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Whenever I'm studying a new topic in mathematics, the question of potential practical application is the one that matters to me the most. While it's relatively easy to come up with hypothetical cases of practical application of, let's say, optimization, related rates etc., in this case I'm confused.

I must underline that I fully appreciate the elegance of the idea of local linearization, but I would really like to see how this technique can be applied to solve real world problems.

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Local linearity lets us approximate a curve about a point with a line, very useful if you're a statistician or numerical analyst. Also, once it is known that a curve is locally linear, we are able to use tools of calculus to perform useful analysis on it. One specific example of local linearity helping us is Newton's method for finding roots of a function.

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