Rates of Change Practice Test

Referring to the context, rate of change means output change per 1-unit input. In a physics test track, a car’s speed is modeled by the polynomial $p(t)= -2t^2+12t$ (m/s) for $0\le t\le 6$. The speed rises from $t=0$ to $t=3$ and then falls from $t=3$ to $t=6$, showing acceleration then deceleration. Over $t=0$ to $t=3$, the average rate of change is $\frac{p(3)-p(0)}{3}$. Over $t=3$ to $t=6$, the average rate of change is $\frac{p(6)-p(3)}{3}$. A sensor delay is modeled by the rational function $r(t)=\frac{10}{t-3}$, which has a vertical asymptote at $t=3$. As $t$ approaches 3, the delay grows without bound in magnitude. Far from $t=3$, the delay approaches 0 seconds. Engineers compare these rates to decide when measurements are least reliable. Based on the passage, at which interval is the car’s average rate of change in speed negative?

Referring to the context, rate of change is the average change in speed per second. A car’s speed is modeled by the polynomial $p(t)=-t^2+10t$ (m/s) for $0\le t\le 10$. The speed increases from $t=0$ because the model starts at $p(0)=0$ and rises quickly. It later decreases because the negative quadratic term eventually outweighs the linear term. For example, $p(3)=21$ and $p(7)=21$, indicating the peak occurs between them. A second measurement is modeled by the rational function $r(t)=\frac{50}{t-12}+10$, approaching $10$ as $t$ increases. Near $t=12$, $r(t)$ has a vertical asymptote and is not usable. Based on the passage, what does a negative average rate of change of $p(t)$ on an interval indicate?