Final answer:
The degree of the Taylor polynomial is determined by the highest power of x in the polynomial. The Taylor polynomial is centered at a specific point, in this case, x=18. To evaluate T(18), substitute the value of x=18 into the polynomial expression. The purpose of the Taylor polynomial is to approximate a function near a given point.
Step-by-step explanation:
a. The degree of the Taylor polynomial T(x) centered at x=18 is determined by the highest power of x in the polynomial. It is equal to n if the Taylor polynomial is of nth degree The Taylor polynomial T(x) is centered at x=18, which means that the polynomial is approximating the original function near the point x=18. To evaluate T(18) for a specific n, you need to substitute the value of x=18 into the Taylor polynomial expression and calculate the result.
purpose of the Taylor polynomial T(x) is to approximate a function near a given point by using a polynomial. This approximation can be used to estimate the value of the function or to simplify calculations.The student's question pertains to Taylor polynomials, which are polynomial approximations of functions. Let's answer each part of the question:The degree of T(x) is n, where n is the highest power of the polynomial. The Taylor polynomial T(x) is centered at x=18. To evaluate T(18) for a specific n, you need the coefficients of the Taylor polynomial. However, as a general property, since the polynomial is centered at 18, T(18) will always be equal to the value of the original function at x=18, regardless of the value of n.. The purpose of T(x) is to approximate the original function near the point where it is centered, making it easier to calculate or estimate the function's value for points close to the center.