4.3 Rates of Change in Applied Contexts

Rates everywhere

Derivatives model any rate: water flow (dV/dt), cooling temperature, population growth, or marginal cost (the derivative of the cost function). Each is a rate with real units.

Interpretation

As always, the sign tells direction: positive = increasing, negative = decreasing. If dA/dt = 3 cm²/s, the area is increasing at 3 cm² per second.

Key terms

Applied rate

A derivative modeling any real-world rate of change.

dV/dt

The rate the volume changes over time.

Marginal cost

The derivative C'(x) of the cost function.

Sign

Positive = increasing; negative = decreasing.

Units

Each applied rate carries real units.

Try an AP-style question

C(x) = 0.01x² + 5x + 200 dollars. Using marginal cost, estimate the cost of producing the 101st item.

1. $7
2. $2
3. $5
4. $302