14 finding taylor or maclaurin series calculus - calculator allowed 1. what is the coefficient of x^9 in the…

Explanation:

Step1: Recall Maclaurin series formula

The Maclaurin series for $e^t=\sum_{n = 0}^{\infty}\frac{t^n}{n!}=1 + t+\frac{t^2}{2!}+\frac{t^3}{3!}+\cdots$. Let $t=x^3$, then $e^{x^3}=\sum_{n = 0}^{\infty}\frac{(x^3)^n}{n!}=\sum_{n = 0}^{\infty}\frac{x^{3n}}{n!}$.

Step2: Find the term with $x^9$

We want to find the coefficient of $x^9$. Set $3n=9$, solving for $n$ gives $n = 3$. When $n = 3$, the term in the series for $e^{x^3}$ is $\frac{x^{3\times3}}{3!}=\frac{x^9}{6}$.

Step3: Consider the function $f(x)=\frac{e^{x^3}}{3}$

Since $f(x)=\frac{e^{x^3}}{3}$, and the coefficient of $x^9$ in $e^{x^3}$ is $\frac{1}{6}$, then the coefficient of $x^9$ in $f(x)$ is $\frac{1}{3}\times\frac{1}{6}=\frac{1}{18}$.

Answer:

$\frac{1}{18}$