Taylor and Maclaurin Polynomial

Taylor polynomials

A Taylor polynomial uses the function value and its derivatives at a chosen centre. The coefficient of degree k is f⁽ᵏ⁾(a)/k!. This tool uses the existing symbolic differentiation engine for orders up to four. Choosing zero as the centre gives a Maclaurin polynomial.

Example

All derivatives of exp(x) equal 1 at zero. The polynomial of degree four is 1+x+x²/2+x³/6+x⁴/24. At x=1 it gives 65/24, an approximation to exp(1).

Interpretation

The reported absolute difference compares the numeric function and polynomial values at one selected point. It is neither an error bound over an interval nor a proof that an infinite series converges. Check differentiability near the centre before using the approximation.

Build the expression around its centre

As an example, choose f(x) = x³, centre 2 and degree 2. At the centre, the function is 8, its first derivative is 12 and its second derivative is 12. The coefficients are 8, 12 and 12/2 = 6, giving 8 + 12(x − 2) + 6(x − 2)². At target 2.5, the polynomial is 15.5. The actual function is 15.625, so the absolute difference is 0.125. Adding the cubic term represents this polynomial function exactly.

Expansion powers apply to x − a, the distance from the centre, rather than to x alone. Reading the coefficients as coefficients of powers of x gives the wrong result away from a zero centre. At the centre itself, the constant term must equal the function value. This is a simple check.

Trigonometric expressions use radians. Enter supported notation and avoid a function undefined at the centre. Raising degree does not guarantee sufficient accuracy at every target. Do not use the single point difference as an error bound for an entire interval. A singularity or definition change between centre and target needs separate examination.

Frequently asked questions

How do you find the Maclaurin series for e^x?

For example, the degree 4 Maclaurin polynomial of exp(x) about zero gives 2.708 at 1, against a true value of 2.718. Each coefficient is the derivative at the centre divided by a factorial, and every derivative equals one at zero here, so this finite expansion does not establish convergence of the infinite series or bound the error over an entire interval.

Calculate with these values