find the average rate of change of f(x)=5x² -...
find the average rate of change of f(x)=5x² - 2 on the interval 4,t. your answer will be an expression involving t. question help: message instructor
Answer
# 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 $\frac{f(b)-f(a)}{b - a}$. Here, $a = 4$, $b=t$, and $f(x)=5x^{2}-2$.
## Step2: Find $f(t)$ and $f(4)$
$f(t)=5t^{2}-2$, and $f(4)=5\times4^{2}-2=5\times16 - 2=80 - 2=78$.
## Step3: Calculate the average rate of change
The average rate of change is $\frac{f(t)-f(4)}{t - 4}=\frac{(5t^{2}-2)-78}{t - 4}=\frac{5t^{2}-80}{t - 4}$.
Factor the numerator: $5t^{2}-80 = 5(t^{2}-16)=5(t + 4)(t - 4)$.
Then $\frac{5(t + 4)(t - 4)}{t - 4}=5(t + 4)$.
# Answer:
$5(t + 4)$