Taylor and Maclaurin polynomials – approximating functions

What it shows

A Taylor polynomial of degree n matches a function and its first n derivatives at a centre a: Pₙ(x) = Σ f⁽ᵏ⁾(a)/k!·(x − a)ᵏ, and with a = 0 it is called a Maclaurin polynomial. Close to a the polynomial hugs the curve; further away the error |Rₙ(x)| grows. The Lagrange bound M|x − a|ⁿ⁺¹/(n + 1)! limits that error, and for ln(1 + x), 1/(1 − x) and arctan x the series converges only on an interval. The Small angles screen compares sin θ ≈ θ, cos θ ≈ 1 − θ²/2 and tan θ ≈ θ, with θ in radians.

How to use

Choose the Function f(x), then move Degree n, Centre a and Point x, or drag on the graph: near the blue diamond you move a, anywhere else you move x. Press Step up n to watch the polynomial approach the curve. Change Tolerance ε and read where the error stays below it. On Small angles, drag the Angle θ slider and compare the values in the table.

Parameters you can change

• Screen Taylor polynomial, Small angles
• Function f(x) sin x, cos x, eˣ, ln(1 + x), 1/(1 − x), arctan x
• Degree n 0–15
• Centre a -3–3
• Point x where the error is checked -6–6
• Tolerance ε 0.1, 0.01, 0.001
• Shade the interval of convergence
• Show the error band ±ε
• Angle θ (Small angles screen) 0–90 °

Questions to explore

1. For sin x about a = 0, which degree n first keeps the error at x = 2 below 0.01?
2. Why does a higher degree n make the approximation of ln(1 + x) worse at x = 1.5?
3. Up to roughly which angle in degrees is sin θ ≈ θ accurate to within 1 %?