Taylor polynomial approximation

Parametric equations

The degree-$n$ Taylor polynomial of a function $f$ at $x=a$:

At $x=a$, $T_n$ and $f$ have the same value and the same first $n$ derivatives. This page uses $\sin x$, $\cos x$, and $e^x$. The classical expansions ($a=0$):

Farther from the center, the higher-degree terms matter more. A finite $T_n$ is only a polynomial approximation. It does not equal $f$ along the whole real line.

Interaction

• sin x / cos x / e^x: switch the function and watch how its coefficients are arranged
• Center a: drag the center on the figure and watch the best-fit region move with it
• Degree n: raise $n$ and watch $T_n$ settle onto the faint graph of the original function
• Error band: show the gap $|f(x)-T_n(x)|$, close near the center and wide farther away
• Terms: turn on Advanced, then lay out each term $\frac{f^{(k)}(a)}{k!}(x-a{)}^k$ and see which degree starts to drive the correction

What to notice

• The Taylor polynomial is most accurate near $x=a$. Farther from the center, the error at a finite degree usually grows.
• For $\sin x$ and $\cos x$, odd and even powers of $x-a$ separate when $a$ is a multiple of $\pi /2$ (only even or only odd powers appear). At other centers both kinds appear. Every derivative of $e^x$ is $e^x$, so the coefficients decrease in the most regular way.
• Raising $n$ improves the local fit, but it does not guarantee that the approximation gets steadily more accurate across the whole window, because a high-degree term can dominate far from the center.

Related

• Tangent approximation
• Fourier series
• Limits and Riemann sums
• Sequences and series

Further reading

• Taylor series (Wikipedia)
• Taylor’s theorem (Wikipedia)