I have traditionally had trouble fitting in Intermediate Value Theorem and Mean Value Theorem into Calculus without them seeming arbitrary. Kind of by accident, I found a nice way to not only talk about both theorems, but introduce early idea for curve sketching. And it was super simple!
Start with a table of a continuous function:
Have students calculate the average rate of change on all the intermediate intervals. In this case, 3 to 7, 7 to 11, etc. In addition, have them calculate the average rate of change for the extreme values of the table, in this case 3 to 26.
We now have a list of slopes: 1.525, 2.975, -8.6, -1.3667, 1.683 with an extreme slope of 0.1227.
Next, have students graph the values from the table.
They did this by hand, but I used Desmos here for demonstration purposes. Have them describe the behaviors of the function. Next, make a graph of the slopes, including the slope we found from the endpoints:
Great moment to talk about why we would graph the slopes this way, and the assumptions involved when taking an average rate of change. Now some questions. How many times is the slope zero? How many times is the slope equal to the overall slope? Could we sketch a function that would output these values?
Go back to their description of behaviors. When the function was increasing, what type of slope did you have? when the function was increasing? when the slope appeared to be zero?
Before you know, you’ve teased out the concepts behind the relationship between K(t) and K’(t) in addition to a good demonstration of the Mean Value Theorem at work. Finally, with a graph of the original function, a discussion of Intermediate Value Theorem comes naturally. Pick some arbitrary y-values and have students decide if they should exist. How would you know?
I don’t even know what made we think of it, but adding the graph of the slopes to the mix really made this an interesting problem. Everything I need to cover in the next few weeks is sitting right there! And students found it all intuitive, we just needed to add some formality to our justifications. I really enjoyed how this turned out.