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4.4 Introduction to Related Rates

Syllabus
2020
Topic
4.4
Level
—

Learning objectives

—CHA-3.D—Calculate related rates in applied contexts• CHA-3.D Calculate related rates in applied contexts.• CHA-3.D.1 The chain rule is the basis for differentiating variables in a related rates problem with respect to the same independent variable.• CHA-3.D.2 Other differentiation rules, such as the product rule and the quotient rule, may also be necessary to differentiate all variables with respect to the same independent variable.• Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

Differentiate the Relationship Between Changing Quantities

In a related-rates problem, two or more quantities change while remaining linked by an equation. Treat each changing quantity as a function of the same independent variable—usually time tt—and differentiate the entire relation with respect to that variable.

  1. Define each changing quantity and its rate.\n2. Write one equation relating the quantities.\n3. Differentiate both sides with respect to the common variable.\n4. Apply the chain rule to every changing quantity; use product or quotient rule when the equation's structure requires it.\n5. Substitute values from the specified instant, solve for the requested rate, and attach units.

A=\pi r^2\quad\Longrightarrow\quad \frac{dA}{dt}=2\pi r\frac{dr}{dt}

A circular region has radius r=3r=3 meters at an instant when dr/dt=0.5dr/dt=0.5 meter per second. Substituting into the differentiated relation gives dAdt=2π(3)(0.5)=3π m2/s.\frac{dA}{dt}=2\pi(3)(0.5)=3\pi\ \mathrm{m^2/s}. At that instant, the area is increasing at 3π3\pi square meters per second.

Do not replace a changing quantity with its instantaneous number before differentiating; doing so can incorrectly turn it into a constant. Also, d(r2)/dt=2r dr/dtd(r^2)/dt=2r\,dr/dt, not merely 2r2r: the rate factor is the chain-rule record that rr changes with time.