Question

1. Find an equation for the contour line of the surface f(x, y) = 3(x + 1)2 sin(y) + 5x – 8xy that goes through the point (0,5). 2. (a) Show that lim does not exist. (b) Does there exist c E R such that the function (x,y) →(0,0) x2 + y 6 ху3 f(x, y) = { x2 + yo if (x, y) = (0,0) lc if (x, y) = (0,0) is continuous at (0,0)? Give reason(s) for your answer. (2)

can question 1 , 2a and 2b be answered

Answer 1

☺ Sol A contour of a function of aturo variables xand - y en a carne along which the function has a - constant value given by the equation f(x,y) = kikü a constant . Here, flee; 4) = 3(x+D sing tsaasny=k.. but it goes through (0, 2) : theore fane i (67) sably the above equation. Thu, 31045 sina + 5.0-8.0.7 =k. => K = 3. . So, the equation of the contain as. 3 (a+j) Y siny +52 804 = 3.

1 (2) (a) alim ny² (2,4)=600) tyto We will are the requen eiad erilisica. Hene, uno yn..->o as mx . So, (2n, Yn) - 6,0). nowy see, of (an, Mn) where, f (2,4) = 24 fanan) = best for a = ŕt į . po ay sina flo,o=0 ५० ००) dain't exist . (3,4) – (0,0) (6) Sl for any cElr, the finction Plan) is not continuous at (eo) , became if it u continwas. lim f (any) should (314>0,0) exint (forget about c), but from the abou explanation in (ala) we have already seen that lim f(x,y) docon't exist.