Taylor Polynomial Lab — Fibo Math Lab

Fibo Math Lab

Taylor Polynomial Lab

Skill goal: Approximate functions with Taylor polynomials and watch the error shrink as degree grows.

1 · Observe

The navy curve is the true function; the teal curve is its Taylor polynomial Pₙ centered at a = 0. The gold dot is the evaluation point — drag it left or right along the graph.

true f(x) Taylor Pₙ(x) evaluation point

2 · Manipulate

Raise the degree and watch the teal curve snap onto the navy curve near the center. Drag the gold dot (or use the slider) to test points far from the center.

3 · Predict

Before testing: for f(x) = eˣ at x = 1, how many terms does it take to get the error below 0.01? Set the degree slider to your guess and read the error below.
Degree n = 4: P₄(1) = 1 + 1 + 1/2 + 1/6 + 1/24 = 2.70833, and e ≈ 2.71828, so the error is ≈ 0.00995 ✓. At n = 3 the error is still 0.0516 — the x⁴/4! term is what pushes it under 0.01.

4 · See

f(x)
—
P₃(x)
—
|error|
—

Error readout updates live as you drag the point or change the degree.

5 · Explain

Why it works: the Taylor polynomial Pₙ matches the function and its first n derivatives at the center a = 0 — same value, same slope, same curvature, and so on. Each new term fixes one more derivative, so the fit near the center keeps improving. But the match is built from information at one point only, so far from the center the polynomial can drift away from the true function.

6 · Challenge

1. For sin x: P₃(0.5) = 0.479167, and the true sin(0.5) = 0.479426, so the error is ≈ 0.00026. Is the error less than 0.01?

2. For eˣ: P₂(0.2) = 1.22 and e0.2 ≈ 1.22140. Enter the absolute error (within 0.0005).

3. What is the smallest degree n (0–6) with error below 10⁻⁴ when approximating e0.5?

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