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IB DP Mathematics: Applications and Interpretation | SL & HL | Calculus — Topic 5.3
📋 Learning Intentions
- State the power rule for differentiating polynomial terms of the form \( ax^n \).
- Find the derivative of a polynomial function by applying the power rule term by term.
- Calculate the gradient of a curve at a specific point by substituting into the derivative.
- Determine the equation of a tangent or normal line to a polynomial curve at a given point.
- Identify stationary points of a polynomial function and classify them as local maxima, minima, or points of inflection.
- Apply differentiation to solve optimisation problems in real-world contexts.
Real-World Introduction
A drone delivery company models the height of a package above the ground during flight using the polynomial \( h(t) = -2t^3 + 9t^2 + 12t \), where \( t \) is time in seconds. To ensure the package does not rise too steeply or descend too fast, engineers need to know the rate of change of height at every moment — not just the height itself.
This rate of change is exactly what differentiation provides. By differentiating \( h(t) \), the engineers obtain a new function \( h'(t) \) that gives the vertical velocity at any instant. When \( h'(t) = 0 \), the package is momentarily neither rising nor falling — it has reached its highest point. Differentiation turns a static formula into a dynamic tool for understanding how quantities change.
🌍 Why This Matters Differentiation of polynomials underpins optimisation in engineering, economics, and biology. Pharmaceutical companies use polynomial models to find the time at which a drug reaches peak concentration in the bloodstream — a direct application of finding where the derivative equals zero.
Core Concepts & Definitions
Before defining the derivative formally, consider what it means to measure steepness on a curve. On a straight line, the gradient is constant — it is the same everywhere. On a curve, the steepness changes from point to point. To measure the gradient at a single point , imagine zooming in on the curve until it looks like a straight line. The slope of that line is the derivative at that point.
📐 Key Definition — The Derivative The derivative of a function \( f(x) \), written \( f'(x) \) or \( \frac{dy}{dx} \), is a function that gives the instantaneous rate of change of \( f \) with respect to \( x \) at every point in its domain. Geometrically, \( f'(a) \) equals the gradient of the tangent to the curve \( y = f(x) \) at the point where \( x = a \).
The notation \( \frac{dy}{dx} \) is read "dee y by dee x" and emphasises that the derivative