1. The problem asks for the average rate of change of a function $g$ over the interval $[-3,3]$ using the given table values. 2. The average rate of change formula over an interval $[a,b]$ is: $$\text{Average rate of change} = \frac{g(b) - g(a)}{b - a}$$ This formula calculates the slope of the secant line between points $(a, g(a))$ and $(b, g(b))$. 3. From the table, $g(-3) = 12$ and $g(3) = -4$. Substitute these into the formula: $$\frac{g(3) - g(-3)}{3 - (-3)} = \frac{-4 - 12}{3 + 3} = \frac{-16}{6} = -\frac{8}{3}$$ 4. Among the options, option B matches the correct formula: $$\frac{(-4) - 12}{3 - (-3)}$$ --- 5. Next, for $f(x) = e^{2x}$, the average rate of change over $[1,b]$ is 20. The formula is: $$\frac{f(b) - f(1)}{b - 1} = 20$$ This corresponds to option C. --- 6. For $f(x) = 2x^3 - x$, the average rate of change over $[1,3]$ is: $$\frac{f(3) - f(1)}{3 - 1}$$ This matches option C. --- 7. For $g$ with values $g(-2) = -3$ and $g(2) = 5$, the average rate of change over $[-2,2]$ is: $$\frac{g(2) - g(-2)}{2 - (-2)} = \frac{5 - (-3)}{2 + 2} = \frac{8}{4} = 2$$ Option B matches this formula. --- 8. For $f(x) = 2 \sin x + \cos x$, average rate of change over $[0,b]$ is 0.05: $$\frac{f(b) - f(0)}{b - 0} = 0.05$$ This is option C. --- 9. For $f(x) = \sec x + \csc x$ over $[\frac{\pi}{4}, \frac{3\pi}{8}]$, average rate of change is: $$\frac{f(\frac{3\pi}{8}) - f(\frac{\pi}{4})}{\frac{3\pi}{8} - \frac{\pi}{4}}$$ This matches option C. Final answers: - First problem: B - Second problem: C - Third problem: C - Fourth problem: B - Fifth problem: C - Sixth problem: C