The table records a cart's position $s(t)$, in meters, at time $t$ in minutes. Using the values that bracket $t=3$, approximate $s^{\prime }(3)$.

The table records the temperature $T(t)$ of a liquid, in degrees Celsius, after $t$ minutes. Use the rows at $t=2$ and $t=4$ to approximate $T^{\prime }(3)$.

The table records a tank's volume $V(t)$, in liters, at time $t$ in minutes. Using the nearest values on either side of $t=5$, approximate $V^{\prime }(5)$.

The table records a plant's height $H(t)$, in meters, after $t$ minutes. Use the rows at $t=2$ and $t=4$ to approximate $H^{\prime }(3)$.

The table records the mass $P(t)$ of a sample, in grams, after $t$ minutes. Using the values that bracket $t=5$, approximate $P^{\prime }(5)$.

The table records a solution's concentration $C(t)$, in grams per liter, after $t$ minutes. Use the rows at $t=2$ and $t=4$ to approximate $C^{\prime }(3)$.

Revenue $R(x)$ is measured in thousands of dollars when $x$ hundred units are sold. If $R^{\prime }(9)=4.2$, what does this mean?

A drone's altitude $A(t)$ is measured in meters after $t$ minutes. If $A^{\prime }(12)=-3.5$, which statement is correct?

Water temperature $T(t)$ is measured in degrees Celsius after $t$ minutes. Suppose $T^{\prime }(5)=0.8$. What does the derivative describe?

The mass $M(t)$ of a dissolving tablet is measured in grams after $t$ seconds. If $M^{\prime }(7)=-0.06$, what is the correct interpretation?

A bacteria population $B(h)$ is measured in thousands of cells after $h$ hours. If $B^{\prime }(3)=1.5$, what does this mean?

An object's distance $D(t)$ from a sensor is measured in meters after $t$ seconds. If $D^{\prime }(8)=0$, what necessarily follows?

A differentiable function $g$ satisfies $g(4)=-3$ and $g^{\prime }(4)=7$. Evaluate ${\lim }_{h\to 0}\frac{g(4+h)-g(4)}{h}$.

The volume $K(t)$ of gas is measured in liters at time $t$ in minutes, and $K^{\prime }(2)=-1.5$. Evaluate ${\lim }_{h\to 0}\frac{K(2+h)-K(2)}{h}$.

A differentiable function $f$ satisfies $f(3)=8$ and $f^{\prime }(3)=-5$. Evaluate ${\lim }_{x\to 3}\frac{f(x)-f(3)}{x-3}$.

A sensor records distance $S(t)$ in centimeters after $t$ minutes. Given $S(2)=18$ and $S^{\prime }(2)=4$, evaluate ${\lim }_{h\to 0}\frac{S(2+h)-18}{h}$.

A differentiable function $p$ satisfies $p(6)=11$ and $p^{\prime }(6)=\frac{1}{2}$. Evaluate ${\lim }_{h\to 0}\frac{p(6+h)-11}{h}$.

The graph of a smooth function $f$ is shown, with points marked at $x=1$, $x=3$, and $x=5$. Which ordering of the tangent slopes is correct?

The graph of smooth $f$ is shown with marked inputs $x=-1$, $x=0$, and $x=2$. At which marked input is $f^{\prime }(x)$ greatest?

The graph of smooth $f$ is shown with marked inputs $x=1$, $x=2$, $x=3$, and $x=4$. At which marked input is $f$ decreasing?

The graph of smooth $f$ is shown. Compare the instantaneous rates at the marked inputs $x=2$ and $x=4$.

The graph of smooth $f$ is shown with marked inputs $x=-1$, $x=0$, $x=1$, and $x=2$. At which pair are the tangent slopes equal?

A hiker's elevation $E(t)$ in meters is graphed against time $t$ in hours. At which shown time does $E^{\prime }(t)$ fail to exist?

A vehicle's displacement $d(t)$ in meters is graphed against time $t$ in seconds. Where does $d^{\prime }(t)$ fail to exist?

The flow rate $Q(t)$ in liters per minute is graphed against time $t$ in minutes. Why does $Q^{\prime }(1)$ fail to exist?

A runner's signed position $p(t)$ in meters is graphed against time $t$ in seconds. At what time does the instantaneous velocity fail to exist?

A student writes: “Since $V(2)=18$ and $V(6)=42$, volume was increasing at $6$ cubic meters per minute at the instant $t=4$.” Which evaluation is correct?

A student claims, “Because $f$ is continuous at $x=2$, $f^{\prime }(2)$ exists.” Which evaluation is correct?

A student notes $h(0)=4$ and $h(5)=19$ and writes, “By the Mean Value Theorem, some $c$ in $[0,5]$ has $h^{\prime }(c)=3$.” Which evaluation is correct?

A student writes, “The average rates of change of $f$ on $[1,3]$ and $[3,5]$ are both $5$, so $f^{\prime }(3)=5$.” Which evaluation is correct?