Question

practice assignment 3.3 rates of change and behavior of graphs
score: 40/120 answered: 4/12
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question 5
find the average rate of change of f(x)=8x^2 - 4 on the interval 1,a. your answer will be an expression involving a.

Explanation:

Step1: Recall the 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, $x_1 = 1$, $x_2=a$, and $f(x)=8x^{2}-4$.

Step2: Find $f(1)$ and $f(a)$

First, find $f(1)$:
Substitute $x = 1$ into $f(x)=8x^{2}-4$, we get $f(1)=8\times1^{2}-4=8 - 4=4$.
Then, find $f(a)$:
Substitute $x = a$ into $f(x)=8x^{2}-4$, we get $f(a)=8a^{2}-4$.

Step3: Calculate the average rate of change

The average rate of change is $\frac{f(a)-f(1)}{a - 1}=\frac{(8a^{2}-4)-4}{a - 1}=\frac{8a^{2}-8}{a - 1}$.
Factor the numerator: $8a^{2}-8 = 8(a^{2}-1)=8(a + 1)(a - 1)$.
So, $\frac{8(a + 1)(a - 1)}{a - 1}=8(a + 1)$ for $a
eq1$.

Answer:

$8(a + 1)$