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1. Find the indicated derivatives: a) u = x+y y+z , x = p + r + t, y = p − r + t, z = p + r − t; ∂u ∂r . b) y 5 + x 2 y 3 = 1 + yex 2 ; dy dx . c) ln(x + yz) = 1 + xy2 z 3 ; ∂z/∂y 2. Let f(x, y) = x 2 y + x 3 y 2 and suppose you dont know what φ(t) = (x(t), y(t)) is, but you know φ(2) = (1, 1), dx dt (2) = 3, and dy dt (2) = 1. Find the derivative of f(φ(t)) when t = 2. 3. Show that the following functions are functionally dependent and find a relation connecting them: f(x, y, z) = x + y + z, g(x, y, z) = x 2 + y 2 + z 2 , h(x, y, z) = xy + yz + xz 4. Find the local maxima, minima, and saddles of the functions h(x, y) = (2x−x 2 )(2y −y 2 ). 5. Find the largest volume of a box with an open top, and surface area 100m2 . 6. Find the absolute minimum of f(x, y) = x 2 + y 2 + 2y − 1 on D = {(x, y)|x 2 + y 2 4 ≤ 1}
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