Part 3/8:
To construct a quadratic approximation, we start with a function ( f(x) ) and define its quadratic polynomial approximation, denoted as ( P_2(x) ). The key conditions for this approximation are:
( P_2(a) = f(a) ): The polynomial value at point ( a ) equals the function value at that point.
( P_2'(a) = f'(a) ): The first derivative (slope) of the polynomial matches the first derivative of the function at point ( a ).
( P_2''(a) = f''(a) ): The second derivative of the polynomial approximates the second derivative of the function at point ( a ).
This method leads us to the general form of a quadratic polynomial, which can be restructured for ease of calculation:
[
P_2(x) = a + b(x - a) + c(x - a)^2
]