use the following information to complete parts a. and b. below. f(x)=sin x, a = π/3 a. find the first three…

Explanation:

Step1: Recall Taylor - series formula

The Taylor series of a function $f(x)$ centered at $a$ is given by $f(x)=\sum_{n = 0}^{\infty}\frac{f^{(n)}(a)}{n!}(x - a)^n=f(a)+f^{\prime}(a)(x - a)+\frac{f^{\prime\prime}(a)}{2!}(x - a)^2+\frac{f^{(3)}(a)}{3!}(x - a)^3+\cdots$. First, find the derivatives of $y = f(x)=\sin x$.

Step2: Find the first - derivative

The derivative of $y=\sin x$ is $y^{\prime}=f^{\prime}(x)=\cos x$. Evaluate it at $a = \frac{\pi}{3}$, so $f^{\prime}(\frac{\pi}{3})=\cos\frac{\pi}{3}=\frac{1}{2}$.

Step3: Determine the second non - zero term

The second non - zero term of the Taylor series is $f^{\prime}(a)(x - a)$. Substituting $a=\frac{\pi}{3}$ and $f^{\prime}(\frac{\pi}{3})=\frac{1}{2}$, the second non - zero term is $\frac{1}{2}(x-\frac{\pi}{3})$.

Answer:

$\frac{1}{2}(x - \frac{\pi}{3})$