In my Calculus class, recently a discussion came up about when a derivative trig function would have a midline that is not 0. We quickly discussed the algebraic result of this question, but I thought afterwards that this was just too juicy of a discussion to let slide by.
So, on our new practice quiz, I decided to throw in one such bonus problem. (Go to the end.) They were AMAZED by the resulting graph when they worked on this in class today. One algebra-whiz kid was like, "This is NOT allowed! You cannot mix trig function with other functions!!" So hilarious!! I LOVE MY KIDS. I was pleased with their sense of wonder and surprise, but even more pleased that they were still able to look at and compare features between the derivative and original function graphs, even though they thought what they were looking at was super weird and not intuitive.
At the end of class, I off-handedly asked a couple of the fastest-working kids today, "So what's the average value of the derivative function?" They said, "...-1?" And then I asked, "How does that show up inside the original graph?" Those kids' eyes got so big and they said, "It has an average derivative of -1! OMG, you can see it!!!"
Cool graphical connections!!! We'll have to revisit this bonus problem as a class tomorrow, to make sure that everyone can appreciate the juiciness of this connection before we move on to other algebra goodness.