Final answer:
To calculate a Taylor polynomial centered at a particular point, one must evaluate the function and its derivatives at that point and then use the Taylor series formula to construct the polynomial. For quadratic functions, only the second-degree Taylor polynomial is needed.
Step-by-step explanation:
To calculate the Taylor polynomials centered at a specific point for a given function, we use the formula:
- T_n(x) = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \dots + \frac{f^{(n)}(c)}{n!}(x-c)^n,
where T_n(x) represents the Taylor polynomial of degree n centered at c, and f^{(k)}(c) denotes the kth derivative of f evaluated at c. For quadratic functions, the Taylor polynomial of degree 2 would be sufficient since higher-order terms will have derivatives equal to zero.
To construct such a polynomial, first, determine the value of the function and its derivatives at the specified point. Then, plug these values into the Taylor polynomial formula. If the function were associated with an Hermite polynomial, as briefly mentioned, you would use the given information about Hermite polynomials to tailor the Taylor polynomial accordingly. Learning about graphing polynomials can help you understand how the constants in the polynomial affect the shape of the graph.