Applications of Differentiation for the ESAT

One of the primary motivations for studying basic calculus techniques is to help sketch curves given an equation. As you progress through these ideas, consider how they relate to the physical shape of a curve. You should have a firm grasp of the general shapes of quadratic, cubic, quartic, and quintic polynomials, and be able to generalise these shapes to polynomials with higher powers of $x$.

Gradients, Tangents, and Normals

You are expected to use differentiation to find the gradient of a tangent at a specific point and use this to find the equations of both the tangent and the normal to a curve. The gradient of the tangent at a point $(x, y)$ is given by $\frac{dy}{dx}$. Since the normal is perpendicular to the tangent, its gradient is the negative reciprocal: $-\frac{1}{dy/dx}$. These are fundamental skills assumed for the ESAT.

Identifying and Classifying Stationary Points

Stationary points occur when the tangent to a curve is horizontal. At these points, the gradient is zero, so we solve for $x$ in the equation:

$\frac{dy}{dx} = 0$

Stationary points are often called local maxima or local minima because they represent the highest or lowest value in a specific neighbourhood, rather than necessarily the absolute greatest or least value of the entire function. There are several methods to classify these points:

1. General Shape of the Curve: If you are sketching a cubic such as $y = 2x^3 - 3x^2 - 12x + 6$, you can differentiate to find the stationary points:

$\frac{dy}{dx} = 6x^2 - 6x - 12 = 0$

$6(x^2 - x - 2) = 0$

$6(x - 2)(x + 1) = 0$

Stationary points occur at $x = 2$ and $x = -1$. Because the coefficient of the $x^3$ term is positive, the graph starts from the bottom left and goes to the top right. This tells us the maximum must be to the left of the minimum. Thus, the maximum is at $x = -1$ and the minimum is at $x = 2$.

1. Testing Values Either Side: You can calculate the $y$ values for $x$ coordinates slightly smaller and slightly larger than the stationary point. For $x = 2$, calculating $y$ at $x = 1.9$ and $x = 2.1$ would show that the $y$ values on either side are larger than at $x = 2$, suggesting a minimum. Be careful not to choose values so far away that you cross another stationary point.

2. The Second Derivative Test: This is often the most efficient method. If $\frac{d^2y}{dx^2} > 0$ at the stationary point, it is a minimum. If $\frac{d^2y}{dx^2} < 0$, it is a maximum.

Why the Second Derivative Works

The second derivative describes how the first derivative (the gradient) is changing. If the gradient goes from negative to positive as $x$ increases, the curve reaches a bottom point (a minimum). In this case, the gradient is increasing, which is why $\frac{d^2y}{dx^2} > 0$.

Conversely, if the gradient goes from positive to negative, the curve reaches a peak (a maximum). Here, the gradient is decreasing, meaning $\frac{d^2y}{dx^2} < 0$.

Logic of the Second Derivative Test

It is vital to understand the logic of sufficiency versus necessity.

• The condition $\frac{dy}{dx} = 0$ and $\frac{d^2y}{dx^2} > 0$ is sufficient for a minimum, but not necessary.
• The condition $\frac{dy}{dx} = 0$ and $\frac{d^2y}{dx^2} < 0$ is sufficient for a maximum, but not necessary.

This is because at a minimum, it is possible for $\frac{d^2y}{dx^2}$ to equal zero (for example, in the function $y = x^4$ at the origin). Therefore, while a positive second derivative guarantees a minimum, a zero second derivative does not rule one out.

Strictly Increasing and Decreasing Functions

A strictly increasing function is one where the $y$ value always gets greater as the $x$ value gets greater. Intuitively, the graph always slopes upwards.

A strictly decreasing function is one where the $y$ value always gets less as the $x$ value gets greater, meaning it always slopes downwards.

In the context of the ESAT, we use these definitions based on the derivative:

• If $f'(x) > 0$, then the function is strictly increasing.
• If $f'(x) < 0$, then the function is strictly decreasing.

Note that these are sufficiency conditions. We cannot say that if a function is strictly increasing, then $f'(x) > 0$ for all $x$. Some strictly increasing functions have corners where the derivative does not exist, or points where the gradient is zero (like $y = x^3$ at $x = 0$). For the ESAT, you should focus on simple, well behaved polynomial functions where these conditions apply clearly over a specified domain $a < x < b$.

Points of Inflexion

You are expected to have a qualitative understanding of points of inflexion. These occur where the curve changes its sense of curvature. A classic example is $y = x^3$ at $x = 0$. While you do not need to identify them using complex differentiation techniques, you should be aware of their existence when sketching polynomials.