Taylor Polynomials

Real Analysis · Axiom Academy

Approximating a function with polynomials that match its derivatives at a point — the best local approximation, hugging the curve more tightly as the degree grows. The degree- n Taylor polynomial of f centered at a takes the value, slope, curvature, and higher derivatives of f at the single point a and packs them into a polynomial. Each coefficient is f^ (k) (a)/k! , so the term of degree k is exactly what is needed to match the k th derivative there. 2. Linear Approximation (n = 1) The first-degree Taylor polynomial is the tangent line at a . It is built to agree with f in exactly two ways at that point: the same value and the same slope . That is the most a straight line can promise — and it is already the best linear approximation to f near a . Since and , the value and slope at a give : This is the famous small-angle approximation , valid for x near 0 . 3. Quadratic Approximation (n = 2) A line cannot bend, so the tangent immediately peels away from a curving function. Adding the degree-2 term lets the polynomial match the concavity at a as well — now it shares value, slope, and second derivative, so it curves the same way f does and stays close much longer. Here , , and , so the value, slope, and second derivative give: The flat tangent T_1(x) = 1 ignores the curvature; the parabola bends down to follow . 4. Higher-Order Approximations

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