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)...

ASSIGNMENT 4 1. [4 marks] Find the natural domains of the functions below. Determine where the functionf is continuous. Justify your answer using the properties of continuous functions. 1=2 (a) f(x;y) = (xy) : xy (b) f(x;y) = cos( ). 2 2 1 +x +y 2. [4 marks] Find the limit, if it exists. 3 3 2 2 x + 8y ln(1 +x +y ) (a) lim (b) lim . 2 2 (x;y)!(0;0) (x;y)!(0;0) x + 2y x +y 3 x y 3. [2 marks] (a) Show that the value of approaches 0 as (x;y)! (0; 0) along 6 2 2x +y 2 any straight line y =mx, or along any parabola y =kx . 3 x y (b) Show that lim does not exist. 6 2 (x;y)!(0;0) 2x +y 3 [Hint: Let (x;y)! (0; 0) along the curve y = x , and then compare the result with the results in (a).] 2 4. [3 marks] Find f ; f and f : f(x;y;z) =z ln(x y cosz). x y z 2 2 5. [4 marks] (a) Show that f(x;y) = ln(x +y ) satises 2 2 @ f @ f + = 0: 2 2 @x @y This is called the Laplace's Equation. x x (b) Show that u(x;y) =e cosy and v(x;y) =e siny satisfy @u @v @u @v = ; = : @x @y @y @x These are called the Cauchy-Riemann Equations. 2 2 2=3 6. [3 marks] Let f(x;y) = (x +y ) . Show that f (0; 0) = 0. [Hint. Use the denition x f(h;0)f(0;0) f (0; 0) = lim .] x h!0 h