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4.1 Interpreting the Meaning of the Derivative in Context

Syllabus
2020
Topic
4.1
Level
—

Learning objectives

—CHA-3.A—Interpret the meaning of a derivative in context• CHA-3.A Interpret the meaning of a derivative in context.• CHA-3.A.1 The derivative of a function can be interpreted as the instantaneous rate of change with respect to its independent variable.• CHA-3.A.2 The derivative can be used to express information about rates of change in applied contexts.• CHA-3.A.3 The unit for f′(x) is the unit for f divided by the unit for x.• Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

Read a Derivative as a Contextual Rate

If y=f(x)y=f(x), then f′(a)f'(a) is the instantaneous rate at which the output yy is changing with respect to the input xx when x=ax=a. Its sign gives the direction of change, and its magnitude gives how quickly the output is changing at that instant.

\text{units of }f'(x)=\frac{\text{units of }f}{\text{units of }x}

A complete interpretation identifies: (1) the input value or instant, (2) the output quantity that is changing, (3) the independent variable it changes with respect to, (4) whether it is increasing or decreasing, and (5) the rate units.

Suppose T(t)T(t) is the temperature of a drink in degrees Celsius, tt minutes after it is poured, and T′(5)=−0.8T'(5)=-0.8. At t=5t=5 minutes, the drink's temperature is decreasing at an instantaneous rate of 0.80.8 degrees Celsius per minute. The negative sign indicates decreasing temperature; the units are ∘C/min^\circ\mathrm{C}/\mathrm{min}.

T′(5)=−0.8T'(5)=-0.8 does not mean the temperature has changed by −0.8∘C-0.8^\circ\mathrm{C} in total, nor that it will keep changing at that rate for a long interval. It describes the local rate at the single instant t=5t=5.