5 · IB Maths
Maclaurin series
A polynomial built from derivatives at zero. Add terms one at a time and watch it hug the curve over a widening interval, then probe a value of x to compare the polynomial with the function.
Predict first
Start from the default settings (sin x, 2 terms, probe x = 1.00). Raise Number of terms n to 3. What happens to the size of the error at x = 1?
With 2 terms P = 1 − 1/6 = 0.8333 and the error is 0.00814. Adding x⁵/5! makes P = 0.84167, so the error is only −0.000196, about 40 times smaller.
Check it in the model: Move the Number of terms n slider from 2 to 3 and compare the Error readout at x = 1.00.
f(x) polynomial Pₙ(x) error f(x) − Pₙ(x)
sin x = x − x³/3! + x⁵/5! − …
cos x = 1 − x²/2! + x⁴/4! − …
eˣ = 1 + x + x²/2! + x³/3! + …
ln(1 + x) = x − x²/2 + x³/3 − … (|x| < 1)
1/(1 − x) = 1 + x + x² + … (|x| < 1)
Try this
- Choose sin(x) and add terms one by one. Which powers of x appear, and how does the interval where the polynomial matches sin(x) grow with each term?
- Choose ln(1 + x) with 8 terms and move the probe x past 1. What happens to the error, and why does adding more terms not repair it?
- Choose eˣ with 3 terms and set x = 0.5, then x = 2.0. Compare the error at each, and say why the series is far more accurate near x = 0.
Exam tip: The Maclaurin series is f(0) + f′(0)x + f″(0)x²/2! + …; each extra term improves the fit near x = 0, and sin and cos contain only odd or only even powers, so check which powers should appear before you write your answer.
More on 5: Chain and product rules · Second derivative and points of inflexion · Trapezoidal rule and Riemann sums · Displacement and distance travelled · Euler's method and differential equations · Implicit differentiation · Related rates of change · Area under a curve · Calculus of motion · Limits · Newton–Raphson method (enrichment) · Optimisation · Slope fields · Tangents and the derivative · Volume of revolution