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Finding Taylor Polynomial of Degree N Calculator

Reviewed by Calculator Editorial Team

A Taylor polynomial is an approximation of a function using a sum of terms calculated from the function's derivatives at a single point. This calculator helps you find the Taylor polynomial of degree n for any function at a given point.

What is a Taylor Polynomial?

A Taylor polynomial is a finite series of terms that approximates a function near a point. It's constructed using the function's derivatives at a single point, called the center of expansion. The general form of a Taylor polynomial of degree n for a function f(x) centered at a is:

Pn(x) = f(a) + f'(a)(x - a) + f''(a)/2!(x - a)2 + ... + f(n)(a)/n!(x - a)n

Taylor polynomials are widely used in mathematics, physics, and engineering for approximation, interpolation, and solving differential equations. They provide a way to understand the behavior of functions near a specific point.

How to Find a Taylor Polynomial

To find a Taylor polynomial of degree n for a function f(x) centered at a, follow these steps:

  1. Calculate the function value at the center point: f(a)
  2. Compute the first n derivatives of the function at the center point
  3. For each derivative, divide by the factorial of its order
  4. Multiply each term by (x - a) raised to the power of its order
  5. Sum all the terms to get the Taylor polynomial

Note: The Taylor polynomial is an approximation that becomes more accurate as the degree n increases, but it may not be exact for all functions.

For many common functions, Taylor polynomials can be derived using known derivative formulas. The calculator on this page automates this process for you.

Example Calculation

Let's find the Taylor polynomial of degree 3 for the function f(x) = ex centered at a = 0.

  1. f(0) = e0 = 1
  2. f'(x) = ex → f'(0) = 1
  3. f''(x) = ex → f''(0) = 1
  4. f'''(x) = ex → f'''(0) = 1

The Taylor polynomial of degree 3 is:

P3(x) = 1 + x + x2/2! + x3/3! = 1 + x + x2/2 + x3/6

This approximation becomes more accurate as we increase the degree of the polynomial.

FAQ

What is the difference between Taylor and Maclaurin polynomials?

A Maclaurin polynomial is a special case of a Taylor polynomial where the center of expansion is at 0. So Maclaurin polynomials are Taylor polynomials centered at 0.

How accurate are Taylor polynomials?

Taylor polynomials provide increasingly accurate approximations as the degree increases. However, they may not be exact for all functions, especially those with singularities.

Can Taylor polynomials be used for complex functions?

Yes, Taylor polynomials can be extended to complex functions. The same principles apply, but derivatives and factorials are generalized to complex numbers.