Calculating Speed and Introduction to Derivatives in Calculus

Introduction

• Context: Physics class example of calculating car speed using a distance vs. time graph.
• Key Formula: Speed = Distance / Time (Gradient of graph = rise over run).

Real-World Application

• Challenge: Real cars don't move at constant speed; speed changes over time.
• Problem: Calculating speed at a specific time (t seconds) is complex due to changing gradients.

Gradient Calculation with Changing Speed

• Initial Approach: Draw a line to find its slope at time t.
• Equation:
  • Gradient = (\frac{f(t_1) - f(t)}{t_1 - t}).
• Simplified Equation:
  • Let (\Delta t = t_1 - t), so Gradient = (\frac{f(t + \Delta t) - f(t)}{\Delta t}).

Concept of Limits

• Inaccuracy: Initial line is not an accurate depiction of car's speed.
• Refinement: As (\Delta t) becomes smaller, approximation gets better.
• Limit Concept:
  • As (\Delta t) approaches zero, the line better represents speed.
  • Special symbol in calculus (limit) represents this process.

Applying Derivative to a Function

• Example Function: (f(x) = x^2) to find gradient at point x.
• Gradient Equation Adaptation:
  • Replace (\Delta t) with (\Delta x) and t with x.
• Algebraic Expansion:
  • Expand ((x + \Delta x)^2).
  • (x^2) terms cancel out.
• Final Simplification:
  • (\Delta x) terms cancel, resulting in limit of (2x + \Delta x) as (\Delta x) approaches zero.
• Result: Gradient of (x^2) graph is (2x).

Key Takeaway: Derivatives

• Concept: Finding the gradient of a function using derivatives.
• Significance: Simplifies complex problems.
• Conclusion: Derivatives allow for understanding and solving a wide range of problems in calculus.

Final Thoughts

• Importance: Calculus transforms difficult problems into manageable ones.
• Encouragement: Motivates further exploration in calculus.