p2 fall 2005 - f at the point ~ r (1) in terms of a and b ....

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Math 192, Prelim 2 October 27, 2005. 7:30-9:00 You are allowed one 8 . 5 × 11 sheet of paper with information on both sides. You are NOT allowed calculators, the text, or any other book or notes. SHOW ALL WORK! 1) (10pts) TRUE or FALSE Instructions: On the first page of your booklet, label five lines a), b), c), d), e). On each line, answer the corresponding question by writing either TRUE or FALSE, or leave the line blank. No abbreviations, please . A correct answer is worth 2pts, an incorrect answer -1pt, no answer 0pts. a) Let f ( x, y ) = 2 xy - x 2 + 4 y and ~v = 6 ~ i + 2 ~ j . TRUE or FALSE: At (1 , 2), ~v gives the direction of greatest increase. b) Let R be the region defined by a x b , g 1 ( x ) y g 2 ( x ). TRUE or FALSE: Z Z R f ( x, y ) dA = Z b a Z g 2 ( x ) g 1 ( x ) f ( x, y ) dydx = Z g 2 ( x ) g 1 ( x ) Z b a f ( x, y ) dxdy. c) TRUE or FALSE: The centroid of a region R must lie in the region R . 2) (12pts) The equation xy 2 + 2 xz + 4 yz 3 = 0 defines z implicitly as a function of x and y . Find the expression for z in terms of x, y and z . 3) (20pts) Let ~ r ( t ) = t 2 ~ i + 3 t ~ j be a smooth curve on the domain of the function f ( x, y ) = ax + by - xy. a) Find an expression for
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Unformatted text preview: f at the point ~ r (1) in terms of a and b . b) Find the unit tangent vector ~ T for ~ r ( t ) at t = 1. c) Find an expression for directional derivative D ~ T f at ~ r (1) in terms of a and b . d) Given that f (1 , 3) =-14 and D ~ T f (1 , 3) =- 13, calculate a and b . 4) (18pts) Consider the ellipse x 2 144 + y 2 25 = 1. Find the points on the ellipse farthest from the point (0 ,-5). 5) (16pts) Let f ( x, y ) = ( x + 2 y ) e-( x 2 +2 y 2 ) . a) Find all the critical points of f . b) Find the local maxima, local minima and saddle points of f . 6) (12pts) Let R be the region of the plane dened by 0 y 2, y 2 x 4. a) Sketch the region R . b) Compute Z 2 Z 4 y 2 y cos( x 2 ) dxdy . 7) (12pts) Let R be the region of the plane dened by | x | + | y | 1. a) Sketch the region R . b) Compute the average of the function f ( x, y ) = x + y 2 over the region R ....
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p2 fall 2005 - f at the point ~ r (1) in terms of a and b ....

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