3. determine the average rate of change over the interval -1.5,2.4 for the function $f(x)=3x^{2}-4x + 2$.

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 $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1.5$ and $b = 2.4$.

Step2: Calculate $f(a)$

Substitute $x=-1.5$ into $f(x)=3x^{2}-4x + 2$. [ \begin{align*} f(-1.5)&=3\times(-1.5)^{2}-4\times(-1.5)+2\ &=3\times2.25 + 6+2\ &=6.75+6 + 2\ &=14.75 \end{align*} ]

Step3: Calculate $f(b)$

Substitute $x = 2.4$ into $f(x)=3x^{2}-4x + 2$. [ \begin{align*} f(2.4)&=3\times(2.4)^{2}-4\times2.4+2\ &=3\times5.76-9.6 + 2\ &=17.28-9.6+2\ &=9.68 \end{align*} ]

Step4: Calculate the average rate of change

[ \begin{align*} \frac{f(2.4)-f(-1.5)}{2.4-(-1.5)}&=\frac{9.68 - 14.75}{2.4 + 1.5}\ &=\frac{-5.07}{3.9}\ &=-1.3 \end{align*} ]

Answer:

$-1.3$