Taylor Polynomials from Derivative Matching

A polynomial that agrees locally

A tangent line approximates a differentiable function near one input by matching its value and first derivative. A Taylor polynomial extends that idea: it matches value, slope, curvature, and as many higher derivatives as its degree allows. The result is still a polynomial, so it is easy to evaluate and manipulate, but it carries carefully chosen local information about a more complicated function.

For a function $f$ centered at $x=a$ , the degree- $N$ Taylor polynomial is

T_N(x)=\sum_{n=0}^{N}\frac{f^{(n)}(a)}{n!}(x-a)^n.

The $n$ th derivative evaluated at $a$ supplies the coefficient of $(x-a)^n$ . The factorial is not an arbitrary decoration: differentiating $(x-a)^n$ exactly $n$ times produces $n!$ , so the denominator makes the $n$ th derivative of that term equal to the desired value at the center. This construction forces $T_N^{(k)}(a)=f^{(k)}(a)$ for every $k$ from $0$ through $N$ .

L3-V01 — Matching derivatives at a center. Draw a smooth curve and three approximating polynomials near a marked center aaa: a tangent line, a quadratic, and a higher-degree curve. Show all touching at the center but separating farther away

Build the polynomial one condition at a time

Suppose $f(x)=e^x$ and the center is $a=0$ . Every derivative of $e^x$ is $e^x$ , so every derivative evaluated at zero is $1$ . The first few Taylor polynomials are

T_0(x)=1,

T_1(x)=1+x,

T_2(x)=1+x+\frac{x^2}{2},

and

T_3(x)=1+x+\frac{x^2}{2}+\frac{x^3}{6}.

At $x=0$ , every one has value $1$ . The linear and higher polynomials also have slope $1$ there, while the quadratic and higher ones additionally have second derivative $1$ . The approximations are not selected by a curve-fitting guess; each new term is required by one more derivative-matching condition.

The special name Maclaurin polynomial is used when the center is $0$ . A Taylor polynomial centered at $a\ne0$ uses powers of $(x-a)$ , not powers of $x$ alone. This shift keeps the approximation organized around the input where the derivatives were measured. Expanding $(x-a)^n$ too early can obscure the center and increase algebraic error.

A nonzero center shows why the shift matters

Let $f(x)=\ln x$ and approximate near $a=1$ . The derivatives are $f(1)=0$ , $f'(1)=1$ , $f''(1)=-1$ , and $f'''(1)=2$ . The cubic Taylor polynomial is

T_3(x) =(x-1)-\frac{(x-1)^2}{2}+\frac{(x-1)^3}{3}.

At $x=1$ , the polynomial gives $0$ , just as $\ln1=0$ . Its derivative at $1$ is $1$ , and its curvature agrees as well. If the same coefficients were attached to powers of $x$ rather than $x-1$ , the resulting polynomial would not be designed to match the function at the intended point. The center is part of the model, not a coordinate label that can be dropped.

Writing the polynomial in centered form also makes error interpretation clearer. The size of $(x-a)$ records exactly how far the target input lies from the point of derivative matching. A small centered difference causes higher powers to shrink rapidly; a large difference causes them to grow. This is the same structural reason that a Taylor approximation is local. It gives a reader a quick way to assess whether an approximation request is being made near its justified center.

L3-V02 — Centered powers. Show a number line with a=1a=1a=1 highlighted and arrows measuring the local input difference x−1x-1x−1. Beside it, contrast (x−1)2(x-1)^2(x−1)2 with x2x^2x2 using a simple algebraic callout

Degree trades simplicity for local accuracy

A higher degree does not promise a better approximation at every possible input. It gives more exact derivative information at the center, so it generally improves the approximation sufficiently close to that center. Far away, omitted terms can become large, and a lower-degree polynomial may even appear closer by accident at a particular point. An approximation should therefore name both its degree and the region where it is being used.

For $e^x$ near zero, $T_1(0.1)=1.1$ , while $e^{0.1}\approx1.10517$ . Adding the quadratic term gives $T_2(0.1)=1.105$ , much closer. At $x=3$ , however, few terms of the Maclaurin polynomial are not automatically reliable. The difference between a local model and a global identity is central to responsible approximation.

The next omitted term is often a useful informal indicator, but it is not a universal error proof. Later lessons develop remainder formulas that bound the difference under stated hypotheses. Until then, compare with a calculator only as a numerical check, not as the reason the polynomial was constructed. The derivative-matching derivation is what explains the coefficients.

L3-V03 — Local accuracy fades with distance. Plot exe^xex, its quadratic Taylor polynomial at zero, and a shaded neighborhood around zero where the curves nearly overlap. Let the separation become visible farther from the center

Retrieval check

Construct the degree-three Taylor polynomial for $\sin x$ centered at $0$ . List the needed derivatives and their values at zero before writing the series terms. Then construct the degree-two Taylor polynomial for $\sqrt{x}$ centered at $4$ , retaining powers of $(x-4)$ . Explain why the two approximations cannot be judged only by their degrees; the chosen center and input distance matter. The next lesson turns repeated derivative patterns into Maclaurin series for key functions that can be used as reusable approximation tools.