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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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