(a)
f(r)=f(x)+f'(x)(r-x)+f''(x)(r-x)2/2+ ...

(b)
r=x-f(x)/f'(x)

(c)
f'(x)=(f(x+h)-f(x-h))/2h

(d)
r=x-2 h f(x)/(f(x+h)-f(x-h))

(e)
f'(x)=(f(x+h)-2 f(x)+f(x-h))/h2

(f)
f''(x)(r-x)2/2+f'(x)(r-x)+f(x)=0

(g)
r=x-(f'(x)+([f'(x)]2-2 f''(x)f(x)))/f''(x)

Example 1: Newton's method is based on the Taylor expansion.