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final 2005 - HKUST MATH 102 Final Examination Multivariable...

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HKUST MATH 102 Final Examination Multivariable Calculus Answer ALL 8 questions Time allowed – 3 hours Problem 1 (a) If f ( x, y ) = ( x 3 + y 2 ) 1 3 , find f x (0 , 0) and f y (0 , 0). (b) Let z = f ( x, y ), where x = g ( t ) and y = h ( t ). (i) Show that d dt ∂z ∂x = 2 z ∂x 2 dx dt + 2 z ∂y∂x dy dt and d dt ∂z ∂y = 2 z ∂x ∂y dx dt + 2 z ∂y 2 dy dt . (ii) Use the formulas in part (i) to help find a formula for d 2 z dt 2 . Problem 2 (a) Find the equation of the tangent plane at the point ( - 1 , 1 , 0) to the surface x 2 - 2 y 2 + z 3 = - e - z . (b) Find the absolute extrema of the function z = f ( x, y ) = xy - 5 3 x - 3 y on the closed and bounded set R , where R is the triangular region with vertices ( - 1 , 0), ( - 1 , 4), and (5 , 0). Problem 3 Let f : R 2 R be defined by f ( x, y ) = x 2 sin 1 x + y 2 for x 6 = 0 and f (0 , y ) = y 2 . (a) Show that f is continuous at (0 , 0). (b) Find the partial derivatives of f at (0 , 0). (c) Show that f is differentiable at (0 , 0). (d) Is the function ∂f/∂x continuous everywhere in the xy -plane?
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