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MATH 1260 - Quiz 6 Solution 2 2 3 (1) Compute √ the directional derivative of f (x, y) = x +y −3xy at the point (1, 2) in the direction u = h1/2, 3/2i. √ √ → (1, 2) = ∇f (1, 2) · u = h−22, −32i · h1/2, Df→ 3/2i = −11 − 16 3. u → (2) √Find the equation for the tangent plane to the surface x2 + 2y 2 + 3z 2 = 10 at the point (1, 3, 1).√ √ ∇f (1, 3, 1) = h2, 4 3, 6i, and the tangent plane is √ 2x + 4 3y + 6z = 20. (3) Let y be a function of x satisfying F (x, y, x + y) = 0, where F (x, y, z) is a given function. Find a formula for dy/dx. Let f : R2 → R be the function f (x, y) = F (x, y, x + y). Then we know that dy/dx = −fx /fy . Consider the function g : R2 → R3 defined by g(x, y) = (x, y, x + y). Then f is the composition of g and F , i.e., f (x, y) = F (g(x, y)). The chain rule tells us that Df = DF · Dg, and therefore 1 0 (fx fy ) = (Fx Fy Fz ) · 0 1 . 1 1 By working out the multiplication on the right we see that fx = Fx + Fz , Therefore, fy = Fy + Fz . dy fx Fx + Fz =− =− . dx fy Fy + Fz 1