ), so ∇ f (1 , 0) = (4 , 2). The unit vector in the direction of u is v = u / k u k = (1 ,- 1) / √ 2. Hence the directional derivative of f at (1 , 0) in the direction of u is D v f (1 , 0) = ∇ f (1 , 0) · v = (4 , 2) · (1 ,- 1) / √ 2 = 2 / √ 2 = √ 2. (b) f will increase fastest in the direction of ∇ f (1 , 0) = (4 , 2), and this fastest rate of increase is equal to k∇ f (1 , 0) k = k (4 , 2) k = √ 20 = 2 √ 5 units. 2. Define f ( x,y,z ) = 2 xz + yz- x 2 y . The surface is then given by the level set { ( x,y,z ) | f ( x,y,z ) =- 10 } . ∇ f ( x,y,z ) = (2 z- 2 xy, z- x 2 , 2 x + y ) and ∇ f (1 ,- 5 , 5) = (20 , 4 ,- 3). An equation of the tangent plane is given by ∇ f (1 ,- 5 , 5) · ( x- 1 ,y +5 ,z...
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