Taylor Polynomial Calculator
Use standard mathematical notation: +, -, *, /, ^, sin(), cos(), etc.
Point around which to expand the function (use 0 for Maclaurin series)
Higher degrees provide better approximations but more complex polynomials
What is a Taylor Polynomial?
A Taylor polynomial is a polynomial approximation of a function centered at a specific point. It uses the function's derivatives at that point to create a polynomial that matches the function's behavior near the center point. Taylor polynomials are powerful tools for approximating complex functions with simpler polynomial expressions.
The Taylor Polynomial Formula
The Taylor polynomial of degree n for a function f(x) centered at a point a is given by:
Pn(x) = f(a) + f′(a)(x-a)/1! + f″(a)(x-a)²/2! + ... + f(n)(a)(x-a)n/n!
When the center point a = 0, the Taylor polynomial is also called a Maclaurin polynomial:
Pn(x) = f(0) + f′(0)x/1! + f″(0)x²/2! + ... + f(n)(0)xn/n!
How to Find a Taylor Polynomial
- Choose the center point a and the degree n of the polynomial
- Calculate the function value f(a) at the center point
- Calculate the derivatives f′(a), f″(a), ..., f(n)(a) at the center point
- Substitute these values into the Taylor polynomial formula
- Simplify the resulting expression
Common Taylor Polynomial Examples
ex centered at x = 0
Maclaurin polynomial of degree 4:
P4(x) = 1 + x + x²/2 + x³/6 + x⁴/24
sin(x) centered at x = 0
Maclaurin polynomial of degree 5:
P5(x) = x - x³/6 + x⁵/120
ln(x) centered at x = 1
Taylor polynomial of degree 3:
P3(x) = (x-1) - (x-1)²/2 + (x-1)³/3
cos(x) centered at x = 0
Maclaurin polynomial of degree 4:
P4(x) = 1 - x²/2 + x⁴/24
Applications of Taylor Polynomials
Function Approximation
Taylor polynomials provide simple approximations of complex functions, making them easier to work with in calculations.
Numerical Methods
They form the basis for many numerical algorithms used to solve differential equations and evaluate integrals.
Error Estimation
Taylor's theorem provides bounds on the error when approximating a function with a Taylor polynomial.
Physics
Used to approximate solutions to physical problems where exact solutions are difficult to obtain.
Computer Graphics
Taylor polynomials help in rendering curves and surfaces efficiently in computer graphics applications.
Signal Processing
Used to approximate and analyze signals in communication systems and audio processing.
Taylor Polynomial vs. Taylor Series
A Taylor polynomial is a finite approximation of a function, while a Taylor series is the infinite sum of all terms in the Taylor polynomial. The Taylor series for a function f(x) centered at a is:
f(x) = f(a) + f′(a)(x-a)/1! + f″(a)(x-a)²/2! + f‴(a)(x-a)³/3! + ...
A Taylor series converges to the function within its radius of convergence. The Taylor polynomial provides an approximation that becomes more accurate as more terms are included, especially near the center point.
Error in Taylor Polynomial Approximation
The error when approximating a function f(x) with its Taylor polynomial of degree n centered at a is given by the remainder term:
Rn(x) = f(n+1)(ξ)(x-a)n+1/(n+1)!
Where ξ is some value between a and x. This remainder term helps quantify how accurate the approximation is at different points.
Pro Tip
When using Taylor polynomials for approximation, remember that the accuracy decreases as you move away from the center point. For best results, choose a center point close to the region where you need the most accuracy.