Applications of derivatives · Definition

Related rates

calculusapplications

Before you start you need to know

Related rates let you find how fast one quantity changes from how fast another changes, when both are linked by an equation.

Definition

In a related rates problem several quantities change with time \(t\) and are linked by an equation. The equation is differentiated with respect to \(t\) (chain rule) and the known values are substituted.

\[A=\pi r^{2}\ \Rightarrow\ \frac{dA}{dt}=2\pi r\,\frac{dr}{dt}\]

If two quantities are tied by an equation and both change over time, their rates of change are tied too. Differentiating the equation with respect to time gives that relation.

Steps: draw and name the variables, write the equation relating them, differentiate with respect to t (using the chain rule) and substitute the values at the instant only at the end.

Typical mistake: plugging in numbers before differentiating; that kills the variation.

Think about it

When are numerical values substituted in a related-rates problem?

After differentiating with respect to time, not before.

Exercises 10

  1. The radius of a circle grows at 2 cm/s. How fast is the area growing when \(r = 5\) cm?
  2. The radius of a sphere grows at \(\frac{1}{2}\) cm/s. How fast is the volume growing when \(r = 3\) cm?
  3. A 13 m ladder leans against a wall. Its base slides away from the wall at 2 m/s. How fast is the top sliding down when the base is 5 m from the wall? (in m/s)
  4. The edge of a cube grows at 1 cm/s. How fast is the volume growing when the edge is 4 cm?
  5. The side of a square grows at \(3\) cm/s. How fast is its area growing when \(s = 2\) cm?
  6. The radius of a circle grows at \(1\) cm/s. How fast is its circumference growing? (in cm/s)
  7. A rectangle has length \(x=4\) m and width \(y=3\) m. The length grows at \(1\) m/s and the width shrinks at \(2\) m/s. How fast is the area changing at that instant? (in m²/s)
  8. The radius of a sphere is \(2\) cm and grows at \(\frac{1}{2}\) cm/s. How fast is its surface area growing? (in cm²/s)
  9. The side of a square grows at \(2\) cm/s. How fast is its perimeter growing? (in cm/s)
  10. One car is \(3\) km east of an intersection and another is \(4\) km north. The first moves away at \(2\) km/h and the second at \(3\) km/h. How fast is the distance between them changing? (in km/h)

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.