what is the average value of cos(x) on the interval -2, 7? choose 1 answer: a (sin(7) + sin(2))/9 b (sin(7)…

Explanation:

Step1: Recall average - value formula

The average value of a function $y = f(x)$ on the interval $[a,b]$ is given by $\bar{y}=\frac{1}{b - a}\int_{a}^{b}f(x)dx$. Here, $a=-2$, $b = 7$ and $f(x)=\cos(x)$, so $\bar{y}=\frac{1}{7-(-2)}\int_{-2}^{7}\cos(x)dx=\frac{1}{9}\int_{-2}^{7}\cos(x)dx$.

Step2: Integrate $\cos(x)$

We know that the antiderivative of $\cos(x)$ is $\sin(x)$. By the fundamental theorem of calculus, $\int_{-2}^{7}\cos(x)dx=\left[\sin(x)\right]{-2}^{7}=\sin(7)-\sin(-2)$. Since $\sin(-x)=-\sin(x)$, then $\sin(-2)=-\sin(2)$, so $\int{-2}^{7}\cos(x)dx=\sin(7)+\sin(2)$.

Step3: Calculate the average value

Substitute the result of the integral into the average - value formula: $\bar{y}=\frac{\sin(7)+\sin(2)}{9}$.

Answer:

A. $\frac{\sin(7)+\sin(2)}{9}$