What quantity did you change?
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*.
Labs / Mathematics / Taylor Series
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*.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
a 0.00N 4P(x*) 0|f−P| 0terms …
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
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?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.