Introduction to Rates of Change

Welcome to one of the most practical chapters in Further Pure Mathematics! In your previous studies, you learned that the derivative \(\frac{dy}{dx}\) represents the gradient of a curve. In this chapter, we take that idea into the real world. Rates of change help us describe how things move, grow, or shrink over time. Whether it is a balloon being inflated, water leaking from a tank, or a metal plate expanding in the heat, calculus allows us to calculate exactly how fast these changes are happening.

Don't worry if this seems a bit abstract at first. The "secret" to this chapter is that it is mostly an application of the Chain Rule, which you have already encountered in the "Product, quotient and function of a function" chapter.


1. Understanding the Notation

In mathematics, the word "rate" almost always means "how something changes with respect to time". We use the letter \(t\) to represent time.

  • \(\frac{dr}{dt}\): The rate of change of the radius \(r\) with respect to time.
  • \(\frac{dV}{dt}\): The rate of change of the volume \(V\) with respect to time.
  • \(\frac{dA}{dt}\): The rate of change of the area \(A\) with respect to time.

Key Rule: If a value is increasing, its rate of change is positive. If a value is decreasing (like water leaking out), its rate of change is negative.


2. Connected Rates of Change

Sometimes, we want to find the rate of change of something that we cannot measure directly. However, we might know how it relates to another variable. This is where Connected Rates of Change come in. We use the Chain Rule to link them together.

For example, if we have variables \(y\), \(x\), and \(t\), the relationship is:

\(\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}\)

The "Gear" Analogy

Think of it like three gears connected in a line. If Gear A moves Gear B, and Gear B moves Gear C, the speed of Gear C depends on the relationship between all of them. To find the final speed, you just multiply the "connections" between them!


3. Step-by-Step Guide to Solving Problems

Most exam questions follow a predictable pattern. Use these steps to stay organized:

  1. Identify what you know: Look for numbers in the question. If it says "the radius is increasing at \(2 \text{ cm/s}\)", write down \(\frac{dr}{dt} = 2\).
  2. Identify what you need to find: If it asks for the "rate of increase of the volume", you are looking for \(\frac{dV}{dt}\).
  3. Find a connecting formula: You need a formula that links the variables involved (e.g., for a sphere, \(V = \frac{4}{3}\pi r^3\)). These are often standard geometric formulas.
  4. Differentiate your formula: Find the derivative (e.g., \(\frac{dV}{dr}\)).
  5. Chain it all together: Set up the Chain Rule equation and substitute your values.

Example: A circular puddle is expanding. Its radius \(r\) increases at a rate of \(0.5 \text{ cm/s}\). Find the rate of increase of the area \(A\) when the radius is \(10 \text{ cm}\).

Step 1: We know \(\frac{dr}{dt} = 0.5\).
Step 2: We want \(\frac{dA}{dt}\).
Step 3: Connecting formula is \(A = \pi r^2\).
Step 4: Differentiate: \(\frac{dA}{dr} = 2\pi r\).
Step 5: Use the Chain Rule: \(\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}\).
\(\frac{dA}{dt} = (2\pi r) \times (0.5) = \pi r\).
When \(r = 10\), \(\frac{dA}{dt} = 10\pi \text{ cm}^2/\text{s}\).


4. Small Approximations

The syllabus also requires you to understand how small changes in one variable affect another. We use the notation \(\delta x\) (read as "delta x") to mean a "very small change in \(x\)".

The formula for small approximations is:

\(\delta y \approx \frac{dy}{dx} \delta x\)

This tells us that the change in \(y\) is approximately the gradient multiplied by the change in \(x\). This is very useful for estimating errors in measurements.

Quick Example:

If \(y = x^2\), then \(\frac{dy}{dx} = 2x\). If \(x\) increases from \(5\) to \(5.01\), then \(\delta x = 0.01\).
The approximate change in \(y\) is: \(\delta y \approx (2 \times 5) \times 0.01 = 0.1\).


5. Common Pitfalls to Avoid

  • Mixing up the "top" and "bottom": Always check your Chain Rule fraction "cancellation". In \(\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}\), the \(dx\) terms diagonally opposite each other "cancel out" to leave \(\frac{dy}{dt}\).
  • Forgetting units: While units aren't always penalized, keeping track of them (like \(\text{cm}^3/\text{s}\) for volume) helps you ensure you are calculating the right thing.
  • Negative signs: If the volume of a container is decreasing, you must use a negative sign (e.g., \(\frac{dV}{dt} = -5\)).

Summary Checklist

Key Takeaways:

  • Rates of change are expressed as derivatives with respect to time (\(t\)).
  • The Chain Rule connects different rates: \(\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}\).
  • Use geometric formulas (Area, Volume) to find the derivative that connects your variables.
  • For small changes, use \(\delta y \approx \frac{dy}{dx} \delta x\).

"Don't worry if this seems tricky at first! The more you practice identifying 'what I have' versus 'what I want', the more these problems will feel like simple puzzles."