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Taylor's theorem is a fundamental concept in calculus and quantitative methods, particularly in the approximation of functions using polynomials. The theorem provides a method to represent a smooth function as an infinite sum of terms calculated from the values of its derivatives at a single point. This representation, known as the Taylor series, is crucial in quantitative methods for simplifying complex functions, solving differential equations, and performing numerical analysis. The basic idea of Taylor's theorem is to approximate a function $f(x)$ _____ ________ __________ ______ _______ _____ _______.
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