Taylor polynomial

College/University

Definition

Used to approximate the function using

Worked examples

\(P_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2\)

A degree-2 Taylor polynomial centered at zero uses the function's value and first two derivatives at that point.

\(e^x \approx 1 + x + \frac{x^2}{2} + \frac{x^3}{6}\)

The third-degree Taylor polynomial for e to the x at zero gives a polynomial that closely matches the exponential near zero.

\(P_1(x) = \sin(\pi) + \cos(\pi)(x - \pi) = -(x - \pi)\)

A first-degree Taylor polynomial centered at π approximates sine using its value and slope at that point.

Common mistakes

\(P_n(x) = f(a) + f'(a) + f''(a) + \cdots\) → \(P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots\) Each term needs the (x - a) power and the factorial divisor; you're building a polynomial in (x - a).
\(P_2(x) = f(a) + f'(a)x + \frac{f''(a)}{2}x^2\) when centered at \(a \ne 0\) → \(P_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2\) Always use (x - a), not just x, unless the center a is zero.
forgetting \(n!\) in the denominator → \(\frac{f^{(n)}(a)}{n!}(x-a)^n\) The factorial in the denominator is essential; without it the approximation fails.

Where you'll use it next

Taylor polynomials lead directly into Taylor and Maclaurin series in calculus, and you'll use them to approximate functions, estimate integrals, solve differential equations, and analyze error in numerical methods throughout higher math, physics, and engineering.

Found in 1 StudyPug lesson

Approximating Functions with Taylor Polynomials and Error Bounds

Calculus

Master the art of function approximation using Taylor polynomials. Learn to assess accuracy with error bounds, and apply these powerful tools in calculus, physics, and engineering.

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