Welcome to Pure Mathematics 2: Differentiation
Hello there! Welcome to one of the most exciting parts of your Pure Mathematics 2 (P2) journey. In Pure Mathematics 1, you learned the basic rules of how to differentiate functions like \(x^n\). In this chapter, we are going to take those skills and use them to solve real problems.
Think of Differentiation as a way to look at "change." By the end of these notes, you’ll be able to find the exact highest and lowest points of a curve, determine if a graph is "climbing" or "falling," and solve practical problems like finding the maximum area of a garden using a fixed amount of fencing. Let's dive in!
1. Increasing and Decreasing Functions
Before we find specific points, we want to know the general "mood" of a graph. Is it going up or is it going down?
What is an Increasing Function?
A function is increasing when the graph moves upwards as you go from left to right. In math terms, this means the gradient is positive.
Key Condition: A function \(f(x)\) is increasing when \(f'(x) \ge 0\).
What is a Decreasing Function?
A function is decreasing when the graph moves downwards as you go from left to right. This means the gradient is negative.
Key Condition: A function \(f(x)\) is decreasing when \(f'(x) \le 0\).
Analogy: Imagine you are hiking. If you are walking uphill, you are on an "increasing" part of the mountain. if you are walking downhill, you are on a "decreasing" part.
Common Mistake: Students often forget to differentiate first! To check if a function is increasing or decreasing at a certain point, you must find \(f'(x)\) (the gradient function) and then plug in your value of \(x\).
Takeaway: Check the sign of the derivative! Positive (\(+\)) means up, negative (\(-\)) means down.
2. Stationary Points
Have you ever thrown a ball straight up in the air? For a tiny, split second at the very top, the ball isn't moving up or down. In calculus, we call these moments Stationary Points.
A stationary point occurs where the gradient of the curve is exactly zero. This is where the tangent to the curve is perfectly horizontal.
How to find Stationary Points:
1. Find the derivative, \(\frac{dy}{dx}\).
2. Set \(\frac{dy}{dx} = 0\).
3. Solve this equation to find the \(x\)-coordinates.
4. Plug those \(x\)-coordinates back into the original equation (\(y\)) to find the \(y\)-coordinates.
Did you know? Stationary points are sometimes called "critical points" because they tell us where the "action" happens on a graph!
3. Maxima and Minima
Not all stationary points are the same. Some are the "peaks" of a hill, and some are the "bottoms" of a valley.
Local Maximum: The highest point in a specific area of the graph. The graph goes from increasing to decreasing.
Local Minimum: The lowest point in a specific area of the graph. The graph goes from decreasing to increasing.
Don't worry if this seems tricky at first! We have a simple mathematical "test" to tell them apart without even looking at the graph.
The Second Derivative Test
The second derivative, written as \(\frac{d^2y}{dx^2}\) or \(f''(x)\), tells us about the "bending" (concavity) of the curve.
How to use it:
• If \(\frac{d^2y}{dx^2} < 0\) at the stationary point, it is a Local Maximum (The curve is "frowning").
• If \(\frac{d^2y}{dx^2} > 0\) at the stationary point, it is a Local Minimum (The curve is "smiling").
• If \(\frac{d^2y}{dx^2} = 0\), the test is inconclusive, and you might have a point of inflection (though P2 usually focuses on Max and Min).
Memory Aid:
• Negative second derivative = Max (Think: Negative thoughts make you frown \(\cap\)).
• Positive second derivative = Min (Think: Positive thoughts make you smile \(\cup\)).
Takeaway: To classify a point, differentiate a second time and check the sign!
4. Sketching Curves
Using everything we've learned, you can now sketch complex cubic or quartic equations like a pro!
Step-by-Step Sketching Guide:
1. Find Intercepts: Set \(x=0\) to find the \(y\)-intercept. Set \(y=0\) to find \(x\)-intercepts.
2. Stationary Points: Find where \(\frac{dy}{dx} = 0\).
3. Classify: Use \(\frac{d^2y}{dx^2}\) to see which are Maxima and which are Minima.
4. End Behavior: Look at the highest power of \(x\) to see where the graph goes as \(x\) gets very large or very small.
5. Connect the dots: Draw a smooth curve through your points.
5. Practical Problems (Modeling)
In your exam, you will often be asked to apply differentiation to real-world scenarios. Usually, this involves a shape or a business cost.
Example Scenario: A farmer has 40m of fencing to make a rectangular pen against a straight wall. Find the maximum area.
The "Game Plan" for Modeling:
1. Write an equation: Use the information given to create an equation for the thing you want to maximize (e.g., Area \(A = xy\)).
2. Substitute: Use other info (like the perimeter) to replace one variable so you only have one (e.g., \(A\) in terms of \(x\) only).
3. Differentiate: Find \(\frac{dA}{dx}\).
4. Optimize: Set \(\frac{dA}{dx} = 0\) and solve.
5. Verify: Use the second derivative to prove it is a maximum.
Quick Review: Whenever a question says "Find the maximum..." or "Find the minimum value of...", it is a hidden instruction to differentiate and set to zero!
Summary Checklist
• Do I know that Increasing means \(f'(x) \ge 0\)?
• Do I know that Stationary Points happen when \(\frac{dy}{dx} = 0\)?
• Can I use the Second Derivative Test to find Maxima and Minima?
• Can I apply these steps to Area and Volume problems?
You're doing great! Differentiation is a powerful tool. Practice these steps, and you'll find that the "scary" algebra starts to make perfect sense.