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Read sections 7.4, 7.5, and 7.6 in the book and do the examples. It's heavy! Newton's Algorithm is a process that takes the parameters for a problem (square-root) and a guess and returns an improvement on that guess. By repeating the algorithm, we get closer to the actual answer (though we may never reach it).
This kind of algorithm is important in scientific computing. We may not have a good process to directly calculate the value for something, but we can get really close. Since we're often dealing with experimental data, it's already a bit off due to human error and measurement. Getting really really close is pretty useful.
Definitely look at the debugging section (7.7) in this chapter. It presents something known as Binary Search, which you'll learn all about in future classes, and relates to all kinds of debugging.