Average vs. Instantaneous Rate

Business Calculus · Axiom Academy

LESSON Average vs. Instantaneous Rate Understanding the difference between overall and moment-by-moment change The Big Idea: In business, we often ask "How fast is something changing?" But this question can mean two different things: the average rate over a period, or the instantaneous rate at a specific moment. Understanding both is crucial for making informed decisions. The average rate of change of f(x) from x = a to x = b is the slope of the secant line connecting the two points: This tells us the overall rate of change across an interval — like your average speed on a road trip. The instantaneous rate of change of f(x) at x = a is the slope of the tangent line at that point: This tells us the rate of change at one specific moment — like your speedometer reading right now. Drag the sliders to see how average rate (secant line) approaches instantaneous rate (tangent line): As point b gets closer to point a, the secant line approaches the tangent line, and the average rate approaches the instantaneous rate. This is the fundamental idea behind the derivative! Measures change over an interval Example: "Sales grew by 20% this quarter" Formula: Δy/Δx = [f(b) - f(a)]/(b - a) Example: "Sales are currently growing at 500/day" Formula: lim(h→0) [f(a+h) - f(a)]/h A company's revenue (in thousands) is modeled by R(t) = t² where t is months since launch. Average rate (months 1 to 3): Revenue grew by 4,000/month on average

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