# _The_Power_of_Exponentials,_Big and Small_

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3 For sure, but when I looked at the moon, things started getting really crazy. The moon is around 3.84 times 10 to the eighth meters from Earth. And I again used the fact that there are 10 to the fourth sheets of paper in a meter to determine that the moon is about 3.84 trillion sheets of paper from the Earth. Seems like a lot, but after checking my handy power table, I saw that after 41 tears, we will have 2.2 times 10 to the 12th sheets. And after 42, we will have 4.4 times 10 to the 12th, more than enough to reach the moon. 42 tears, wow, John must have already made it to the moon. Let's go and check on him. Ah man, tearing paper is so hard. I've only torn it five times, and I just can't tear it anymore. I'll never make it to the moon at this rate. Boy, he gives up really fast. Well, I think as the stack gets taller and taller, it becomes harder and harder to tear it in half. Yeah, plus I guess when we tear the paper into half each time, the area the stack covers decreases exponentially. I wonder if we'll be even able to see the stack after 42 tears. Oh, now I see why we can't tear it 42 times. Well, I'm not a queen. We can't make it to the moon. I wonder what exponentials are good for. Hmm, well, let me show you a neat trick I learned when I was a little kid. Let's go. Hey, John. Hey, John. Oh hey. How are you guys? Geez. Tootsie rolls, I love these things. We got you 10,000 tootsie rolls. 10,000 tootsie rolls. You know, because you could not make it to the moon. Oh my goodness. Guys, tootsie rolls are my absolute favorites. Yeah, we'll give you 10,000 every day of the month. 10,000 tootsie rolls every day of the month. Swati, what do I have to do to get 10,000 tootsie rolls every day of the month? You just have to give us one today, double tomorrow, double the next day, and so on, for a month. You know just so that we have a little bit for ourselves. Yeah, you're on. 10,000 a day. Thanks, guys. Here's your tootsie roll for today. Have a good one. Thank you. Geez, I really do think exponential growth is pretty quick. And I bet we're going to end up having more tootsie rolls than John by the end of the month.

7 manipulate this problem is to know the number of grains of rice that have been played down by the nth square on the board. We suggest that if students are having a tough time understanding exactly how the problem is set up, that you might start them off with a two by two, a three by three matrix instead of an eight by eight, which is a bit large, and have them actually work out for each square the number of pieces of rice that have been laid out on the board. Perhaps, listing those payouts for n equals 1, 2, 3, 4, 5 next to the numbers 2 to the 1, 2 to the 3, 2 to the 4, 2 to the 5, might help the students understand that the general formula by the nth square is 2 to the n minus 1, which is again the key comprehension to be able to proceed and answer a question like could the queen actually pay out 64 squares of the chessboard. The second segment deals with the problem of tearing and stacking paper on top of each other. And specifically, we ask a number of questions related to just how many times you need to do this operation of tearing paper in half and stacking it to get a stack of paper to a certain height. So the key understanding to be able to answer any question of this sort is exactly how many papers we have in our stack after the nth tear. Students may quickly get this depending on their level of mathematical sophistication, but if they don't, we suggest just having students tear sheets of paper, put them on top, and they should be able to get the trend pretty quickly. The fact that after n tears, there are 2 to the n pieces of paper in the stack. Once they understand that, the questions we think require a bit of insight in the students into the kind of key base question, which is just how many pieces of paper do you need in a stack to get to a certain height. We've given a piece of data, which is that a piece of paper is 0.1 millimeters thick, which is 10 to the negative fourth meters. But students should be able to figure out for themselves that they need to know the typical height of a human being and also the distance to the moon. If the students have a way to easily-- they should know roughly how tall human beings are, about 1.75 meters tall, but if they have trouble figuring out how far it is to the moon, you could probably give them that piece of information, it's right around 384,000 kilometers away. So once students have figured out exactly how many sheets of paper they need to reach a certain height, they're in the home stretch. But they still need to actually convert that to a number of tears to answer the problem. So of course, this is really a logarithm, the log base 2 of the number of sheets of paper that they need, and that would give them the answer, the number of tears. And students may be able to compute this with a graphing calculator. We found that many graphing calculators don't have a log base 2 functionality, so there would need to be some sort of change of base, perhaps taking a log base 10 and then dividing that result by the log base 10 of 2. So it might actually be easier to just do the look up in the powers of 2 table to figure out the number of tears. In this case, students would just be looking for the number n, at which finally we exceed the number of sheets of paper that we need to reach a certain height. And this is how we present the material at the beginning of the next section. So in the third and fourth sections, we introduce the concept of which function grows faster. Is it an exponential or is it a polynomial function such as a linear or a quadratic function. And when asked on a range, like on the time range of a 30-day month, this is probably best addressed by plotting the functions. We particularly chose functions that are somewhat easy for students to plot, and we think that students should be able to plot the graph of 2 to the n versus 10,000n without too much trouble. Of course, if they have graphing calculators available to them, another option would be to plot on that domain these two functions and to use the graphing calculator's ability to intersect functions to find an exact point of intersection, which of course would correspond to the exact day at which there's a payout kind of break even point and when we start making more tootsie rolls, Swati starts making more tootsie rolls than John.

8 So the fifth segment is certainly the one that's most mathematically involved. We introduce the concept of in general for an arbitrary polynomial function and an arbitrary exponential function with base greater than one, which one ends up growing faster as n approaches infinity. So there are a couple of concepts here that are very intimately related to calculus. And we recognize that some of the students listening to the lecture might not have already taken a calculus course. But if so, there's kind of natural links to limits, for instance, and to the ratio test. We also think that if there were teachers of calculus classes, who wanted to look at alternate derivations of this fact, for instance, using repeated application of L'Hopital's rule that could be a really interesting and useful exercise for students, who are in that calculus class. Thanks for your time, and we hope you and your students enjoy exploring the world of exponential functions with us.

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