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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 $f(x)$ around a point $a$ using polynomials. The function $f(x)$ is expressed as the sum of its value at $a$, the _____ ____ ________ ___ __________ ________ _________ ________ ___ __________ __________ _____.
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