Taylor Series Interval Calculator

A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This calculator helps you compute Taylor series approximations over a specified interval and visualize the results.

What is Taylor Series?

The Taylor series of a function is a sum of terms that are expressed in terms of the function's derivatives at a single point. For a function f(x) that is infinitely differentiable at a point a, the Taylor series centered at a is:

f(x) ≈ f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)² + (f'''(a)/3!)(x-a)³ + ...

Taylor series are used in various fields of mathematics, physics, and engineering to approximate functions, solve differential equations, and analyze systems. The calculator allows you to compute these approximations over a specific interval.

How to Use the Calculator

Enter the function you want to approximate (e.g., "sin(x)", "e^x", "x^2").
Specify the center point "a" where the Taylor series is centered.
Enter the interval [x_min, x_max] over which you want to compute the approximation.
Select the number of terms in the Taylor series.
Click "Calculate" to see the approximation and visualization.

Formula

The nth-order Taylor series approximation of a function f(x) centered at a is given by:

f(x) ≈ Σ [f^(k)(a) / k!] * (x - a)^k for k = 0 to n

Where:

f^(k)(a) is the kth derivative of f evaluated at a
k! is the factorial of k
(x - a)^k is the (x - a) raised to the power of k

Example Calculation

Let's approximate the function f(x) = e^x centered at a = 0 with 5 terms over the interval [-2, 2].

Example Input:

Function: e^x
Center point (a): 0
Interval: [-2, 2]
Number of terms: 5

The Taylor series approximation will be:

e^x ≈ 1 + x + (x²)/2! + (x³)/3! + (x⁴)/4!

The calculator will display the approximation curve along with the original function for comparison.

FAQ

What is the difference between Taylor and Maclaurin series?

A Maclaurin series is a special case of Taylor series where the center point a is 0. So, Maclaurin series are Taylor series centered at 0.

How many terms should I use for a good approximation?

The number of terms needed depends on the function and the interval. Generally, more terms provide a better approximation but may lead to numerical instability. Start with 5-10 terms and adjust as needed.

Can I use complex functions with this calculator?

This calculator currently supports real-valued functions. Complex functions are not supported in the current version.