Concept 01: Rates of Change & Derivatives

▶ Interactive Demo: Rate of Change & Derivative Visualizer

Open the interactive demo below to shrink the time step dt and watch the average speed converge into the exact instantaneous tangent slope.

1. The Real-World Problem: How Fast Are We Moving?

A robot’s wheel encoder does not measure velocity directly. It only reports how many rotations (or meters) the wheel has turned.

Every 20 milliseconds (dt = 0.02s), the robot’s control loop reads the position:

• At t = 1.00s: Position x₁ = 2.00 meters.
• At t = 1.02s: Position x₂ = 2.08 meters.

To find the robot’s speed, we calculate the rate of change:

   Speed = (Distance Traveled) / (Time Taken) = (2.08 - 2.00) / 0.02 = 4.0 meters per second

2. Solving It in Code (Java & WPILib)

First-Principles Java: Numerical Derivative

// Sensor position readings (meters) at two timestamps
double x1 = 3.00, t1 = 1.00;
double x2 = 3.42, t2 = 1.05;

// Finite difference derivative: v = dx / dt
double dt = t2 - t1; // 0.05 seconds
double velocity = (x2 - x1) / dt; // 0.42 / 0.05 = 8.40 m/s

System.out.printf("Instantaneous Velocity: %.2f m/s%n", velocity);

3. Bridge to Machine Learning: The Loss Slope

In machine learning:

• When training a model, we measure how much the prediction error changes when a weight w is tweaked slightly:

  Slope = dLoss / dw
  

• If the derivative is positive, increasing w increases error (bad!). If the derivative is negative, increasing w decreases error (good!).

4. Review Checkpoints

Checkpoint 1

An encoder reports x = 5.0m at t = 2.0s, and x = 5.15m at t = 2.05s. What is the average velocity over this interval?

Solution: v = Δx / Δt = (5.15 - 5.0) / (2.05 - 2.0) = 0.15 / 0.05 = 3.0 m/s.

Checkpoint 2

If a robot’s position curve is flat (horizontal line, x(t) = 3.0m constant), what is its velocity derivative dx/dt?

Solution: Because position is not changing (dx = 0), the slope is zero: v = dx/dt = 0 m/s (The robot is stationary).