Tangents, normals, stationary points, increasing and decreasing functions

Introduction: Exploring the Shape of Curves

Welcome! In this chapter, we are going to move beyond just calculating derivatives and start using them to uncover the "secrets" of a graph. Think of differentiation as a high-powered microscope: it tells us exactly what is happening at any single point on a curve. We will learn how to find the equation of lines that graze the curve, identify where a graph is "climbing" or "falling," and find those crucial turning points where everything stands still for a moment.

Don't worry if this seems tricky at first! We will break every concept down into simple steps. If you can differentiate basic powers of \(x\), you already have the most important tool in your kit.

1. Tangents and Normals

Imagine a roller coaster track. At any specific point on that track, there is a straight-line direction the coaster is pointing. That line is the tangent. The normal is simply a line that sticks straight up out of the track at a 90-degree angle.

The Tangent

A tangent is a straight line that just touches a curve at a specific point and has the same gradient (slope) as the curve at that point.

Key Rule: The gradient of the tangent at \(x = a\) is the value of \(\frac{dy}{dx}\) when you substitute \(a\) into it.

The Normal

A normal is a straight line that is perpendicular (at right angles) to the tangent at the point of contact.

Memory Aid: Flip and Change! To find the gradient of the normal, take the tangent's gradient (\(m\)), flip it upside down, and change the sign: \(-\frac{1}{m}\).

Step-by-Step: Finding Equations

To find the equation of a tangent or normal at a point \((x_1, y_1)\):

1. Differentiate the function to find \(\frac{dy}{dx}\).
2. Substitute the \(x\)-value of your point into \(\frac{dy}{dx}\) to find the gradient, \(m\).
3. For a Tangent, use \(m\). For a Normal, use \(-\frac{1}{m}\).
4. Plug your point and your chosen gradient into the straight-line formula: \(y - y_1 = m(x - x_1)\).

Example: Find the tangent to \(y = x^2\) at the point (2, 4).
\(\frac{dy}{dx} = 2x\).
At \(x = 2\), the gradient \(m = 2(2) = 4\).
Equation: \(y - 4 = 4(x - 2)\), which simplifies to \(y = 4x - 4\).

Quick Review: Tangents and Normals

• Tangent gradient = \(\frac{dy}{dx}\)
• Normal gradient = \(-\frac{1}{\text{gradient of tangent}}\)
• Perpendicular gradients always multiply to give \(-1\).

2. Increasing and Decreasing Functions

Graphs are like hills. Sometimes you are walking uphill (increasing), and sometimes you are walking downhill (decreasing).

Increasing Functions

A function is increasing when the graph is going up as you move from left to right. At these points, the slope is positive.

Condition: \(\frac{dy}{dx} > 0\)

Decreasing Functions

A function is decreasing when the graph is going down as you move from left to right. At these points, the slope is negative.

Condition: \(\frac{dy}{dx} < 0\)

Did you know? A function can be increasing in some areas and decreasing in others. We use differentiation to find the specific intervals (ranges of \(x\)) where this happens.

Common Mistake: Students often forget that "increasing" means the gradient is strictly greater than zero. If the gradient is zero, the function is momentarily stationary, not increasing!

Key Takeaway

To find where a function is increasing or decreasing, differentiate it and solve the inequality \(\frac{dy}{dx} > 0\) or \(\frac{dy}{dx} < 0\).

3. Stationary Points

A stationary point is a point on the curve where the gradient is zero. If you were standing there, you would be on perfectly flat ground.

The Golden Rule: At any stationary point, \(\frac{dy}{dx} = 0\).

Types of Stationary Points

1. Local Maximum: The "top of the hill." The graph stops going up and starts going down.
2. Local Minimum: The "bottom of the valley." The graph stops going down and starts going up.

How to Find Stationary Points

1. Find \(\frac{dy}{dx}\).
2. Set \(\frac{dy}{dx} = 0\).
3. Solve this equation to find the \(x\)-coordinates.
4. Important: Substitute these \(x\)-values back into the original \(y = ...\) equation to find the corresponding \(y\)-coordinates.

Quick Review Box:
• Stationary means not moving up or down.
• Always find both \(x\) and \(y\) unless the question only asks for \(x\).

4. Classifying Stationary Points

Once you’ve found a stationary point, you need to decide if it's a Maximum or a Minimum. The easiest way is using the Second Derivative, written as \(\frac{d^2y}{dx^2}\).

The Second Derivative Test

The second derivative tells us how the gradient is changing (the "curvature").

1. The "Happy Face" (Minimum)

If \(\frac{d^2y}{dx^2} > 0\) (positive), the curve is bending upwards. This is a Minimum.

Analogy: A positive person smiles; a smile looks like a cup that holds water (a minimum point at the bottom).

2. The "Sad Face" (Maximum)

If \(\frac{d^2y}{dx^2} < 0\) (negative), the curve is bending downwards. This is a Maximum.

Analogy: A negative person frowns; a frown looks like a hill (a maximum point at the top).

What if the second derivative is zero?

If \(\frac{d^2y}{dx^2} = 0\), the test is inconclusive! You should check the gradient slightly to the left and right of the point to see how the sign of \(\frac{dy}{dx}\) changes.

Summary Table for Classification

Condition: \(\frac{d^2y}{dx^2} > 0\) → Nature: Minimum
Condition: \(\frac{d^2y}{dx^2} < 0\) → Nature: Maximum

Key Takeaway

Think of the second derivative as "acceleration." If the acceleration is positive, you are being pushed into the "floor" of a valley (Minimum). If it's negative, you are being pushed away from the "ceiling" (Maximum).

Final Checklist for Success

• Check your Differentiation: Most mistakes happen in the very first step. Double-check your powers and signs!
• Original vs. Derivative: Use \(y\) to find coordinates. Use \(\frac{dy}{dx}\) to find gradients. Use \(\frac{d^2y}{dx^2}\) to find the nature of the point.
• Read the Question: Does it ask for the coordinates (x and y) or just the value of x?
• Normal Gradients: Remember the negative reciprocal! If the tangent gradient is 5, the normal is \(-\frac{1}{5}\). If the tangent is \(-\frac{2}{3}\), the normal is \(+\frac{3}{2}\).

You've got this! Keep practicing these steps, and soon identifying the behavior of curves will feel like second nature.