The slope field is shown. Let $y=f(x)$ satisfy $f(0)=0$. What does the field indicate about ${\lim }_{x\to \infty }f(x)$?

Which differential equation could generate the displayed slope field?

The slope field is shown. A solution has $y(0)=1$. What long-term behavior does the field suggest?

The slope field is shown. If $y(0)=4$, which value does the solution approach as $x$ increases?

In the displayed slope field, on which horizontal line are the slope segments horizontal?

What feature of the displayed slope field shows that the differential equation depends only on $x$?

For the displayed slope field, a solution passes through $(0,0)$. What is the slope of its tangent line at that point?

A medication amount $M$, in milligrams, satisfies $\frac{dM}{dt}=-0.15M$, where $t$ is in hours and $M(0)=80$. How much remains at $t=6$, to the nearest milligram?

A bacteria culture has population $P$ satisfying $\frac{dP}{dt}=0.4P$, where $t$ is in hours and $P(0)=150$ bacteria. What is $P(3)$, to the nearest whole bacterium?

A liquid cools according to $\frac{dT}{dt}=-0.3(T-20)$, where $T$ is in degrees Celsius and $t$ is in minutes. If $T(0)=80$, what is $T(2)$, to the nearest degree Celsius?

A contaminant mass $C$, in grams, satisfies $\frac{dC}{dt}=-0.25C$, where $t$ is in hours and $C(0)=40$. What is $C(4)$, to the nearest gram?

An algae population $A$ satisfies $\frac{dA}{dt}=0.12A$, where $t$ is in days and $A(0)=300$ organisms. What is $A(10)$, to the nearest organism?

A radioactive sample has mass $N$, in grams, satisfying $\frac{dN}{dt}=-\frac{\ln 2}{4}N$, where $t$ is in days and $N(0)=64$. What is $N(12)$?

A culture has volume $V$, in milliliters, satisfying $\frac{dV}{dt}=0.5V$, where $t$ is in hours and $V(0)=12$. Which particular solution models the volume?

The salt mass $S$, in grams, in a rinsing tank satisfies $\frac{dS}{dt}=-0.1S$, where $t$ is in minutes and $S(0)=50$. What is $S(5)$, to the nearest gram?

A tumor volume $V$, in cubic centimeters, satisfies $\frac{dV}{dt}=0.3V\left(1-\frac{V}{40}\right)$, where $t$ is in weeks. What does $\frac{dV}{dt}=0$ at $V=40$ mean?

A reservoir volume $Q$, in liters, satisfies $\frac{dQ}{dt}=-3$ at $t=5$ hours. What does this rate say about the reservoir at that instant?

The table shows a plant population over time. What is the average rate of change from $t=2$ to $t=4$ hours?

A fish population $P$, in fish, satisfies $\frac{dP}{dt}=0.5P\left(1-\frac{P}{100}\right)$, where $t$ is in months. What is the instantaneous rate when $P=50$?

A chemical amount $A$, in liters, satisfies $\frac{dA}{dt}=-0.08A$, where $t$ is in hours. When $A=200$ liters, what does the model predict for the instantaneous rate?

The table shows a reservoir's volume. What is the average rate of change from $t=2$ to $t=4$ hours?

A student writes: “Since $\frac{dP}{dt}=0.4P$ and $P(0)=20$, the population is increasing for all $t\ge 0$ because a positive population has a positive rate.” Is the justification complete?

A student writes: “For $\frac{dV}{dt}=V(10-V)$ with $V(0)=12$, $V^{\prime }(0)=12(10-12)<0$. Therefore, $V$ is initially decreasing.” Is the justification complete?

A student writes: “Because $\frac{dN}{dt}=-kN$ and $k>0$, $N$ is decreasing for all $t\ge 0$.” The initial value is $N(0)=80$ grams. Is the justification complete?

A student writes: “Since $\frac{d^{2}P}{d{t}^2}>0$, the population $P$ is increasing.” Does this argument establish that $P$ is increasing?

A student writes: “Since $M(0)=50$ and $\frac{dM}{dt}=-0.2M$, $M$ remains positive because it is decreasing.” Does this establish the claim?

A bacteria population satisfies $\frac{dP}{dt}=kP$ and triples in 5 hours. Which expression gives $k$ in reciprocal hours?

A culture satisfies $\frac{dA}{dt}=kA$ and grows from 20 to 70 organisms in 3 hours. Which expression gives $k$ in reciprocal hours?

A radioactive material satisfies $\frac{dN}{dt}=-kN$ and has half-life $h$ hours. Which expression gives $k$ in reciprocal hours?

A tank's volume $V$, in liters, satisfies $\frac{dV}{dt}=-0.08V$, where $t$ is in hours and $V(0)=500$. Which expression gives the volume at $t=6$ hours?