Taylor Expansion
Taylor expansion approximates a function as a truncated polynomial series about a point x=a based on the function’s derivatives at a.
The Taylor expansion about a up to degree n is:
f(x) ≈ f(a) + f’(a)(xa) + f’’(a)(xa)^2/2! + … + f^n(a)(xa)^n/n!
Where f^(n)(a) is the nth derivative of f evaluated at a.
Java example  e^x approximation:


C++ example  sin(x) approximation:


Python example  log(1+x) approximation:


Taylor expansion provides a polynomial form for approximating functions, useful for numerical analysis.
Taylor expansion is a way to represent a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. It provides a polynomial approximation for functions that are differentiable. The idea is to expand the function around a point, ( a ), using its derivatives:
[ f(x) = f(a) + f’(a)(x  a) + \frac{f’’(a)(x  a)^2}{2!} + \frac{f’’’(a)(x  a)^3}{3!} + \cdots ]
Each term involves a derivative of the function ( f(x) ) evaluated at the point ( a ). Taylor expansion is widely used in numerical methods, simulations, and calculations involving functions that are complex to deal with directly.
Java Code for Taylor Expansion


In Java, the code calculates the Taylor expansion of ( e^x ) using a for
loop. The Math.pow()
method is used for exponentiation, and the factorial is calculated with a separate factorial()
function.
C++ Code for Taylor Expansion


In C++, the code for Taylor expansion is quite similar to the Java code. The pow()
function from <cmath>
is used for exponentiation.
Python Code for Taylor Expansion


In Python, the math.pow()
and math.factorial()
functions are used for exponentiation and calculating the factorial, respectively.
Taylor expansion is a fundamental concept in calculus and has numerous applications in engineering, physics, and computer science. It serves as a basis for approximating functions and is essential for various computational techniques.