mid2bpractice

mid2bpractice - z = e 3 x 2 y y = ln(3 u-w x = u 2 v...

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PRACTICE PROBLEMS FOR MIDTERM II Problem 1. Find the critical points of the function f ( x,y ) = 2 x 3 - 3 x 2 y - 12 x 2 - 3 y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Problem 2. Determine the global max and min of the function f ( x,y ) = x 2 - 2 x + 2 y 2 - 2 y + 2 xy over the compact region - 1 x 1 , 0 y 2 . Problem 3. Using Lagrange multipliers, optimize the function f ( x,y ) = x 2 + ( y + 1) 2 subject to the constraint 2 x 2 + ( y - 1) 2 18 . Problem 4. Consider the function w = e x 2 y where x = u v, y = 1 uv 2 . Using the chain rule, compute the derivatives ∂w ∂u , ∂w ∂v . Problem 5. (i) Compute the degree n Taylor polynomial of the function f ( x,y ) = e x 2 - y around (0 , 1) . (ii) The second degree Taylor polynomial of a certain function f ( x,y ) around (0 , 1) equals 1 - 4 x 2 - 2( y - 1) 2 + 3 x ( y - 1) . Can the point (0 , 1) be a local minimum for f ? How about a local maximum? Problem 6. Suppose that
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Unformatted text preview: z = e 3 x +2 y , y = ln(3 u-w ) , x = u + 2 v. Calculate ∂z ∂v , ∂z ∂w . Problem 7. Find the point on the plane 2 x + 3 y + 4 z = 29 that is closest to the origin. 1 Problem 8. Calculate the derivative of the function f : U → Mat n × n , A → ( I + A 2 )-2 defined on the open subset where I + A 2 is invertible. Problem 9. Show that the function f ( x,y ) = ( x 3 x 2 + y 2 if ( x,y ) 6 = (0 , 0) if ( x,y ) = (0 , 0) admits all directional derivatives, it is continuous but it is not differentiable. Problem 10. Show that a differentiable function f : R n → R m is continuous. Problem 11. Find the minimal distance between two points on the ellipse x 2 + 3 y 2 = 9 and the circle x 2 + y 2 = 1....
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This note was uploaded on 03/28/2011 for the course MATH 31B taught by Professor Dragosoprea during the Winter '11 term at UCSD.

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mid2bpractice - z = e 3 x 2 y y = ln(3 u-w x = u 2 v...

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