4 10 pts assume f 1 2 4 and f x y is differentiable

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______________________________________________________________________ 4. (10 pts.) Assume f (1,-2) = 4 and f ( x , y ) is differentiable at (1,-2) with f x (1,-2) = 2 and f y (1,-2) = -3. Using an appropriate local linear approximation, estimate the value of f (0.9,-1.950). f (0.9,-1.950)
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TEST3/MAC2313 Page 3 of 5 ______________________________________________________________________ 5. (10 pts.) (a) Use an appropriate form of chain rule to find z / v when z = cos( x )sin( y ) when x = u - v and y = u 2 + v 2 . z v (b) Assume that F ( x , y , z ) = 0 defines z implicitly as a function of x and y . Show that if F / z 0, then z y F / y F / z . ______________________________________________________________________ 6. (10 pts.) Locate all relative extrema and saddle points of the following function. f ( x , y ) x 4 y 4 4 xy Use the second partials test in making your classification. (Fill in the table below after you locate all the critical points.) Crit.Pt. f xx @ c.p. f yy @ c.p. f xy @ c.p. D @ c.p. Conclusion
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TEST3/MAC2313 Page 4 of 5 ______________________________________________________________________ 7. (10 pts.) (a) Evaluate the following iterated integral. ln(3) 0 ln(2) 0 e x y dydx (b) Very carefully write down, but do not evaluate , an iterated integral whose numerical value is the volume under the plane z = 2 x + y and over the rectangle R = {(x,y) : 3 x 5, 1 y 2}. V ______________________________________________________________________ 8. (10 pts.) Evaluate the integral below by first reversing the order of integration. Sketching the region is a key piece of the puzzle. 4 0 2 f8e5 y e x 3 dxdy
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TEST3/MAC2313 Page 5 of 5 ______________________________________________________________________ 9. (15 pts.) Let f ( x , y ) = x 2 - y 2 on the closed unit disk defined by x 2 + y 2 1. Find the absolute extrema and where they occur. Use Lagrange multipliers to analyze the function on the boundary.
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