6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Who (aside from Ross Geller) doesn't love ice cream? As much as humans love ice cream, we all know that it's a penguin's favorite dessert. They eat, breathe, and sleep ice cream. Well, actually they just eat it. If they breathed ice cream, they'd probably be extinct by now.

The point is that penguins eat ice cream right out of the carton—and fast! Let's say h(t) = h₀ – At represents the height of the ice cream in its carton after t hours, where h₀ is the initial ice cream height. A is the amount of ice cream that a penguin eats every hour.

In that case, the average rate of change between two values is just the change in the ice cream level (the values of the function) over the change in time (the values of the domain). In other words:

Students should know that, in general, we can write this for any function f(n):

For instance, if a penguin eats ice cream at a rate of A = 1 inch per hour and the initial ice cream level is 8 inches, the rate of change between time t₁ = 3 hours and t₂ = 5 hours, we can calculate the average rate of change:

If we plotted h(t), we'd see that this rate of change is just the slope of the line since h(t) is in the form y = mx + b. We also had this information right from the start, since A is the slope of the linear equation, and A was already given as 1 inch per hour.

Students should know that we can find the rate of change given a table of x and y values. Really, it's the exact same thing as finding the slope:

However, for non-linear functions, the rate of change between any two points might not be the same as the rate of change of the overall function. For instance, let's say we have the following table of data.

In this case, we note that the slope of the line, or the rate of change between the measurements is not constant. This is why we define rate of change over a specified range, say, between each pair of adjacent points.

In this case, we don't have the same slope for all points on the line. That's not a mistake. That's because life is sometimes unpredictable, and changes at odd rates. If we're more interested in the overall meaning of life (it's 42, by the way) we could calculate the overall rate of change (in this case, between x = 0 and x = 7) and this would give us a slightly bigger picture.

Students should know that we can calculate the average rate of change for any function. Having a linear relationship is super nice, but not necessary; as long as we have initial and final values for x and f(x), then they shouldn't have any problems. If they're confused or struggling, we recommend relating rate of change to the slope of a line. Every function has a "slope" between any two points, and the rate of change is how we find that slope.

Of course, we can expand the idea of rate of change to non-linear functions and real-life situations, but they all come back to the same basic principle: ice cream.