ASSIGNMENT 4 1. [4 marks] Find the natural domains of the functions below. Determine where the function f is continuous. Justify your answer using the properties of continuous functions. (a) f(x; y) = (x ?? y)1=2: (b) f(x; y) = cos( xy 1 + x2 + y2 ). 2. [4 marks] Find the limit, if it exists. (a) lim (x;y)!(0;0) x3 + 8y3 x + 2y (b) lim (x;y)!(0;0) ln(1 + x2 + y2) x2 + y2 . 3. [2 marks] (a) Show that the value of x3y 2×6 + y2 approaches 0 as (x; y) ! (0; 0) along any straight line y = mx, or along any parabola y = kx2. (b) Show that lim (x;y)!(0;0) x3y 2×6 + y2 does not exist. [Hint: Let (x; y) ! (0; 0) along the curve y = x3, and then compare the result with the results in (a).] 4. [3 marks] Find fx; fy and fz: f(x; y; z) = z ln(x2y cos z). 5. [4 marks] (a) Show that f(x; y) = ln(x2 + y2) satises @2f @x2 + @2f @y2 = 0: This is called the Laplace’s Equation. (b) Show that u(x; y) = ex cos y and v(x; y) = ex sin y satisfy @u @x = @v @y ; @u @y = ??@v @x: These are called the Cauchy-Riemann Equations. 6. [3 marks] Let f(x; y) = (x2 + y2)2=3. Show that fx(0; 0) = 0. [Hint. Use the denition fx(0; 0) = limh!0 f(h;0)??f(0;0) h .]