Let W be the three dimensional region under the graph of f(x, y) = exp[-2(x² + y)] and over the region in the plane z = 0 defined by 2 < x? + y < 5. (a) Use the divergence theorem to calculate the flux of F = (x y +2 x) i+ yj-yzk out of the region W. (b) Find the flux of F out of the part of the boundary of W for which z > 0, i.e. excluding the contribution across the boundary in the plane z = 0 defined by 2 < x² + y < 5.

a) Is the function continuous at (0,0)? b) Do the partial derivatives (D₁f)(0,0) and (D₂f)(0,0) exist? c) For which u in ℝ² \ {0} does the directional derivative (D_uf)(0,0) exist? d) Is f differentiable at (0,0)? e) Determine the partial derivatives D₁f(x,y) and D₂f(x,y) for (x,y) ≠ (0,0). f) Is f a C¹ function?HINT: use the following standerd limit: see picture.You could either solve these questions, or a detailed explanationThank you!

A function f has partial derivatives fx = 4x + y² - 6y +5, fy = 2xy - 6x. (a) Find the stationary points of f. tion (b) Classify the stationary point in the region x > 0, y > 0.

Let f be a function of two variables that has continuous partial derivatives and consider the points  A(5, 2),  B(13, 2),  C(5, 13), and D(14, 14). The directional derivative of f at A in the direction of the vector AB is 4 and the directional derivative at A in the direction of AC is 9.   Find the directional derivative of f at A in the direction of the vector  AD.