Explain the difference between the average rate of change and the instantaneous rate of change of a function f.

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1. The average rate of change of a function f over an interval [a, b] is calculated using the formula: (f(b) - f(a)) / (b - a), which represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).
2. In contrast, the instantaneous rate of change of a function f at a specific point x = c is defined as the limit of the average rate of change as the interval shrinks to zero, expressed mathematically as: lim (h -> 0) [(f(c + h) - f(c)) / h].
3. The average rate of change gives a general idea of how the function behaves over a larger interval, while the instantaneous rate of change provides a precise measure of the function's behavior at a single point.
4. The instantaneous rate of change is also interpreted as the derivative of the function f at the point c, denoted as f'(c), which represents the slope of the tangent line to the curve at that point.
5. Understanding the difference between these two rates of change is crucial in calculus, as it lays the foundation for concepts such as derivatives and the behavior of functions.