2. Use the Divergence Theorem to calculate the flux of F = (y– x)i+(z– y)j+(y– x)k across the boundary of the region D where D is the cube bounded by the planes r = ±1, y = ±1 and z = ±1 by performing the following steps (a) Calculate ▼ · F SI. (b) Evaluate V · F dV

Let F(x, y, z) = (14xyz + 7sin(x))i + 7x²zj + 7x²yk. Find a function f such that F = Vf. f(x, y, z) =

Find a linearization L(x, y, z) of f(x, y, z) = x²y + 4z at (1,−1,2).

Consider the following approximations for a function f(x, y) centered at (0, 0). Linear approximation: P,(x, y) = f(0, 0) + f,(0, 0)x + f,(0, 0)y Quadratic approximation: P2(x, v) = F(0, 0) + f,(0, 0)x + f,(0, 0)y +ica(0, 0)x² + f„(0, 0)xy + fy,(0, 0)y² [Note that the linear approximation is the tangent plane to the surface at (0, 0, f(0, 0)).] (a) Find the linear approximation of f(x, y) = ex -) centered at (o, 0). P;(0, 0) = (b) Find the quadratic approximation of f(x, y) = ex - y) centered at (0, 0). P2(0, 0) = (c) If x = 0 in the quadratic approximation, you obtain the second-degree Taylor polynomial for what function? O eY O ex O e-Y Answer the same question for y = 0. O e