Question

and behavior of graphs due monday by 11:59pm points 10 submitting an external tool practice assignment 3.3 rates of change and behavior of graphs score: 30/120 answered: 4/12 save progress done question 4 0/10 pts 2 99 details score on last try: 0 of 10 pts. see details for more. next question get a similar question you can retry this question below find the average rate of change of y(x)=7x³ + 2/x⁴ on the interval -2,3. question help: video message instructor post to forum submit question

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $a=-2$, $b = 3$, and $y(x)=7x^{3}+\frac{2}{x^{4}}$.

Step2: Calculate $y(b)$

Substitute $x = 3$ into $y(x)$:
$y(3)=7\times3^{3}+\frac{2}{3^{4}}=7\times27+\frac{2}{81}=189+\frac{2}{81}=\frac{189\times81 + 2}{81}=\frac{15309+2}{81}=\frac{15311}{81}$.

Step3: Calculate $y(a)$

Substitute $x=-2$ into $y(x)$:
$y(-2)=7\times(-2)^{3}+\frac{2}{(-2)^{4}}=7\times(-8)+\frac{2}{16}=-56+\frac{1}{8}=\frac{-56\times8 + 1}{8}=\frac{-448 + 1}{8}=-\frac{447}{8}$.

Step4: Calculate the average rate of change

$\frac{y(3)-y(-2)}{3-(-2)}=\frac{\frac{15311}{81}-(-\frac{447}{8})}{5}=\frac{\frac{15311}{81}+\frac{447}{8}}{5}$.
First, find a common denominator for the numerator. The common denominator of 81 and 8 is $81\times8 = 648$.
$\frac{15311}{81}+\frac{447}{8}=\frac{15311\times8+447\times81}{648}=\frac{122488+36207}{648}=\frac{158695}{648}$.
Then, $\frac{\frac{158695}{648}}{5}=\frac{158695}{648\times5}=\frac{158695}{3240}=\frac{31739}{648}\approx49$.

Answer:

$\frac{31739}{648}$