Use 2nd order Newton`s Interpolating polynomial to estimate f(3.5).x 1 3 5 y 50 138 179

Using Newton's Interpolating Formula we construct the
following
table:

x0=1...y[x0]=50

x1=3...y[x1]=138...y[x0,x1]
= 44

x2=5...y[x2]=179...y[x1,x2] =
20.5...y[x0,x1,x2]=-5.875

P2(x) = y[x0]+ y[x0,x1]*x +
y[x0,x1,x2]*x(x-2)

P2(x) =50 + 44x -
5.875*x*(x-2)

for x = 3.5, P2(x) = 50 +
44*3.5 - 5.875*3.5*1.5 = 173.156

Therefore
for x = 3.5, y is approximately equal to 173.156.