Math241_Fall08_2PM_Exam2 - Math 241 Exam 2 2PM V1 55 points...

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Math 241 – Exam 2 – 2PM V1 October 20, 2008 55 points possible 1. Let F ( x, y, z ) = ( x + e z + y, yx 2 ). (a) Find the best linear approximation of f ( x, y, z ) at ~x = (1 , 1 , 0). (b) Explain what we mean when we say that the linearization in part (a) is the best linear approximation at ~x = (1 , 1 , 0). 2. (a) Let f and g be two scalar-valued functions. State the Product Rule for ( fg ). (b) Let f ( x, y, z ) = xy + e z and g ( x, y, z ) = y 2 sin ( z ). Compute ( fg )( x, y, z ). 3. Let f ( x, y, z ) = xe y + ye z + ze x . (a) Find the directional derivative of f at the point (1 , 1 , 1) in the direction of v = (5 , 1 , - 2). (b) In what direction does f have the maximum rate of change? What is this maximum? 4. Let F ( x, y, z ) = y 2 z 3 i + 2 xyz 3 j + 3 xy 2 z 2 k . (a) Find ∇ × F . (b) Show that F is a gradient field by finding a potential function f . 5. Let F ( x, y ) = (cos y + x 2 , e x + y ) and G ( u, v ) = ( e u 2 , u - sin v ). (a) Write the formula for F G in terms of ( u, v ). (b) Calculate D ( F G )(0 , 0) using the multivariable Chain Rule. (Hint: Don’t use your answer from part (a).) 6. Let f ( x, y ) = ( x 4 + y 4 ) 1 / 3 (a)
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