Voiceover: The Cozy Car Company ships some of their new cars to Japan and Vietnam. The number of cars that will be shipped to Japan during the next t months is modeled by the function j of t is equal to 2 to the tth power. The number of cars that will be shipped to Vietnam during the next t months is modeled by the function v of t is equal to 2t squared. Which country had received more cars from the Cozy Car Company after 5 months, or will have received after 5 months? Let's see how much Japan is going to receive after 5 months. t is in months, so j of 5 is going to be equal to 2 to the 5th power, which is equal to 2 times 2 times 2 times 2 times 2, and let's see, 2 times just 4, 8, 16, 32. Japan will have received 32 cars, and Vietnam, so v of 5 is going to be 2 times 5 squared, which is going to be 2 times 25, which is equal to 50. Based on these 2 models for how much they're going to receive after t months, after 5 months, Vietnam is going to receive, Vietnam is going to receive more cars. I guess the answer to that is Vietnam. Vietnam will have received more cars after 5 months. Which country had received more cars from the Cozy Car Company, or will have received more cars after 7 months? Once again, let's try this out. j of 7 is equal to 2 to the 7th power. Let's see. 2 to the 6th is going to be 32 times ... we can read this as 2 to the 5th times 2 times 2, which is going to be equal to, this is going to be equal to 32 times 4, which is 128 cars after 7 months will have gone to Japan, and to Vietnam, v of 7 is going to be equal to 2 times 7 squared, so that's equal to 2 times 49, which is equal to 98 cars. After 7 months, Japan would have received more cars, so Japan, Japan will have received more cars after 7 months. This is interesting. We see that the exponential function, notice where you have the t as the exponent, that although it might start off a little bit slower than this, what's essentially a quadratic function when you have something squared, it starts off slow. After 5 months, you would have shipped fewer cars than using the quadratic model right over here. But then, it more than catches up, and it starts to increase at a faster and faster rate, and even by 7 months, it's able to pass up the quadratic function. Which country will have received more cars from the Cozy Car Company ... Will the country which received more cars from the Cozy Car Company after 7 months continue to receive more cars than the other country in future months? Yeah, absolutely. Once the exponential function passes up the quadratic function, it just goes faster. It just keeps increasing at a faster and faster rate. You could see that if we wanted to compare 8 months, so j of 8, this is the exponential function, this would be 2 to the 8th power, which would be this times 2, they'd get 256 cars, and v of 8, v of 8 is going to be 2 times 8 squared, which is 2 times 64, which is 128. Notice now we would have shipped twice as much to Japan as Vietnam, which isn't what the case right over here. We shipped more to Japan than Vietnam but not twice as much. We could keep going. We could, if you want, you could go to j of 9. j of 9 is 2 to the 9th power, which is going to be 256 times 2 or 512 cars, while v of 9, v of 9 is going to be 2 times 9 squared, which is 2 times 81, which is 162, so now it's more, way more than double, actually more than triple. So you see that once you get past those initial few months, the exponential function is increasing at a much, much, much faster rate. We could actually visualize that. Let's actually get out a graphing calculator to visualize these 2 things to see how that is happening. Let's graph it. The first one, let me graph the exponential, so 2 to the, well, I'll just say x power. We'll say x is our independent variable here, so 2 to the x power. Then let's do the quadratic one. This is y of 2, although it will be v of 2. Let's say 2 times x, 2 times x squared. Now let me set the range. Let me set the range here. Let's see. Let's say x starts at 0, and then let's say it goes up to 10. Let's say it goes up to 10. The x-scale could be 1. X-scale could be 1. Now y's minimum, let's say we'll start at 0, and then y-max, let's say, let's go to 1,000. Let's go to 1,000, and let's make the y-scale 100, 100, and now I think we're ready to graph. Let's graph this. Let's see what happens. It's munching on things. That, right over there, that's the exponential, and then there, you see right over there, you have the quadratic. Actually, let me zoom in a little bit on this so that we can see where they pass ... or actually, let me zoom in a little bit on this. I'll do it with a box so that we can really see, we can see where they, or attempt to see where they pass each other up. I'm going to start there. I'm going to make my box, let's see, go ... Whoops. It's weird using a calculator on a computer like this. But you see, you definitely see, even what we've already graphed, that the exponential really just starts to shoot up while the quadratic is just going ... Well, it's still increasing at a decent pace but nowhere near as fast, and the difference is becoming more and more pronounced as time increases. Let me just make sure that's as low as that, and let's graph over in this range right over here. Let's see. That, right over there, that's the exponential function. That's 2 to the t power. Then that right over there is the quadratic. You see, you definitely see ... Actually let me ... You definitely see that earlier on, the quadratic has higher values, and you see that right over here, after 5 months, we ship more cars to Vietnam. But then the exponential passes it up and then just keeps shooting faster at an ever increasing pace.