One of the most useful mathematical insights I gained from working as an astronomer is that the bounds in an integral are inextricably linked to the variable of integration, and that this plays an absolutely critical role in integration by substitution. Changing the bounds when you change the variable isn’t just a shortcut or a convenience: it is an essential part of the process of substitution. And it makes total sense if we look at substitution in a physical context in which we change between two variables that actually mean something to us. The problems linked here introduce substitution by placing it in a physical context and clarifying what it really means to change the variable of integration.
One of my favorite start-of-the-year activities is to have students produce a growth rate graph from the CDC growth chart for children. Here is a link to a student-facing version of the activity, which you are free to copy and adapt for use in your class1. And here is a quick-start guide to using this activity in class.
This activity has students compute slopes using both difference quotients and tangent lines. It asks them to abstract the difference quotient (in spreadsheet notation), so that it can be computed quickly for hundreds of data points. It gets them warmed up to the idea of viewing slope as a function, illustrating a situation where graphing slope as its own function is quite illuminating. It helps them see how a function relates to its derivative and prepares them for interpreting derivatives in the future, both in applied contexts and in curve sketching.
The graphs below2 give a quick sense of how this activity turns out. Students start with the graph on the left (data from the CDC), and end up producing the graph on the right. Features that are barely visible on the left become obvious on the right, for example the incredibly rapid growth of 2-year-olds and the growth spurt around age 12. Truly, the growth rate graph gives a much richer sense of how human growth varies throughout childhood.
The right edge of the growth rate graph also illustrates how a function can continue to increase even if its derivative decreases. Every year, I say to my students after they do this: “the rate of growth graph drops off pretty fast after age 12. Does that mean the person is shrinking?” And they immediately say, “no, it means that they’re growing more slowly.” In this context, students see that this makes sense: the declining growth rate corresponds to a flattening slope, not to a decreasing height. Later in the year, I remind them of this every time someone gets confused about the distinction between decreasing f’ and decreasing f.
