1. The problem gives two functions $R(t)$ and their first and second derivatives $R'(t)$ and $R''(t)$. 2. The function $R(t)$ represents a quantity depending on time $t$ (e.g., revenue, position, etc.). 3. The first derivative $R'(t)$ represents the rate of change of $R(t)$ with respect to time $t$. It tells us how fast $R(t)$ is increasing or decreasing at any time $t$. 4. The second derivative $R''(t)$ represents the rate of change of the first derivative, or the acceleration of $R(t)$. It tells us whether the rate of change is increasing or decreasing. 5. For the first function: $$R(t) = -2.8t^3 + 11.6t^2 - 6.01t + 6.50$$ $$R'(t) = -8.4t^2 + 23.2t - 6.01$$ $$R''(t) = -16.8t + 23.2$$ - $R'(t)$ shows how fast $R(t)$ changes at time $t$. - $R''(t)$ shows how the rate of change itself changes. 6. For the second function: $$R(t) = -55t^3 + 133t^2 - 8.7t + 178.7$$ $$R'(t) = -165t^2 + 266t - 8.7$$ $$R''(t) = -330t + 266$$ - Similarly, $R'(t)$ is the instantaneous rate of change of $R(t)$. - $R''(t)$ is the acceleration or concavity of $R(t)$. 7. Interpretation summary: - $R(t)$ is the main quantity. - $R'(t)$ tells if $R(t)$ is increasing (positive) or decreasing (negative) at time $t$. - $R''(t)$ tells if the increase/decrease is speeding up (positive) or slowing down (negative). This helps understand the behavior of $R(t)$ over time, such as finding maxima, minima, or inflection points.