So the notion of rate of change serves as a launchpad into the study of limits and derivatives, the heart of differential Calculus! Relation to the Mean Value Theorem If you replace a by x (to make it more like a function), and allow the difference h to approach 0 by taking a limit, then you get the definition of derivative of the function.Īfter we apply the limit, we often call this formula the instantaneous rate of change, or instantaneous velocity. Suppose now we specify that the point b is exactly h units to the right of a. In fact, they are the same formula! The average rate of change of a function on the interval is exactly the slope of the secant line between the points at x = a and x = b. Notice how close this is to another important formula, the slope of a secant line. The average rate of change of a function f on a given interval is: To simplify formulas, we often use the Greek capital delta ( Δ ) to stand for change. So, if rate = distance/time, then let’s define the (average) rate of a function to be the change in y-values divided by the change in x-values on a given interval. The output, y = f( x), will be considered analogous to distance. Let’s agree to treat the input x as time in the rate of change formula. In Calculus, most formulas have to do with functions. Ok, next let’s talk about the precise formula. You’ve just computed my average rate of change, or average velocity. Multiplying by the conversion factor, 60 min./hr., we find a speed of:īut what does this number represent? I may not have been traveling a constant 30 mph the entire trip, but if you smoothed out all of the slow and fast parts, then on average I was going 30 mph.
Rate = (Change in Distance) / (Change in Time) = (10 miles) / (20 minutes) On the other hand, if you did use the rate formula, you could still find out useful information. You’d have no idea if I stopped at a red light 2 minutes into the trip, sped up to pass another car at the halfway point, or slammed on the brakes when the highway patrol showed up near the end of the trip.
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