Taylor Polynomials

Definition

A Taylor Polynomial $T_n(x)$ is a finite sum of terms from a Taylor series that provides a polynomial approximation of a more complex function $f(x)$ near a point $a$. $T_n(x) = \sum_{i=0}^{n} \frac{f^{(i)}(a)}{i!} (x-a)^i$

• How to read: “T-n of x equals the sum from i equals zero to n of the i-th derivative of f at a, over i factorial, times (x minus a) to the i.”
• Meaning: The best degree-$n$ polynomial approximation to $f$ near $x = a$, built from derivatives at the center point.

Why It Matters

Approximation is the heart of engineering. We rarely need the ‘exact’ answer; we need an answer that is ‘good enough’ within a certain range. Taylor polynomials provide a rigorous way to trade off computational complexity for accuracy.

Core Concepts

• Degree ($n$): The highest power in the polynomial. As $n$ increases, the approximation generally fits the original function over a larger interval and with higher precision.
• Center ($a$): The point where the approximation is exact (i.e., $T_n(a) = f(a)$).
• Agreement of Derivatives: By construction, $T_n(x)$ and its first $n$ derivatives match the original function and its first $n$ derivatives at the center $a$.
• Truncation Error: The difference between the actual function and the polynomial, $R_n(x) = f(x) - T_n(x)$.
  • How to read: “R-n of x equals f of x minus T-n of x.”
  • Meaning: The remainder (error) from cutting off the infinite series at degree $n$. Bounded by Taylor’s Inequality.

Connected Concepts

• Taylor Series
• Maclaurin Series
• Taylor’s Inequality (Remainder Estimation)
• Linearization
• Power Series
• Information Theory
• Exponential Growth Model
• Exponential Decay Model
• Map-Territory Relation