Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Unit 5: Calculus | Topic 5.1
📋 Learning Intentions
- Understand the concept of a limit as the foundation of differential calculus, and interpret it geometrically as the gradient of a tangent line.
- Define the derivative \( f'(x) \) as the gradient function of \( f(x) \), and state what it measures at any point on a curve.
- Apply the power rule to differentiate polynomial functions, and use the GDC to find numerical derivatives at specific points.
- Use correct derivative notation: \( f'(x) \), \( \frac{dy}{dx} \), and \( \frac{d}{dx}[f(x)] \), selecting the appropriate form for a given context.
- Interpret the value of a derivative in context — as an instantaneous rate of change with correct units.
- Distinguish between average rate of change (gradient of a secant) and instantaneous rate of change (gradient of a tangent).
Real-World Introduction
A car's speedometer does not show how far the car has travelled — it shows how fast the distance is changing at that precise instant. This is not the same as average speed over a journey. If a car travels 120 km in 2 hours, the average speed is 60 km/h, but the speedometer may have read anything from 0 to 120 km/h at different moments.
The mathematical tool that captures this idea of an instantaneous rate of change is the derivative. Calculus was developed independently by Newton and Leibniz in the 17th century precisely to answer questions like: at this exact moment, how quickly is this quantity changing? The derivative is the answer.
🌍 Why This Matters In pharmacokinetics, the rate at which a drug concentration in the bloodstream decreases is modelled using derivatives. Doctors use this to determine safe dosing intervals. The derivative of the concentration–time function tells them exactly how fast the drug is leaving the body at any given moment — not just on average.
Core Concepts & Definitions
Before defining the derivative, it is essential to understand what it replaces. Given two points on a curve, the gradient of the straight line connecting them — the secant line — gives the average rate of change of the function over that interval. This is a familiar idea: it is simply rise over run.
📖 Key Definition — Average Rate of Change The average rate of change of \( f(x) \) over the interval \( [a, b] \) is the gradient of the secant line joining the points \( (a, f(a)) \) and \( (b, f(b)) \): \[ \text{Average rate of change} =