Related Rates of Change (DP IB Applications & Interpretation (AI): HL): Revision Note

Related rates of change

What is meant by rates of change?

• A rate of change is a measure of how a quantity is changing with respect to another quantity

• Mathematically rates of change are derivatives

  • $\frac{\mathrm{d}V}{\mathrm{d}r}$ could be the rate at which the volume of a sphere changes relative to how its radius is changing

• Context is important when interpreting positive and negative rates of change

  • A positive rate of change would indicate an increase

    • e.g. the change in volume of water as a bathtub fills

  • A negative rate of change would indicate a decrease

    • e.g. the change in volume of water in a leaking bucket

What is meant by related rates of change?

• Related rates of change are connected by a linking variable or parameter

  • This is usually time, represented by$t$

    • Seconds is the standard unit for time but this will depend on context

• E.g.  water running into a large bowl

  • Both the height and volume of water in the bowl change with time

  • Time is the linking parameter

How do I solve problems involving related rates of change?

• Use the chain rule

$y=g\left(u\right),\mathrm{where}u=f\left(x\right)\Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\times \frac{\mathrm{d}u}{\mathrm{d}x}$

• The chain rule is given in the formula booklet in the form above

  • Different letters may be used relative to the context of a question

    • e.g. $V$ for volume,$S$ or $A$ for surface area,$h$ for height,$r$ for radius

• Problems often involve one quantity being constant

  • This means another quantity can be expressed in terms of a single variable

    • which makes finding a derivative a lot easier

• STEP 1
  Write down the rate of change given and the rate of change required
   

• STEP 2

  Use the chain rule to form an equation connecting these rates of change with a third rate

  • E.g. if you know rate of change of radius, $\frac{\mathrm{d}r}{\mathrm{d}t}$, and want to find out the rate of change of volume, $\frac{\mathrm{d}V}{\mathrm{d}t}$

    • You can write a chain rule equation with 'blanks'

      • $\frac{\mathrm{d}V}{\mathrm{d}t}=\frac{\mathrm{d}r}{\mathrm{d}t}\times \frac{▯}{▯}$

    • then fill in the blanks to make the equation valid

      • $\frac{\mathrm{d}V}{\mathrm{d}t}=\frac{\mathrm{d}r}{\mathrm{d}t}\times \frac{\mathrm{d}V}{\mathrm{d}r}$

• STEP 3

  Write down the formula for the related quantity, and differentiate to find the missing derivative

  • E.g. the volume of a sphere is $V=\frac{4}{3}\pi r^3$

    • So differentiating with respect to $r$ gives $\frac{\mathrm{d}V}{\mathrm{d}r}=4\pi r^2$
       

• STEP 4

  Substitute the derivative and known rate of change into the equation and solve