Dick said:Read this http://en.wikipedia.org/wiki/Taylor_series and try it out yourself. It's just a cookbook formula. You have to add various derivatives of U(x) at x=4 times powers of (x-4) up to quadratic. It's not that mysterious. We'll be glad to look at your efforts.
A simple Taylor expansion is a method used in calculus to approximate a function using a polynomial. It is based on the Taylor series, which is an infinite sum of terms that can represent a function at a specific point.
A simple Taylor expansion is calculated by finding the derivatives of a function at a specific point and then plugging them into the Taylor series formula. The formula is:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
The purpose of a simple Taylor expansion is to approximate a function at a specific point using a polynomial. This can be useful in situations where the function is complex, but its derivatives are easier to calculate.
A Taylor series is an infinite sum of terms that represents a function at a specific point, while a simple Taylor expansion is a finite sum that approximates a function at a specific point. A Taylor series provides a more accurate representation, but a simple Taylor expansion is easier to calculate.
A simple Taylor expansion is not a good approximation when the function is not smooth or when it has a singularity at the point of expansion. In these cases, the Taylor series may not converge and the simple Taylor expansion will not provide an accurate approximation.