Taylor Polynomial

The central idea

The degree-$n$ Taylor polynomial for $f$ centered at $a$ is $T_n(x)=\sum_{k=0}^n f^{(k)}(a)(x-a)^k/k!$. It is the unique polynomial of degree at most $n$ matching $f$ and its first $n$ derivatives at $a$. Taylor's theorem writes $f(x)=T_n(x)+R_n(x)$.

Definitions, hypotheses, and notation

The Lagrange remainder is $R_n(x)=f^{(n+1)}(\xi)(x-a)^{n+1}/(n+1)!$ for some $\xi$ between $a$ and $x$, under the theorem's differentiability hypotheses. Bounding the derivative on that whole interval—not merely at the center—turns the expression into a guaranteed error bound.

A higher-degree polynomial is not automatically more accurate far from the center. Accuracy depends on derivative size, distance, and convergence of the full series. Centering near the evaluation point often matters as much as adding terms.

Conceptual meaning

Taylor polynomials extend linearization by retaining curvature and higher local derivative information. Each added term makes the approximation error flatter at the center and usually improves nearby accuracy.

A dependable method and decision rule

1. Choose the center and desired degree.
2. Compute derivatives only through that degree.
3. Evaluate each at the center.
4. Insert coefficients with factorials and powers of $x-a$.
5. Use a remainder bound or numerical check to justify accuracy.

Fully worked example

Graphical or geometric meaning

At the center, the polynomial and function share value, tangent, curvature, and the third-order bending data. Their graphs separate gradually as distance from the center makes the remainder more significant.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate $T_n$ at $a$ and recover derivative data.
• Verify $T_n(a)=f(a)$.
• Compare actual error with a valid remainder bound.

The center controls coefficients and accuracy

The degree-$n$ polynomial at $a$ is $P_n(x)=\sum_{k=0}^n f^{(k)}(a)(x-a)^k/k!$. It matches $f$ and its first $n$ derivatives at that single center, so every coefficient must use the same $a$. Approximation is usually strongest nearby. Under suitable hypotheses, $R_n(x)=f^{(n+1)}(\xi)(x-a)^{n+1}/(n+1)!$ for some intermediate $\xi$; bounding that derivative gives a guaranteed error. Check $P_n(a)=f(a)$ immediately. A finite polynomial can approximate well even where an infinite Taylor representation is not established, so keep the approximation theorem separate from the assertion that a series equals the function.

When symmetry forces all odd or all even derivatives at the center to vanish, write the missing powers explicitly in the degree count. The polynomial's degree is the largest retained exponent, not the number of nonzero terms.

Practice

1. Find $T_2$ for $\sin x$ at zero.
2. Find $T_2$ for $\ln x$ at $a=1$.
3. What does $T_1$ equal?

Connections and next steps

• Linear Approximation
• Taylor Series
• Maclaurin Series