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As I progress into advanced mathematics I have noticed that [f(b) - f(a)] / (b - a) appears all the time, representing different things. For instance, it can represent slope, and if seen as Δposition/Δtime, it represents average velocity.

How can I read this expression, moreso the numerator and denominator separately, intuitively? For instance, the formula for a mean is (∑x)/N, which makes sense to me. However, I'm not quite there yet with [f(b) - f(a)] / (b - a), and as I encounter different versions of it throughout calculus, I think it would help me to truly understand and read it in a general way, and know why it applies to so many different applications.

Thanks!