Taylor series expansion
Taylor Series expansion Calculator
This online Taylor Series expansion calculator is designed to facilitate the approximation of functions around a specific point. It allows for the decomposition of a function into a polynomial series, offering a clear view of its local behavior. This tool is especially useful for students, teachers, and professionals looking to analyze complex functions in a simplified manner.
How to Use the Taylor Series expansion Calculator
To use this tool, fill in the following fields:
- Main Variable: The function's variable (e.g., x).
- Function: The function to develop (e.g., e^x, sin(x)).
- x0 (Point): The point around which the development is performed (e.g., 0, 1).
- Order (n): The order up to which the development is calculated (e.g., 2, 3).
Data Entry Guide
This guide helps you correctly enter data into the calculator:
| Variables | A function can have one or more variables, but only one main variable. A variable is a single lowercase or uppercase letter. Examples: A function f with one main variable : f(x) = 4*x A function g with one main variable x and a secondary parameter m, g(x) = 4*x*m + x + 1, In this case, enter x in the “main variable” field |
|---|---|
| Numbers | Use a dot as decimal separator |
| Operators |
+ (addition), - (substration), * (multiplication), / (division), ^ (power), For multiply operator, enter a*b not a.b nor ab. Example: 2*x. |
| Constants | You may use these constants : pi (approx. 3.14), e (approx. 2,72) Examples: f(x) = pi * x or f(x) = e * (x+1+2*e)2 |
| Common Functions |
You may use theses functions in the expression of f(x) sqrt(x) (square root), exp(x) (exponential function), log(x) or ln (natural logarithm), |
| Trigonometric functions |
You may use theses functions in the expression of f(x) sin (sine), cos (cosine), tan (tangent), cot (cotangent), sec (secant), csc (cosecant), |
| Inverse trigonometric functions |
You may use theses functions in the expression of f(x) arcsin (arcsine), arccos (arccosine), arctan (arctangent), arccot (arcotangent), arcsec (arcsecant), arccsc (arccosecant), |
| Hyperbolic Functions |
You may use theses functions in the expression of f(x) sinh (hyperbolic sine), cosh (hyperbolic cosine), tanh (hyperbolic tangent), coth (hyperbolic cotangent), sech (hyperbolic secante), csch (hyperbolic cosecant) |
| Inverse hyperbolic functions |
You may use theses functions in the expression of f(x) asinh (inverse hyperbolic sine), acosh (inverse hyperbolic cosine), atanh (inverse hyperbolic tangent), acoth (inverse hyperbolic cotangent), asech (inverse hyperbolic secant), acsch (inverse hyperbolic cosecant) |
What Is a Taylor Series expansion?
Taylor Series expansion is a polynomial approximation of a function around a given point, often used to simplify the analysis of local behaviors of complex functions. It involves expressing a function as a sum of polynomial terms and a remainder, usually as a function of the power of `(x - a)`, where `a` is the development point.
Importance and Use of Taylor Series expansion
Taylor Series expansion are essential in analysis and calculus for estimating the behavior of functions near a specific point. They are particularly useful for calculating limits, derivatives, and integrals. These approximations also help simplify complex expressions in asymptotic analyses or convergence studies.
Aspects and Rules for Calculating Taylor Series expansion
To calculate a Taylor Series expansion, it is necessary to know the successive derivatives of the function at the development point. The general formula (or Taylor's formula) for a Taylor Series expansion of order `n` around `a` is given by:
`f(x) ≈ f(a) + f'(a)*(x-a) + (f''(a))/(2!)*(x-a)^2 + ... + f^n(a)/(n!)*(x-a)^n`
Taylor Series expansion of Common Functions
Here are some examples of Taylor Series expansion for commonly used functions:
- `e^x ≈ 1 + x + x^2/2! + x^3/3! + ...` around 0.
- `sin(x) ≈ x - x^3/3! + x^5/5! - ...` around 0.
- `cos(x) ≈ 1 - x^2/2! + x^4/4! - x^6/6! + ...` around 0.
- `tan(x) ≈ x + x^3/3 + 2x^5/15 + 17x^7/315 + ...` around 0.
- `ln(1+x) ≈ x - x^2/2 + x^3/3 - ...` around 0.
- `sqrt(1+x) ≈ 1 + x/2 - x^2/8 + ...` around 0.
- `(1+x)^a ≈ 1 + ax + a(a-1)x^2/2! + a(a-1)(a-2)x^3/3! + ...` around 0 (a is a real number).
Examples of Taylor Series expansion Calculation
Here are several examples illustrating the calculation of Taylor Series expansion for different functions using the Taylor method:
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Development of `cos^2(x)` around 0 up to the 2nd order:
For `f(x) = cos^2(x)`, calculate `f(x_0)`, `f'(x_0)`, and `f''(x_0)` for `x_0 = 0`.
We have `f(x_0) = cos^2(0) = 1`,
`f'(x) = -2*cos(x)*sin(x)` so `f'(x_0) = 0`, and
`f''(x) = -2*cos^2(x) + 2*sin^2(x)` so `f''(x_0) = -2`.
The Taylor Series expansion is therefore `cos^2(x) ≈ 1 - x^2 + o(x^2)`.
-
Development of `1/(1 - x)` around 0 up to the 2nd order:
With `f(x) = 1/(1 - x)`, determine `f(x_0)`, `f'(x_0)`, and `f''(x_0)` for `x_0 = 0`.
Here, `f(x_0) = 1/(1 - 0) = 1`,
`f'(x) = 1/(1 - x)^2` so `f'(x_0) = 1`, and
`f''(x) = 2/(1 - x)^3` so `f''(x_0) = 2`.
The Taylor Series expansion is then `1/(1 - x) ≈ 1 + x + x^2 + o(x^2)`.
Conclusion
The Taylor Series expansion calculator is a valuable tool for studying the local behavior of functions. It facilitates the approximation of complex functions and enhances the understanding of fundamental function properties in mathematical analysis.
See also
Derivative of a function
Primitive calculator
Function limit
Value of a function
Definite Integral