Taylor polynomial

The polynomial of degree $n$ that matches $f$ and its first $n$ derivatives at a point $a$. Degree 1 is the tangent line, degree 2 adds curvature; the higher the degree, the wider the region where it is a good approximation. At $a = 0$ it is called a Maclaurin polynomial.

Polynomials are what we can compute: additions and multiplications. Other functions — $\sin$, $e^x$, the solution of a differential equation — are not. Taylor's idea gives a systematic way to trade any smooth function for a polynomial near a point, with a precise error estimate.

Matching more derivatives means matching the shape more closely: value (same height), first derivative (same slope), second (same bending), third (same change of bending)… Each extra term corrects what the previous polynomial got wrong, and the corrections get smaller near $a$ because they carry higher powers of $(x - a)$. Move the slider in the demo and watch the approximation "peel" away from $a$ more slowly as the order grows.

If $f$ is $n$ times differentiable at $a$, its Taylor polynomial of order $n$ is

the unique polynomial of degree $\le n$ with $T_n^{(k)}(a) = f^{(k)}(a)$ for $k = 0, \dots, n$. Moreover $f(x) = T_n(x) + o\big((x - a)^n\big)$ as $x \to a$.

Compute $f(a), f'(a), f''(a), \dots$, divide the $k$-th by $k!$. For compositions it is usually faster to combine known series (substitute, multiply, integrate term by term) than to differentiate repeatedly. Software can do it exactly with Taylor-mode arithmetic on truncated power series — which is how the demo computes the coefficients.

Small-angle approximation: $\sin\theta \approx \theta$ (order 1) turns the pendulum equation $\ddot\theta = -\frac gL\sin\theta$ into $\ddot\theta = -\frac gL\theta$, solvable by hand: period $2\pi\sqrt{L/g}$. The error at $\theta = 10° \approx 0.1745$ is $\theta^3/6 \approx 0.0009$ — about 0.5%.

Gradient descent is "trust the order-1 Taylor model a little"; Newton's method is "jump to the minimum of the order-2 model". Numerical differentiation, ODE solvers (Euler, Runge–Kutta) and their error analysis are derived from Taylor expansions. Physics linearizes with them, and math libraries start from them.