Let f:R2→R be defined by f(x,y) =|xy|e−(x2+y2). Evaluate
∫R2
f, if it exists


Let f:R2→R be defined by f(x,y) =|xy|e−(x2+y2). Evaluate ∫R2f, if it exists Let f : R2...
Let f : R2 + R be defined by f(x,y) = |xy|e-(z?+y?). Evaluate SR2 f, if it exists
5. Let f : R2 + R be defined by f(x,y) = xyle=(x2+zº). Evaluate Sr2 f, if it exists (8 points).
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2 If(x,y)| 〈 M(x2 + y2)· for all (a·y) E R2 Prove that:
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2...
1. Consider the unit circle: (x,y) : x2 y2 = 1. T. Let f R2 ->R be defined by f(x,y) = x2-y, and let F : R2 -> R be defined by F(x, y) Compute the integral of f and F around the unit circle. For the integral of F, proceed in the standard (anticlockwise) direction
Let f: R2 + R be defined by f(x, y) = xyle=(22+y?). Evaluate Sr2 f, if it exists
Let f : R2 + R be defined by f(L,y) = |kyle=(2²+y?). Evaluate /ik2 f, if it exists
2. Let f:R2 + R be defined by gry, if (x, y) + (0,0) f(x,y) := { x2 + y2 + 1 0 if (x, y) = (0,0). Show that OL (0, y) = 0 for all y E R and f(x,0) = x for all x E R. Prove that bebu (0,0) + (0,0).
Let f: R2 + R be defined by f(x, y) = xyle=(22+v?). Evaluate Sg2 f, if it exists
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 : z? + y2 < 4} of radius 2. (a) Find the maximu and minimu of Duf at (0,0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk(,y) E R2 r2 + y2 < 4} and all unit vectors u. (llint. Think of IvJF as a function ofェand y in the disk.)
Let f(x, 2) Va r], (x1, X2) E R2. Determine all directions E R2 along (0,0) exists. which
Let f(x, 2) Va r], (x1, X2) E R2. Determine all directions E R2 along (0,0) exists. which