Part 4/8:
The coefficients ( a ), ( b ), and ( c ) can be determined using the above conditions, thus allowing us to effectively approximate the function in a neighborhood around point ( a ).
Introducing Taylor Polynomials
Taylor polynomials extend the concept of approximating functions beyond quadratics. Named after mathematician Brook Taylor, this approach allows us to create a polynomial ( T_n(x) ) of any degree ( n ) that serves as an approximation for ( f(x) ) around the point ( a ):
[
T_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \ldots + \frac{f^{(n)}(a)}{n!}(x - a)^n
]