Taylor expansions and applications
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The Taylor expansion of a function around a real point x 0 is the representation of the map as sum of a polynomial of a certain degree and an infinitesimal function of order bigger than the degree. It provides an extremely effective tool both from the qualitative and the quantitative point of view. In a small enough neighbourhood of x 0 one can approximate the function, however complicated, using the polynomial; the qualitative features of the latter are immediate, and polynomials are easy to handle. The expansions of the main elementary functions can be aptly combined to produce more involved expressions, in a way not dissimilar to the algebra of polynomials.