Thinking Flexibly About Exponential Form

I had a great discussion with my Calculus class about how the exponential sequence 2^x is
1, 2, 4, 8, 16, ... but that there is nothing special about the base 2. There has to be a number "a", such that (3^a)^x = 2^x, because there has to be an exponent "a" such that 3^a = 2. Together they found that number, and we said that g(x) = 3^(0.631x) is therefore an approximation of f(x) = 2^x. Similarly, there must be an exponent k such that (e^k)^x = 2^x. So, the kids took a minute or so to find that k = ln(2), so h(x) = e^(0.693x) must also be an approximation of f(x) = 2^x. We discussed how h is easily differentiable now that we are Chain Rule pros.

I then set the kids loose on an activity that asks them to think flexibly about the exponential form. The reason why I like this activity is because the kids distinguish between an exact form that is "natural" for the situation versus the usual e^(kx) approximation. (The kids know that they can substitute k with an exact expression without losing precision, however.) For example, for #1 in the worksheet, where the kids are looking at bacteria that double every 6 minutes, both the kids and I think that it's natural to think of the sequence as {10, 20, 40, 80, ...} and to observe the general form y = 10(2)^x. To fix the problem that we want it to double only ONCE by x = 6 minutes, not doubling 6 times, we divide our exponent by 6 to get y = 10(2)^(x/6). For me, this is the natural way of writing the equation and testing it initially to make sure that it fits the bill. If they then want to differentiate it, they then turn it into base e, where e^k = 2^(1/6), so that y = 10(e)^(kx) replaces y = 10(2)^(x/6). The kids figure out that k = (1/6)*ln(2) or around k = 0.116. So, y = 10(e)^(0.116x) is an approximation of our exact function, and it has the benefit of being easily differentiable.

In thinking about exponential form as being fluid, the kids can consider equivalent compound-interest scenarios. I gave them a couple of scenarios to play with and to explain, in order to get at that idea. I am pretty happy with the level of understanding they have with this concept, seeing that it's the second time this year we've seen exponential compounding. After that, they didn't seem to have much trouble working through our practice quiz on the exponential topic. Overall, I am pretty happy with the way our differentiation technique unit has gone. We're moving a little bit slower than I had hoped, but their understanding of the connections between concepts has been really great!!! Both they and I are still excited to walk into this class everyday, and that's a good feeling. I anticipate that by January, we'll be wrapped up with all the differentiation techniques (including related rates problems, which I've been sprinkling into the mix periodically), and the kids will be ready to start thinking backwards and/or to do a differentiation project.

Stay tuned!