TITLE: Why does Newton's method work?
QUESTION [36 upvotes]: I find many sites explaining how to use Newton's method, but none explaining why it works. Could someone give me the intuition behind it? Thanks.

REPLY [3 votes]: Most root finding methods work by replacing the function, which is only known at a few points, with a plausible model, and finding the root of the model.
For instance, the chord and regula falsi methods work from two known points and hypothetise a linear behavior in between. Newton uses a single known point and the direction of the tangent, and also hypothesizes a linear behavior. Brent uses three points and a parabolic interpolation.

In all cases, the reason why it works is simple: because the new estimate is closer to the root.
For this property to hold, the functions must satisfy certain criteria, which are established in the frame of calculus, and essentially mean that the function can be well fitted by the model. In particular, when the function has Taylor develomments, it locally behaves like a polynomial of some degree.