What Is Rate of Change? Explained Visually | Calculus From Zero #02

What does rate of change actually mean? Before learning derivatives, we need to understand one simple idea: How much does one quantity change per change in another? In Episode 2 of Calculus From Zero, we return to the moving car from Episode 1 and break down what “how fast” really measures. The car moves from 20 metres to 60 metres while time moves from 2 seconds to 4 seconds. Two things changed: Position changed by 40 metres. Time changed by 2 seconds. So the rate is: 40 metres ÷ 2 seconds = 20 metres per second. And that little word “per” contains the key idea behind every rate. But rate of change isn't only about speed. Temperature can rise by degrees per hour. Prices can change by dollars per item. Water can flow in litres per second. Populations can change by people per year. A positive rate means the quantity is rising. A zero rate means it isn't changing. A negative rate means it is falling. Then we move from the real-world examples to the graph. The horizontal change is Delta X. The vertical change is Delta Y. So: Rate of change = Delta Y / Delta X And that ratio has another familiar name: Slope. That's the central idea of this episode: Rate = Slope. Two words, one idea. Finally, we discover the limitation of an average rate. Two cars can have exactly the same starting position, ending position, travel time and average speed—while moving completely differently in between. So an average can hide what is actually happening inside the interval. That leaves us with the question for Episode 3: How do we measure the rate of change at one exact moment? Next Episode: Average Rate vs Instantaneous Rate of Change #calculus #CalculusFromZero #rateofchange #slope #AverageRateOfChange #derivatives #mathexplained #learnmath #mathematics #mathvisualization