https://github.com/worthant/taylor-tan-visualization
Interactive visualization of Taylor series, approximating tg(x)
https://github.com/worthant/taylor-tan-visualization
animation p5 p5js taylor-approximation taylor-polynomial taylor-series
Last synced: 3 months ago
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Interactive visualization of Taylor series, approximating tg(x)
- Host: GitHub
- URL: https://github.com/worthant/taylor-tan-visualization
- Owner: worthant
- License: apache-2.0
- Created: 2025-02-16T16:26:50.000Z (3 months ago)
- Default Branch: main
- Last Pushed: 2025-02-17T13:21:15.000Z (3 months ago)
- Last Synced: 2025-02-17T14:23:49.887Z (3 months ago)
- Topics: animation, p5, p5js, taylor-approximation, taylor-polynomial, taylor-series
- Language: TypeScript
- Homepage: https://imtjl.github.io/taylor-tan-visualization/
- Size: 271 KB
- Stars: 0
- Watchers: 0
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
## Maths behind this repo
**Taylor series** of any real or complex-valued function f(x), that is
infinitely differentiable at a real or complex number a, looks like this:
$$
f(x) = f(a) + f'(a) \cdot (x-a) + \frac{f''(a)}{2!} \cdot (x-a)^2 +
$$
$$
,+ \frac{f^{(3)}}{3!} \cdot (x-a)^3 + \dots + \sum_{n=0}^\infty \frac{f^{(n)} (a)}{n!} (x - a)^n
$$
I want to visualize $tan(x)$ - it's basically a $\frac{sin(x)}{cos(x)}$. To
approximate those functions we can just use Taylor series at $a=0$, which is
called **Maclaurin series**:
$$
sin(x) = sin(0) + sin'(0) \cdot (x - 0) + \frac{sin''(0)}{2!} \cdot (x - 0)^2 + \dots
$$
$$
sin(x) = sin(0) + cos(0) \cdot x + \frac{-sin(0)}{2!} \cdot x^2 + \dots
$$
$$
sin(x) = 0 + 1 \cdot x + 0 - 1 \cdot \frac{x^3}{3!} + 0 + \dots
$$
$$
sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots
$$
Performing same operations for cos(x):
$$
\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots
$$
Now, as i said above, $tan(x)$ function can be simply calculated from those
approximations of $cos(x)$ and $sin(x)$ by deviding them:
$$
tan(x) = \frac{sin(x)}{cos(x)} = \frac{x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots}{1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots}
$$
Also, the sin(x) and cos(x) approximation formulas can be summarized as this:
$$
sin(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}
$$
$$
cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}
$$
This repo visualizes this approximation of tan(x) and shows how adding more
terms improves the accuracy, using those formulas for $sin(x)$ and $cos(x)$.