Warm-up¶

Pick one of the scenarios below, and talk about what the graph of the function of interest could look like. Sketch a possible graph, and label a feature you find interesting.

Filling a water tank¶

You're pumping water into the bottom of a cylindrical tank. You want to know $y(t)$, the water level after $t$ seconds of pumping. The tank starts out empty. The pump pushes water into the tank with a constant pressure. As the water level rises, the pressure working against the pump grows proportionally.

Wearing out truck tires¶

You order a batch of new tires for a very large fleet of trucks. You want to know $n(t)$, the number of tires that are still intact after $t$ distance units of driving.

Swinging a pendulum¶

You attach a heavy weight to the ceiling using a light string. You pull the weight 60° back from vertical, keeping the string taut, and let it go. You want to know $\theta(t)$, the angle of the string after $t$ seconds of swinging.

You're giving a hospital patient an antibiotic through an IV line. You want to know $x(t)$, the amount of antibiotic in the patient's blood $t$ minutes after you start the IV drip. The IV adds antibiotic at a constant rate. As the antibiotic builds up, the rate at which the patient's kidneys remove it grows proportionally.
After coming home from the hospital, the patient from earlier takes an antibiotic pill, which dissolves almost instantly in their intestines. Their blood absorbes the antibiotic from their intestines, and their kidneys remove the antibiotic from their blood. The absorption rate is proportional to the amount of antibiotic in their intestines, and the removal rate is proportional to the amount of antibiotic in their blood. At $t$ minutes after the pill dissolves, you want to know $w(t)$, the amount of antibiotic in the patient's intestines, and $x(t)$, the amount of antibiotic in the patient's blood.