Question

Question 1. Start on a new page. Total marks: 5+6+5+5 = 21. (a) Sketch the natural domain of the function f(x,y) = In (9 - p²-72). Use solid lines for portions of the boundary included in the domain and dashed lines for portions not included. (b) Suppose w = rs'. r = ln(uv)s = re. Use the chain rule, after drawing an appropriate "tree diagramme", to find w/ou in terms of u, v and a (c) Find parametric equations of a line through the origin and parallel to the line r=1-ty= 2.2 = 3 +11. (d) Determine whether or not the following limit exists. If the limit exists, find its value, lim ?

Answer 1

& foxy- ln (9-2292) Coprithan is defined logarithen for positive g. ²270 & lety 22g Domain of fie Inside region of Cale x2 + y = excluding the boundary. (6) w=1254 ; r = leluh ; s=x²²" (1354) en la boos + (134) 2(229)

30 = 2559 .v +5r4 xzeu UV - 2mCw) (eg 2 + 4 (ar Cowg (2 24 Po = zate cu bulun) +4 kelele na liny (9 Required line is Poradel to the line n-1-t; g = 2, 7-37ht > Placement & Both lines have same direction natios - (-1,0,4) equation of line passing through (0,0,0) with direction Cosines (-10,4) x = yo = z-get (1=-;y0; tast

lim (xy (20) x2 + y2 17 w Consider live the pathe nat; yo 4to - firma ( to +2+0 0 Consider the path get; não lin ott lim t = to ott2 to 2) Since bothe limits are not lim x²y3 (ny)-(0,0) p2ty2 does not exist.