In this video, we use exponential functions to model exponential growth and decay, and explore some contrasting examples in applications. We draw on techniques from recent videos to help us successfully perform calculations and make predictions using these models. Recall the shape of the curve for the natural exponential function y equals e to the x. What happens if we reflect this curve in the y-axis? Here are the two graphs on the same diagram. The pink curve is the result of reflecting the green curve in the y-axis. To get the reflected image, we're keeping the same y-values but just switching each input x with its negative. This means that the rule for the pink curve must be y equals e to the minus x. Observe that e to the minus x is one of the e to the x, which is one over e all to the x. If you typed the reciprocal of e into your calculator, you get about 0.3678. So, we get an exponential function with a base one over e which is less than one. It's the case that every exponential function y equals a to the x with base a less than one is a scaled version of this particular shape. In the earlier videos, all of the graphs that we exhibited in fact, had a base a greater than one, like a equals e, or a equals two. They become extremely steep as you move to the right, which models something called exponential growth. But just for now, I want to focus on the case where the base a is less than one, and concentrate on this part of the graph where the curve looks like it's hugging the positive x axis. The behavior of the y values getting closer and closer to zero very rapidly models what we call exponential decay. Physical examples includes, decay of radioactive elements such as uranium and certain isotopes of carbon usually carbon dating. Decay or dissipation of drugs in the bloodstream, and reduction in air pressure as you increase your altitude above the Earth's surface. To capture the mathematics as flexibly as possible, we scale in both the x and y directions by considering functions of the form y equals A e to the kx, where A and k are constants. Note that A to the kx can be rewritten as a to the x by setting a equal to e to the k. So, the rule for the function can also be written as y equals capital A times little a to the x. If little a is greater than one, then we have exponential growth, and if little a is less than one then we have exponential decay. In many applications, the input variable x denotes time. But let's apply the exponential decay to model air pressure. In this application, the variable x will denote altitude above sea level. So, y written as a function of x measures air pressure at altitude x, and the altitude x equals zero just means at sea level. The Summer Olympics were held in Mexico City in 1968 and there was a huge controversy about athletes who tried near sea level being disadvantaged and having to acclimatize the high altitude and thinner air of Mexico City. What in fact was all the fuss about? Well, let's find the percentage reduction in air pressure as you move from sea level to Mexico City which has an altitude of 2,237 meters. There's a useful scientific fact we can use that air pressure decays exponentially at about 0.4 percent per 30 meters. Put y equal to air pressure in some appropriate units as a function of altitude x meters above sea level. Using an exponential decay model y equals A e to the kx, for some constants A and k, and in fact k will turn out to be negative. Capital A equals y evaluated at zero, and represents the air pressure at sea level. But we never need to know the actual value of A, which you might find surprising. However, we do need to get information about the constant k. The first step is to find k. Observe that 0.4 percent is 0.004 as a real number. If you take that away from one, you get 0.996. So, the science tells us that y of 30 is 0.996 times y and zero. Which is a direct mathematical translation of the fact that air pressure at 30 meters is 99.6 percent of air pressure at zero meters, that is at sea level. Putting this together with the rule for the function, we get A times e to the 30k, putting A is equals 30, is 0.996 times y of zero, which just becomes 0.996 times A. Canceling A from both sides, gives e to the 30k is 0.996. Taking natural logs of both sides, we get 30k is the log of 0.996 because the log undoes the exponential on the left-hand side. So, we get k is log of 0.996 divided by 30. You can approximate k on a calculator if you wish, but it's best to keep k in exact form for now rather than risk introducing rounding errors. This completes the first step. Remember, we're trying to find the percentage reduction in air pressure as we go from sea level to Mexico City. It will help to know what fraction of the air pressure at sea level when x equals zero is represented by the air pressure in Mexico City, when x is 2,237. The second step therefore, is to calculate the fraction y of 2,237 divided by y of zero, the ratio of the air pressure at Mexico City to the air pressure at sea level. Feeding these values to x into the formula for y, the ratio becomes this expression. But again, the A's cancel and e to the zero is one, so this simplifies to e to the 2,237 times k. We then fade in our expression for k which we found from the first step, rearrange the order of the exponents to get a in the natural log next to each other, which simplifies because the exponential undoes the logarithm. Hence, the ratio we're looking for becomes this fractional power 0.996 to 2,237 divided by 30, which we can type into our calculator and see that it equals 0.74 to two decimal places. This completes the calculation for the second step. We can now convert this information into a percentage reduction. The ratio of the air pressure is 0.74 which equals 74 percent. So, the percentage reduction must be a 100 minus 74 percent, which is 26 percent. This solves our original problem. The answer we estimate to be about 26 percent. In fact, the true reduction is about 25 percent, so the model gives quite a good estimate. You can appreciate why athletes might have had trouble breathing whilst competing in the Mexico City Olympics. What about examples of exponential growth. In fact, if your bank account grows exponentially, the base a on default savings accounts with poultry interest rates is so close to one these days that you might not notice the difference. Examples where the base is large occur commonly in population dynamics and the spread of diseases and early stages of epidemics. For example, if food is prepared and left out in the sun exposed and a fly lands on it and deposits some bacteria, it might be safe to eat straight away if your body's immune or defense system is robust enough to handle relatively small numbers of bacteria. But if the food is sitting there for an extended period of time and the bacteria has a chance to propagate and multiply, there can be an exponential explosion in the population. If you then eat the food, then your body's defenses could be overwhelmed and you might get sick. A more benign example of exponential explosion occurred in a previous video. When we drew part of the graph of y equals two to the x and contemplated what would happen if we increased the sizes of the inputs. I'd like to take that example two to the x a bit further, and tell you a story about Aladdin, you know the guy with a lamp and the genie. Well, Aladdin and princess Jasmine wanted to get married, but the Sultan wasn't keen on the idea and didn't immediately grant permission. The Sultan said, "Aladdin, to obtain my consent, you must perform the following simple task spread over 64 days. One of my court yards is designed as a gigantic chess board. They're 8 times 8 is equal to 64 huge empty squares. Tomorrow is the first day of your task and you must bring one grain of rice and place it on the first square in the courtyard. On the next day, you must bring two grains of rice and place them on the second square. Then after that, you must bring four grains of rice have and place them on the third square. Every subsequent day up to and including the 64th day, you must bring twice as many grains of rice as the previous day and place them on the next empty square of the courtyard. If you managed to do this for 64 days, then you will have my consent" Aladdin thought he had it made. This seems such an absurdly simple task. Each square was big enough to take sacks and sacks of rice and the Sultan was only talking about just a few grains. Let's look at the mathematics and also something closer to home. Bondi beach has a lot of sand. How many grains of sand do you think there are? The beach is about one kilometer long and about 200 meters wide. Supposedly the sand goes down about 10 meters. A grain of sand occupies about 10 to the minus nine cubic meters. So, a rough estimate of the number of grains of sand you can calculate. It turns out to be about 2 times 10 to the 15. Getting back to Aladdin. He has to bring twice as many grains of rice each successive day to the Sultan's courtyard. So, the amount he brings on any given day is a power of two with a variable exponent. So, it becomes an exponential function of the number of days that have passed. Let n equal to n of x, be the number of grains of rice that Aladdin brings on day x, to put on square x of the Sultan's courtyard chess board. If you think about the pattern of doubling from one day to the next, you get the formula n of x is 2 to the x minus 1, which is to 2 to the x divided by 2. The reason for the minus one in exponent is that on the first day Aladdin brings one grain of rice, and one equals two to the zero which is two to the 1 minus 1. Do you think in completing his task that Aladdin might ever have to bring to the Sultan more grains of rice than there are grains of sand on Bondi beach? The answer might surprise you. Let's do the math and work out the first day x on which the number of grains of rice must exceed the number of grains of sand on Bondi beach. We want 2 to the x divided by 2 to be greater than 2 times 10 to the 15. So, 2 to the x is greater than 4 times 10 to the 15. We want to find x but it appears as an exponent. To bring it down, we apply the natural logarithm to both sides and use some log laws to get that x log 2 is greater than log of 4 plus 15 log 10. So, dividing by log of 2, we get that x is greater than this fraction which evaluates to approximately 51.8 on a calculator. So, the 52nd day will be the first day in which Aladdin is lugging in more grains of rice to put on one square of the courtyard than there are grains of sand on Bondi Beach. Aladdin hasn't even reached the last row of the chessboard. The Sultan was cunning. This looks even to tough a task for Aladdin's genie. Such is the deceptive and mind boggling nature of exponential growth. The moral of the story, don't eat food left out in the sun if there's any risk of flies and bacteria playing games with exponential explosion. In this video, we looked at two important aspects and the behavior of exponential functions. This can be used to model exponential growth when the base is moving one, and exponential decay when the base is less than one, and we analyze the mathematics system contrasting examples. Please read the notes accompanying this video and when you're ready, please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
