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If the tangent line is approaching a function, we expect the tangent line to have a root approaching the function's root.

Using Newton's rule to calculate square root:

For

f(x)=x^2-a

f`(x)=2x

x_n+1=x_n-((x_n)^2-a)/2x_n=1/2(x_n+a/x_n)

tangent line: y=f(x_0)+f`(x)(x-x_0)

to find x_0, let y=0

0=f(x_0)+f`(x)(x_1-x_0)

-f(x_0)=f`(x_0)(x_1-x_0)

-f(x_0)/f`(x_0)=x_1-x_0

If f(x_1)!=, then take a tangent line here

y=f(x_1)+f`(x_1)(x-x_1)

Find the x-int(or find x_2)

0=f(x_1)+f`(x_1)(x_2-x_1)

x_2=x_1-f(x_1)/f`(x_1)

Herative Formula for Newton's Method:

x_(n+1)=x_n-f(x_n)/f`(x_n)