current objective find the average value of a function over an interval question what is the average value…

Explanation:

Step1: Recall average - value formula

The average value of a function $y = h(x)$ over the interval $[a,b]$ is given by $\frac{1}{b - a}\int_{a}^{b}h(x)dx$. Here, $a=-3$, $b = 4$, and $h(x)=-6x + 7$.

Step2: Calculate the integral

First, find $\int(-6x + 7)dx=-6\times\frac{x^{2}}{2}+7x=-3x^{2}+7x$. Then, evaluate the definite - integral $\int_{-3}^{4}(-6x + 7)dx=\left[-3x^{2}+7x\right]_{-3}^{4}$. [ \begin{align*} &(-3\times4^{2}+7\times4)-(-3\times(-3)^{2}+7\times(-3))\ =&(-3\times16 + 28)-(-3\times9-21)\ =&(-48 + 28)-(-27-21)\ =&-20-(-48)\ =&-20 + 48\ =&28 \end{align*} ]

Step3: Calculate the average value

The length of the interval $b - a=4-(-3)=7$. The average value is $\frac{1}{7}\times28 = 4$.

Answer:

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