Labs / Mathematics
Taylor Series
Every smooth curve can be rebuilt from a single point using nothing but powers of x. Drag across the graph to move the expansion center a, then add terms with the degree slider and watch the pink polynomial hug the true curve ever more tightly. The shaded band is the approximation error, and the dashed line reads it off at your test point x*.
a 0.00N 4P(x*) 0|f−P| 0terms …
What to try
- With sin x centered at 0, how many terms do you need before the wave stays glued to the curve all the way out to x = 6?
- Slide the center a toward your test point x*. Why does the error shrink even when the degree stays the same?
- Try 1/(1−x) and push the degree up. Why does the approximation refuse to follow the curve past x = 1, no matter how many terms you add?
- For eˣ, every term is positive — but for cos x the terms alternate sign. What is it about each function that decides this?
- Turn on the error band and move the center. Where is the error always exactly zero, and can you make the whole band vanish?