find the average rate of change of f(x)=4x² - 3 on the interval 5,a. your answer will be an expression…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[x_1,x_2]$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$. Here, $x_1 = 5$, $x_2=a$, and $f(x)=4x^{2}-3$.

Step2: Find $f(5)$ and $f(a)$

First, find $f(5)$: $f(5)=4\times5^{2}-3=4\times25 - 3=100 - 3=97$. Then, find $f(a)$: $f(a)=4a^{2}-3$.

Step3: Calculate the average rate of change

Substitute $f(5)$ and $f(a)$ into the average - rate - of - change formula: $\frac{f(a)-f(5)}{a - 5}=\frac{(4a^{2}-3)-97}{a - 5}=\frac{4a^{2}-100}{a - 5}$. Factor the numerator: $4a^{2}-100 = 4(a^{2}-25)=4(a + 5)(a - 5)$. So, $\frac{4(a + 5)(a - 5)}{a - 5}=4(a + 5)$.

Answer:

$4(a + 5)$