#AP Calculus AB/BC: Differentiation - Rates of Change 🚀

Hey there, future calculus champ! Let's get you prepped for Unit 2 with a super focused review of average and instantaneous rates of change. Think of this as your ultimate cheat sheet for acing the exam. Let's dive in!

#2.1: Average vs. Instantaneous Rate of Change

# ✈️ Average Rate of Change

Think of the average rate of change as the slope of a secant line between two points on a curve. It tells you the overall trend of a function over an interval. It's like calculating your average speed on a road trip – total distance divided by total time.

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For a function $f(x)$ over an interval $[a, b]$, the average rate of change is:

$$\frac{f(b) - f(a)}{b - a}$$

# ⛷️ Instantaneous Rate of Change

The instantaneous rate of change is the slope of the tangent line at a single point. It tells you how the function is changing at that exact moment. Imagine looking at your speedometer – that's your instantaneous speed at that precise time. In calculus, we use the derivative to find this.

The instantaneous rate of change of $f(x)$ at $x = c$ is denoted as $f'(c)$ and is given by:

$$f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}$$