Question

homework assignment 3.3 rates of change and behavior of graphs score: 3/11 answered: 3/11 question 5 find the average rate of change of f(x)=4x^2 - 5 on the interval 1,b. your answer will be an expression involving b. question help: video written example message instructor 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 $\frac{f(b)-f(a)}{b - a}$. Here, $a = 1$ and $f(x)=4x^{2}-5$.

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

First, find $f(1)$:
Substitute $x = 1$ into $f(x)=4x^{2}-5$, we get $f(1)=4\times1^{2}-5=4 - 5=-1$.
Then, find $f(b)$:
Substitute $x = b$ into $f(x)=4x^{2}-5$, we get $f(b)=4b^{2}-5$.

Step3: Calculate the average rate of change

Using the formula $\frac{f(b)-f(1)}{b - 1}$, substitute $f(1)=-1$ and $f(b)=4b^{2}-5$:
\[

$$\begin{align*} \frac{f(b)-f(1)}{b - 1}&=\frac{(4b^{2}-5)-(-1)}{b - 1}\\ &=\frac{4b^{2}-5 + 1}{b - 1}\\ &=\frac{4b^{2}-4}{b - 1}\\ &=\frac{4(b^{2}-1)}{b - 1}\\ &=\frac{4(b - 1)(b + 1)}{b - 1}\\ &=4(b + 1) \end{align*}$$

\]

Answer:

$4(b + 1)$