how do the average rates of change for the pa...
how do the average rates of change for the pair of functions compare over the given interval? f(x)=0.4x² g(x)=1.2x² 5≤x≤10 the average rate of change of f(x) over 5≤x≤10 is . the average rate of change of g(x) over 5≤x≤10 is . the average rate of change of g(x) is times that of f(x). (simplify your answers. type integers or decimals.)
Answer
# Explanation:
## Step1: Recall average - rate - of - change formula
The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$.
## Step2: Calculate average rate of change of $f(x)$
For $f(x)=0.4x^{2}$, $a = 5$, $b = 10$.
$f(5)=0.4\times5^{2}=0.4\times25 = 10$.
$f(10)=0.4\times10^{2}=0.4\times100 = 40$.
The average rate of change of $f(x)$ is $\frac{f(10)-f(5)}{10 - 5}=\frac{40 - 10}{5}=\frac{30}{5}=6$.
## Step3: Calculate average rate of change of $g(x)$
For $g(x)=1.2x^{2}$, $a = 5$, $b = 10$.
$g(5)=1.2\times5^{2}=1.2\times25 = 30$.
$g(10)=1.2\times10^{2}=1.2\times100 = 120$.
The average rate of change of $g(x)$ is $\frac{g(10)-g(5)}{10 - 5}=\frac{120 - 30}{5}=\frac{90}{5}=18$.
## Step4: Find the ratio
The ratio of the average rate of change of $g(x)$ to that of $f(x)$ is $\frac{18}{6}=3$.
# Answer:
The average rate of change of $f(x)$ over $5\leq x\leq10$ is $6$. The average rate of change of $g(x)$ over $5\leq x\leq10$ is $18$. The average rate of change of $g(x)$ is $3$ times that of $f(x)$.