Question

State the definition of an \(n\) th-degree Taylor polynomial of \(f\) centered at \(c .\)

Short answer

The nth degree Taylor polynomial of a function \(f\) centered at point \(c\) is defined by the formula: \( P_n(x) = f(c) + f'(c)(x - c) + f''(c)\frac{(x - c)^2}{2!} + \ldots + f^n(c)\frac{(x - c)^n}{n!} \)

Step-by-step solution

Understand key terms

A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. A Taylor polynomial is a certain finite number of terms of the Taylor series. The nth-degree Taylor polynomial is a polynomial of degree ‘n’ which best fits the function. ‘Centered at c’ means that the Taylor polynomial is being calculated about the point ‘c’.

Define nth-degree Taylor polynomial

The nth degree Taylor Polynomial for a function 'f' centered at a point 'c' can be generalised by the formula: \[P_n(x) = f(c) + f'(c)(x - c) + f''(c)\frac{(x - c)^2}{2!} + \ldots + f^n(c)\frac{(x - c)^n}{n!}\]. Here, the 'n' represents the nth derivative of function 'f' at point 'c', and the symbol \(!\) denotes the factorial of the number preceding it.

Provide explanation of the Taylor polynomial form

The representation appears complicated but logically presents the process of approximation. 'n' defines the degree of the polynomial. Each term represents successive derivatives at point 'c', multiplied by the difference of 'x' and 'c' raised to the corresponding term's degree, all over the factorial of that degree.