What's the difference between linear approximation and newton's method? both are similar in a way.
+33615 Yes, the linear approximation they are talking about isn't the bisection method I've illustrated above (Newton's method is the same).
Their linear method uses a tangent line, as does Newton's method, but it's just used to find an approximation to the value of the function at some "ugly" number (where "ugly" means it is not easy to calculate the function at that point) by using a nearby "non-ugly" number! I suspect this is a very old-fashioned method, where calculations were done either in your head, or with a slide rule, or with log tables. In these days of calculators and computers, there aren't really any such "ugly" numbers!
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+33615 If, by linear, you mean linear bisection, then the following pictures might help:
In the linear bisection method, two x values are chosen, such that the signs of the corresponding y-values are different, and a straight line is drawn between them. The next value of x is chosen to be where this line hits the y-axis. It replaces one or other of the previous x values (depending on the sign of the new y-value), and the process repeated to convergence. Just one step is illustrated in the first graph above - the open circle indicates the next x-value.
In Newton's method, just one point is chosen and the gradient of the curve at that point is calculated. The gradient is then extended until it crosses the y-axis. The x-value at that point is used as the next x-value and the process repeated etc. Again just one step is illustrated in the second graph above.
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+118608 That makes good sense. Thanks Alan.
I still haven't looked at the article properly but it seems to be talking about something different. 
+33615 Yes, the linear approximation they are talking about isn't the bisection method I've illustrated above (Newton's method is the same).
Their linear method uses a tangent line, as does Newton's method, but it's just used to find an approximation to the value of the function at some "ugly" number (where "ugly" means it is not easy to calculate the function at that point) by using a nearby "non-ugly" number! I suspect this is a very old-fashioned method, where calculations were done either in your head, or with a slide rule, or with log tables. In these days of calculators and computers, there aren't really any such "ugly" numbers!
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+23246 My understanding is a little different:
A linearization gives approximate y-values near a given point on a graph, while Newton's method is used to find a zero of a function by repeated approximations, starting at a given point.
A linearization will give an approximate y-value anywhere on a function while Newton's method is used to find an x-value that gives a y-value of zero.
