4.3 Rates of Change in Applied Contexts Other Than Motion

Learning objectives

—CHA-3.C—Interpret rates of change in applied contexts• CHA-3.C Interpret rates of change in applied contexts.• CHA-3.C.1 The derivative can be used to solve problems involving rates of change in applied contexts.• Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

For a contextual model $Q(x)$ , the derivative $Q'(a)$ measures the instantaneous change in the output quantity $Q$ with respect to the input $x$ when $x=a$ . Its units are output units per input unit.

State what the input and output represent, including units.\n2. Differentiate the given model with respect to its input.\n3. Evaluate the derivative at the requested input.\n4. Interpret the sign, magnitude, instant, and quotient units in one contextual sentence.

As an illustrative model, let $P(t)=1000+50t-2t^2$ represent a population $t$ years after observation begins. Then $P'(t)=50-4t\quad\text{people per year},$ so $P'(6)=50-4(6)=26\quad\text{people per year}.$ At $t=6$ years, the model's population is increasing at an instantaneous rate of $26$ people per year.

A positive derivative means the modeled output is increasing at that input; a negative derivative means it is decreasing; a zero derivative means its instantaneous rate of change is zero. The sign belongs to the rate, not automatically to the output value itself.

$P'(6)=26$ is a local rate, not a claim that exactly 26 people are added every future year. The interpretation is also only as valid as the supplied model and the input interval for which that model is intended.

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—4.4 Introduction to Related RatesRead