AP Calculus AB/BC: Applications of Derivatives — Optimisation, Related Rates, Linearisation

TL;DR

Derivatives help you find maximums/minimums (optimisation), relate changing quantities (related rates), and approximate functions with lines (linearisation). These are practical applications where calculus moves beyond just calculating slopes. Mastering these will significantly boost your AP score.

1. The Mental Model

Think of derivatives as a powerful tool for understanding how things change. Optimisation uses them to find the "best" outcome, related rates uses them to track intertwined changes, and linearisation uses them to make complex functions behave like simple lines near a specific point.

2. The Core Material

Optimisation: Finding Maxima and Minima

Optimisation problems ask you to find the absolute maximum or minimum value of a function over a given interval. This usually involves finding critical points (where the derivative is zero or undefined) and checking endpoints.

Here's a general approach:
1. Understand the problem: Identify the quantity you need to maximise or minimise.
2. Formulate the function: Express that quantity as a function of one variable. This often requires using constraints to eliminate other variables.
3. Find critical points: Calculate the first