Question

1. A river starts at the origin (0,0) and flows along the path of y = Vx. A house is located at the point (4,0). Determine what point on the river is closest to the house, and how far the house is from that closest point (assume units are in km). (8 marks) 2. Discuss the convergence of the following integrals and a calculate them when possible: 1 (4 marks) el/xrzdy OSS (4 marks) o Vxer da (4 marks)

Answer 1

(1) River starts from 200) and goes on path y=50 Let P be any point on y=c then P is (x, 5x) for some x 98 ya so Now, (40) distance of p from Po is flox) = PP. = v 04²+ (55-072 We want to minimize f(x) where x=0 f(x) is minimum iff gox) = (f(x)}" is minimum gesc) = (5(x-4)2 + x 2 = (x-4)2 + x gex) = 2 (x-4) + 1 g(x)= 2 > 0 Nov glese) = 0 - 2(x-4) + 1 = 0 » x-4 = £ = x= 4 - 2 2 = 2 As 9"(7) >0 so At x = 7 gox) has minimum so f(s) has minimum at x = 7 So Nearest point of river from house is ( 5) where x= 3 ie point nearest is , distance minimum is £ () - J (29) 24 7 = 11 12 15 kms

22 net i = t - 1 13 dx = at so schon das a ja - d jetit -[-24101 Henre de: 1 Dr

0 (By Jitendre la tarda a leerde Now for xxl 1 <1 = tex. so Barea de a ser de (**). *= 0+1 a gastre dx is convergent For sterde we have tax so base = bintas dix Strada 10x=1 Hence Sex s 1+ ε = Intex is convergent coni Hence с е 50 Sex is

80 a Let 550 = t - so ist es das fadd t doc=dt e - -6-)] = 2 Hence 8P doc is convergent