Question

Consider the function f(x)=x^2+1.
List the first five approximations (i.e. through x5) Newton’s Method gives for this function using the initial guess x1=2. Does Newton’s Method appear to be converging? If not, what appears to be happening?

Answer 1

newtons method is simply to use equation of the tangent line:

newX = f'(x) -f'(x)(x) + f(x)

Answer 2

f'(x) = 2x in this case; so...

newX = 2(2) -2(2) + 5 ;

Answer 3

that seems a bit off...

Answer 4

slope = f'(2) = 2(2) = 4 , at point(2,5)

newX = 4x -4(2) + 5 = 0
= 4x -3 = 0
= 3/4

Answer 5

t{n} = 2(t{n-1})x -2(t{n-1}) + (t{n-1})^2 + 1 ; looks a bit convoluted, but I think its the recurrsion for this :)