Introduction to Stationary Points and Kinematics

Welcome! In the previous chapters, you learned how to differentiate functions to find their gradients. Now, we are going to use those skills for something incredibly useful: finding the "peaks" and "troughs" of a graph and calculating how objects move in the real world. Whether you are designing a roller coaster or predicting the path of a rocket, the math in this chapter is your secret weapon!

1. What are Stationary Points?

Imagine you are hiking up a hill. At the very top, for just a split second, you aren't going up or down—you are on flat ground. The same thing happens at the bottom of a valley. In mathematics, these flat spots on a graph are called stationary points.

Because the graph is flat at these points, the gradient is zero. Since the derivative \( \frac{dy}{dx} \) represents the gradient, we can find stationary points by solving this simple equation:

\( \frac{dy}{dx} = 0 \)

Types of Stationary Points

There are two main types you need to know for your exam:

  • Maximum: The "top of the hill." The gradient goes from positive (uphill) to zero, then to negative (downhill).
  • Minimum: The "bottom of the valley." The gradient goes from negative (downhill) to zero, then to positive (uphill).

Both of these are also known as turning points because the graph "turns" around and changes direction.

Quick Review: To find any stationary point, always start by setting your derivative to zero!

2. How to Find Stationary Points (Step-by-Step)

Don't worry if this seems tricky; just follow these four steps every time:

Example: Find the stationary point of the curve \( y = x^2 - 6x + 10 \).

Step 1: Differentiate the function.
Using the power rule we learned earlier: \( \frac{dy}{dx} = 2x - 6 \)

Step 2: Set the derivative to zero.
\( 2x - 6 = 0 \)

Step 3: Solve for \( x \).
\( 2x = 6 \)
\( x = 3 \)

Step 4: Find the \( y \)-coordinate.
Plug \( x = 3 \) back into the original equation:
\( y = (3)^2 - 6(3) + 10 \)
\( y = 9 - 18 + 10 = 1 \)
The stationary point is at \( (3, 1) \).

3. Maxima or Minima? (The Nature of the Point)

Sometimes the exam will ask you to "determine the nature" of the stationary point. This just means "Is it a maximum or a minimum?"

A great way to check is to look at the gradient just before and just after the point:

  • If the gradient changes from negative to positive, it is a minimum.
  • If the gradient changes from positive to negative, it is a maximum.

Analogy: If you are going downstairs (negative) and then start going upstairs (positive), you must have been at the lowest point (minimum)!

Key Takeaway: Stationary points occur where \( \frac{dy}{dx} = 0 \). To find the full coordinates, you must substitute \( x \) back into the original \( y = ... \) equation.

4. Kinematics: Calculus in Motion

Kinematics is the study of how things move. In Specification B, we focus on three things: Displacement (\( s \)), Velocity (\( v \)), and Acceleration (\( a \)).

The Connections

Calculus connects these three concepts beautifully. If you have an equation for displacement in terms of time (\( t \)):

1. Velocity is the rate of change of displacement:
\( v = \frac{ds}{dt} \)

2. Acceleration is the rate of change of velocity:
\( a = \frac{dv}{dt} \)

Did you know? This means if you differentiate displacement twice, you get acceleration!

Important Keywords in Questions

Exam questions often use specific words that tell you exactly what to do with your equations:

  • "At rest": This means the velocity is zero (\( v = 0 \)).
  • "Initial": This means time is zero (\( t = 0 \)).
  • "Maximum height" or "Stationary": This means the velocity is zero (\( v = 0 \)).
  • "Constant velocity": This means the acceleration is zero (\( a = 0 \)).

5. Working with Kinematics Graphs

The syllabus requires you to interpret distance-time and speed-time graphs. Here is the "cheat sheet" for interpreting them:

Distance-Time Graphs
  • The gradient represents the speed (or velocity).
  • A flat horizontal line means the object is stationary.
  • A straight diagonal line means constant speed.
Speed-Time (or Velocity-Time) Graphs
  • The gradient represents the acceleration.
  • A flat horizontal line means constant speed (zero acceleration).
  • A straight diagonal line means constant acceleration.

Common Mistake to Avoid: Don't confuse the two! On a distance-time graph, a horizontal line means "stopped." On a speed-time graph, a horizontal line means "moving at a steady pace." Always check the labels on the axes!

Summary and Key Tips

  • Stationary Points: Set \( \frac{dy}{dx} = 0 \) and solve for \( x \).
  • Turning Points: These are just maxima and minima.
  • Velocity: Differentiate displacement (\( s \)) with respect to time (\( t \)).
  • Acceleration: Differentiate velocity (\( v \)) with respect to time (\( t \)).
  • Calculators: You can use your calculator to check your arithmetic, but always show your differentiation steps to get full marks!

Don't worry if this seems tricky at first! The more you practice differentiating these expressions, the more "automatic" it will become. Just remember the ladder: Displacement \( \rightarrow \) Velocity \( \rightarrow \) Acceleration. You move down the ladder by differentiating!