Polynomial Approximation of $e^x$

Overview

This was built to visualize how Maclaurin polynomial approximations $P_n(x)$ hold up against the real $e^x$. You can drag a slider to change the polynomial degree.

Runs on NumPy and Matplotlib, nothing else.

Features

• Degree slider: Adjust $n$ from 0 to 25. The plot redraws when you click update plot.
• Grid step input: Change the step size for the calculation grid. Garbage inputs (text, out-of-range numbers) just reset to 0.01.
• Error plot: A second chart shows the absolute error $e^x - P_n(x)$ across the domain so you can actually see where things go wrong.
• Error stats: Shows the max absolute error over $[-2, 2]$ and the exact $x$ where it's worst.

Requirements

Python 3, plus:

• numpy
• matplotlib

pip install numpy matplotlib

Usage

python3 app.py

Controls

1. Degree $n$: Drag the slider (bottom left). Plots update automatically.
2. Grid step: Type a value like 0.01 or 0.05 in the text box (bottom right).
3. Update plot: Click to apply. Anything invalid or out of range (0.0–0.5) falls back to 0.01.

Math

The Maclaurin series for $e^x$ (Taylor series centered at $a = 0$) is:

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^n}{n!}$$

The tool computes the partial sum $P_n(x)$ up to whatever degree $n$ you pick and plots it against np.exp(x) over $[-2, 2]$.

License

This project is licensed under the MIT License - see the LICENSE file for details.