Understanding Derivatives

Introduction

• Main Goal: Explain what a derivative is.
• Secondary Goal: Appreciate the paradoxes associated with derivatives.
• Common Misconception: Derivative measures an "instantaneous rate of change," an oxymoron, since change implies a difference over time.

The Concept of Derivative

• Definition: Derivative is a mathematical concept capturing the idea of change over an infinitesimal period, not just a single instant.
• Example Setup: Car moving from point A to point B, 100 meters in 10 seconds.
• Graph Representation:
  • Vertical axis: Distance traveled.
  • Horizontal axis: Time.
  • Distance function: Often represented as ( s ).

Distance and Velocity

• Velocity Function:
  • Related to the steepness of the distance graph.
  • Reflects the car's speed, increasing as the car speeds and decreasing as it slows.
• Velocity Calculation:
  • Requires comparing two separate points in time.
  • Velocity at a single moment seems paradoxical because it needs two points for measurement.

Resolving the Paradox

• Real-world Speed Measurement:
  • Speedometers measure over small intervals (e.g., 3s to 3.01s).
  • Use ( ds ) for small distance change, ( dt ) for small time change.
  • Velocity is ( \frac{ds}{dt} ).

The Derivative in Mathematics

• True Derivative:
  • Ratio ( \frac{ds}{dt} ) as ( dt \to 0 ).
  • Slope of the tangent line to the graph at a point.

Conceptual Understanding

• Approaching Zero:
  • Not an infinitely small ( dt ), but as ( dt ) approaches zero.
  • Visualized as the tangent slope.
  • Defines derivative as a "best constant approximation for rate of change" around a point.

Notation and Computation

• Notation: ( \frac{ds}{dt} ) indicates intention to approach zero.
• Example:
  • Distance function ( t^3 ).
  • Derivative ( 3t^2 ) from algebraic simplification.

Understanding Paradoxes

• Example: Car at ( t = 0 ).
  • Derivative ( 3t^2 = 0 ) implies zero velocity.
  • Paradox in motion: If zero at ( t=0 ), when does it start?
  • Resolution: Derivative shows "best approximation," not actual movement in an instant.

Conclusion

• Reframe "Instantaneous Rate": Think of it as "best constant approximation for rate of change."
• Future Content: Visual intuitions for derivatives, their computation, and applications.