Question

Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function, centered at a?

Answer 1

The Taylor series is a representation of a function as an infinite sum of terms involving derivatives. The first four nonzero terms can be found by calculating derivatives and plugging them into the series formula.

Step-by-step explanation:

The Taylor series for a function centered at a is a representation of the function as an infinite sum of terms involving the derivatives of the function at a. The first four nonzero terms of the Taylor series can be found by calculating the derivatives of the function at a and plugging them into the series.

Each term in the Taylor series is defined by the formula:

`Tn(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... + f^n(a)(x-a)^n/n!`where n is the degree of the term, f(a) is the value of the function at a, and
`f^n(a)` is the nth derivative of the function at a.

For example, if we want to find the first four nonzero terms of the Taylor series for the function f(x) centered at a, we start by calculating f(a), f'(a), f''(a), and f'''(a). Then, we plug these values into the formula to find the first four terms of the series.