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 $y = f(x)$ over the interval $[a,b]$ is given by $\bar{y}=\frac{1}{b - a}\int_{a}^{b}f(x)dx$. Here, $a=-5$, $b = 4$, and $f(x)=-5x - 4$.

Step2: Calculate the integral

First, find $\int(-5x - 4)dx=-\frac{5}{2}x^{2}-4x + C$. Then, evaluate the definite - integral $\int_{-5}^{4}(-5x - 4)dx=\left(-\frac{5}{2}x^{2}-4x\right)\big|_{-5}^{4}$. [ \begin{align*} \left(-\frac{5}{2}(4)^{2}-4(4)\right)-\left(-\frac{5}{2}(-5)^{2}-4(-5)\right)&=\left(-\frac{5}{2}\times16-16\right)-\left(-\frac{5}{2}\times25 + 20\right)\ &=(-40-16)-\left(-\frac{125}{2}+20\right)\ &=-56-\left(-\frac{125}{2}+20\right)\ &=-56+\frac{125}{2}-20\ &=\frac{-112 + 125-40}{2}\ &=\frac{-27}{2} \end{align*} ]

Step3: Calculate the average value

Now, use the average - value formula. $\bar{y}=\frac{1}{4-(-5)}\int_{-5}^{4}(-5x - 4)dx$. Since $4-(-5)=9$ and $\int_{-5}^{4}(-5x - 4)dx=-\frac{27}{2}$, then $\bar{y}=\frac{1}{9}\times\left(-\frac{27}{2}\right)=-\frac{3}{2}$.

Answer:

$-\frac{3}{2}$