Modeling Rates of Change

Mathematical Modeling · Axiom Academy

INTRO Modeling Rates of Change Discover how differential equations emerge from describing how quantities change over time. From Average to Instantaneous Rates Before we can build differential equation models, we need to understand the difference between average and instantaneous rates of change. Drag the slider to see how the average rate approaches the instantaneous rate. Make the interval smaller to see the average rate approach the instantaneous rate. Common Rate Relationships in Nature Many natural phenomena follow predictable patterns. The rate at which something changes is often related to how much of it exists. Select a scenario to explore: Why? More bacteria means more reproduction happening. Each individual contributes to the growth rate. Mathematical modeling often starts with verbal descriptions. Learning to translate these descriptions into equations is a crucial skill. Match the verbal phrases to their mathematical meaning: Setting Up Differential Equations Let us practice setting up a differential equation from a verbal description. Adjust the parameters to see how the model changes: Model: "The rate of change of P is proportional to P" Why Instantaneous Rates Matter Compare what happens when we model with average rates (difference equations) versus instantaneous rates (differential equations): Increase the number of time steps to see the discrete model approach the continuous model.

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