Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Calculus — Topic 5.2
📋 Learning Intentions
- Identify intervals on which a function is increasing or decreasing using the sign of the first derivative.
- Find the derivative of polynomial, rational, and composite functions and evaluate it at specific points.
- Interpret the value of \( f'(x) \) as the instantaneous rate of change and connect its sign to the behaviour of the graph.
- Determine the \( x \)-values at which a function changes from increasing to decreasing (or vice versa) by solving \( f'(x) = 0 \).
- Justify whether a function is increasing or decreasing on a given interval using derivative evidence, with correct mathematical notation.
- Use the GDC to graph \( f'(x) \) and read off intervals of increase and decrease directly from the graph.
Real-World Introduction
A hospital monitors a patient's temperature over 12 hours after surgery. The temperature rises steeply in the first two hours, levels off around hour four, then gradually falls back toward normal. A doctor does not just want to know the temperature at a single moment — they want to know whether it is still rising or has started to fall. That rate of change is precisely what the derivative measures.
In financial markets, analysts track whether a stock's price is in an upward or downward trend. In ecology, population models distinguish between phases of growth and decline. In every case, the underlying mathematical question is the same: is the function increasing or decreasing, and where does it change direction?
🌍 Why This Matters The sign of the derivative \( f'(x) \) tells us the direction of change at every point on a curve. Engineers use this to identify when a structure's stress is increasing; economists use it to find when profit switches from growing to shrinking. Interpreting \( f'(x) \) is the foundation of all optimisation in the IB course.
Core Concepts & Definitions
Before defining increasing and decreasing formally, consider what happens to the \( y \)-values of a function as \( x \) moves from left to right. If the graph climbs upward, the output is getting larger. If it falls, the output is getting smaller. The derivative captures exactly this behaviour — it measures how steeply and in which direction the function is changing at any instant.
📖 Key Definition — Increasing and Decreasing Functions
- A function \( f \) is increasing on an interval if, for any two values \( x_1
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