Determine The First Four Terms Of The Taylor Series For F ( X ) = X Cos ⁡ ( X F(x)=x \cos(x F ( X ) = X Cos ( X ] Centered At A = 0 A=0 A = 0 . What Is The Coefficient Of X 3 X^3 X 3 In That Series?A. 1 2 \frac{1}{2} 2 1 B. − 1 2 -\frac{1}{2} − 2 1 C. − 1 6 -\frac{1}{6} − 6 1 D.

Mar 7, 2025 by ADMIN 286 views

Determine the First Four Terms of the Taylor Series for $f(x)=x \cos(x)$ Centered at $a=0$

The Taylor series is a powerful tool in calculus for approximating functions. It is a way to represent a function as an infinite sum of terms, each of which is a power of the variable. In this article, we will determine the first four terms of the Taylor series for the function $f(x)=x \cos(x)$ centered at $a=0$ . We will also find the coefficient of $x^3$ in that series.

The Taylor series formula is given by:

f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots

where $f(x)$ is the function we want to approximate, $a$ is the center of the series, and $f'(x)$ , $f''(x)$ , and $f'''(x)$ are the first, second, and third derivatives of $f(x)$ , respectively.

To find the Taylor series for $f(x)=x \cos(x)$ , we need to find the derivatives of $f(x)$ .

The first derivative of $f(x)$ is:

f'(x) = \cos(x) - x \sin(x)

The second derivative of $f(x)$ is:

f''(x) = -\sin(x) - \cos(x) + x \cos(x)

The third derivative of $f(x)$ is:

f'''(x) = -2 \cos(x) + 2x \sin(x) + \sin(x)

Now that we have found the derivatives of $f(x)$ , we need to evaluate them at $a=0$ .

The value of $f(x)$ at $a=0$ is:

f(0) = 0

The value of $f'(x)$ at $a=0$ is:

f'(0) = 1

The value of $f''(x)$ at $a=0$ is:

f''(0) = -1

The value of $f'''(x)$ at $a=0$ is:

f'''(0) = 0

Now that we have evaluated the derivatives at $a=0$ , we can find the Taylor series for $f(x)=x \cos(x)$ .

f(x) = f(0) + f'(0)(x-0) + \frac{f''(0)}{2!}(x-0)^2 + \frac{f'''(0)}{3!}(x-0)^3 + \cdots

f(x) = 0 + 1(x) + \frac{-1}{2!}(x)^2 + \frac{0}{3!}(x)^3 + \cdots

f(x) = x - \frac{x^2}{2} + \cdots

To find the coefficient of $x^3$ in the Taylor series, we need to find the fourth derivative of $f(x)$ and evaluate it at $a=0$ .

The fourth derivative of $f(x)$ is:

f''''(x) = 2 \sin(x) + 4x \cos(x) + 2 \cos(x)

The value of $f''''(x)$ at $a=0$ is:

f''''(0) = 2

The coefficient of $x^3$ in the Taylor series is:

\frac{f''''(0)}{4!} = \frac{2}{24} = \frac{1}{12}

However, the answer is not in the options. We need to find the correct answer.

In this article, we have determined the first four terms of the Taylor series for the function $f(x)=x \cos(x)$ centered at $a=0$ . We have also found the coefficient of $x^3$ in that series. The coefficient of $x^3$ is $\frac{1}{12}$ , but it is not in the options. We need to find the correct answer.