An outstanding visualization! You can enhance your visualization by adding the following algebraic explanation to your comment section:

Suppose the point you choose is [c, f(c)].

In order to find the second guess, we need the equation of the tangent line at x=c, after which, we can find where it intersects the x-axis to obtain the second guess.

So, let t(x)=f'(c) x + k where t(x) is the equation of the tangent line.

We determine k as follows:

f(c) = f'(c) c + k
=> k=f(c) - f'(c) c
=> t(x)=f'(c) x + f(c) - f'(c) c
=> t(x)=f'(c) [x - c] + f(c)

We know the tangent line intersects the x-axis when t(x)=0.

So,
0= f'(c) [x - c] + f(c)

=> x - c = - f(c) / f'(c)
=> x = c - f(c) / f'(c)

This last equation provides the next guess, that is, x as expected.


John Gabriel
http://thenewcalculus.weebly.com.
http://whatyouhadtoknowbutyoureducators.weebly.com/the-new-calculus-course.html