Exam 2 Fall 2007

# Exam 2 Fall 2007 - Math 318 Exam 2 November 9 2007 Show all...

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• rivera.alexander
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Math 318, Exam # 2, November 9, 2007 Show all your work for full credit! 50 points maximum. 1. (10) (a) Find the maximum and minimum value of x 2 +2 y subject to x 2 +4 y 2 = 8. Use the method of Lagrange multipliers. (b) Find the maximum and minimum value of 2. (8) Let g : R 2 R 2 be defined by g ( u, v ) = ( u 2 +2 uv, v ) and let f : R 3 R 2 be differentiable and satisfy f (1 , 2 , 3) = (4 , 5) and f (1 , 2 , 3) = 1 0 2 - 1 - 1 0 . What is h (1 , 2 , 3)? Here h = g f , the composition of g and f . 3.(6) True or False? No partial credit. (a) If a symmetric 2 × 2 matrix A is negative definite, then det A < 0. (b) If a differentiable function f : R n R has an absolute maximum, then it will be attained at a point where f has a local extremum. 4. (8) Consider the curve γ of intersection of the two level surfaces x 2 + 3 y 2 + 2 z 2 = 6 x 2 + y 2 - z 2 = 1 . (a) Use implicit differentiation to determine

Unformatted text preview: ∂x ∂z and ∂y ∂z at (1 , 1 , 1). (b) Use the results from (a) to determine a tangent vector to the curve γ at (1 , 1 , 1). 5. (5) Consider the matrix A = 2-1-1 1 3 Is A positive-, negative- or indeﬁnite? 6. (8) Let f ( x,y ) = x 3 y + 4 y 2 and let-→ u be a unit vector in the direction of (1 , 2). (a) Calculate the Hessian H f (1 ,-1) (b) Calculate ∂f ∂-→ u (1 ,-1) and ∂ 2 f ∂-→ u 2 (1 ,-1). 7. (5) Suppose f : R 2 → R is a diﬀerentiable function that satisﬁes f (0 , 0) = 1 and f ( x,y ) ≥ 5 for all ( x,y ) ∈ R 2 with x 2 + y 2 &gt; 3. Explain why f must attain its minimum value. Typeset by A M S-T E X 1...
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• Spring '14
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• Optimization, differentiable function, minimum value

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