exercises 6 - 4 2 3 ( , ) f x y x y x y =-decreases fastest...

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Exercises – 6 1) Let f (x,y) = y 2 – x 2 . Show that f has a saddle point at (0,0). 2) Find and classify all critical points of f(x,y) = x 4 + y 4 – 2x 2 + 4xy – 2y 2 . 3) Find the directional derivative of ( , , ) sin( ) cos( ) f x y z xz xz π = + at (1,1,1) in the direction of 2i-2j+2k. 4) Let f be a differentiable function. Answer the following questions: i ) When is the directional derivative of f is a maximum ? ii) When is it a minimum? iii) When is it 0? iv) When is it half of its maximum value? 5) Let f be differentiable with (1) 1, '(1) 2 f f = = . Let 2 y z xf x = . a) Find a vector u a so that (1,1) u D z a is a minimum. b) If 2 (1,1) 1 j i D z + = then find 1 2 f . c) If (2,2) 1 z = , find 1 ' 2 f 6) a) Show that a differentiable function f decreases most rapidly at x a in the direction opposite to the gradient vector, i.e. in the direction of ( ) f x -∇ a . b) Use the result of part (a) to find the direction in which the function
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Unformatted text preview: 4 2 3 ( , ) f x y x y x y =-decreases fastest at the point (2,-3). 7) Find the equation of the tangent plane and the normal line to the surface given by the equation sin( ) 2 3 xyz x y z = + + at the point (2,-1,0). 8) a) Find the shortest distance from the point (2,1,-1) to the plane x + y – z = 1. b) Find the point on the plane x – y + z = 4 that is closest to the point (1,2,3). c) Find the points on the surface z 2 = xy + 1 that are closest to the origin. 9) The dimensions of a closed rectangular box are measured as 80 cm , 60 cm and 50 cm with a possible error of 0,2 cm in each dimension. Use differentials to estimate the maximum error in calculating the surface area of the box. 10) Find linear approximation of the function f(x,y)=ln(x-3y) at (7,2) and use it to approximate f(6.9 , 2.06)....
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