Rates of change in context - Mathematics: Applications & Interpretation IB Study Notes

Rates of change describe how one quantity changes with respect to another, forming the foundation of differential calculus in real-world contexts. The derivative $\frac{dy}{dx}$ represents the instantaneous rate of change of $y$ with respect to $x$.

Key Definitions:

• Average rate of change between two points: $\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$

• Instantaneous rate of change at a point: $\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Velocity: rate of change of displacement $(v = \frac{ds}{dt})$

• Acceleration: rate of change of velocity $(a = \frac{dv}{dt} = \frac{d^2s}{dt^2})$

Essential Formulas:

Related Rates: When variables are connected through an equation, their rates of change are related. If $V = f(r, h)$, then $\frac{dV}{dt} = \frac{\partial V}{\partial r} \cdot \frac{dr}{dt} + \frac{\partial V}{\partial h} \cdot \frac{dh}{dt}$ (using chain rule).

Sign Convention: • Positive derivative → increasing quantity • Negative derivative → decreasing quantity • Zero derivative → stationary point (maximum, minimum, or inflection)

Memory Aid (DRIP): Derivative = Rate = Instantaneous = Prime notation

Units Matter! If $s$ is in metres and $t$ in seconds, then $\frac{ds}{dt}$ has units m/s. Always include units in contextual problems—Cambridge mark schemes specifically award marks for correct units.

Common Contexts: Population growth, chemical reactions, economics (marginal cost/revenue), motion problems, and optimization scenarios all utilize rates of change principles.

Understanding rates of change transforms abstract calculus into powerful real-world problem-solving. Think of derivatives as speedometers for any changing quantity, not just cars!

Example 1: Water Tank Drainage

Imagine a cylindrical water tank draining. The volume $V$ (litres) at time $t$ (minutes) follows $V(t) = 500 - 20t + 0.5t^2$. The rate of drainage is $\frac{dV}{dt} = -20 + t$ litres/minute. Notice: • At $t = 0$: $\frac{dV}{dt} = -20$ (draining at 20 L/min) • At $t = 20$: $\frac{dV}{dt} = 0$ (momentarily stopped) • At $t = 25$: $\frac{dV}{dt} = 5$ (filling up!)

This tells a story: the tank drains initially, reaches minimum volume at $t = 20$ minutes, then begins refilling.

Example 2: Bacterial Growth

A bacterial population $P(t) = 1000e^{0.3t}$ (where $t$ is hours) grows exponentially. The growth rate $\frac{dP}{dt} = 300e^{0.3t}$ bacteria/hour is itself increasing exponentially. At $t = 5$ hours, the population is growing at $300e^{1.5} \approx 1346$ bacteria/hour.

Analogy: Think of a derivative as a zoom lens on a curve. From far away (average rate), you see the overall trend. Zooming in (instantaneous rate) shows what's happening right now at a specific moment.

Example 3: Economics - Marginal Cost

If production cost is $C(x) = 1000 + 50x + 0.1x^2$ dollars for $x$ units, then marginal cost $\frac{dC}{dx} = 50 + 0.2x$ represents the approximate cost to produce one additional unit. When $x = 100$, producing the 101st unit costs approximately $50 + 20 = $70$.