Algebra (all content)
Sal discusses two functions that model the shipment rate of cars. One function is quadratic and the other is exponential. Which one will eventually exceed the other? Created by Sal Khan.
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- How could we have proved that Japan will have received more cars after the 7th month?
I tried doing it in many ways, but I always got to a point where I couldn't solve the problem without a graphing calculator, here's what I tried to find the values for which j(t) will be greater than v(t).
2^t - 2t^2 > 0
2 - ((2t^2)^(1/t)) > 0
-((2t^2)^(1/t)) > -2
((2t^2)^(1/t)) < 2
Then I inputed ((2t^2)^(1/t)) and looked what values were less than 2 to know when j(t) will be greater.
Please show me how to solve it manually.(10 votes)
- Wow what a question! Sounds simple, but it is actually extremely advanced. So, you can find the exact moment Japan passes Vietnam by setting
j(t) = v(t)and solving. So
2^t = 2*t^2. Might look simple, and, well, you can use a calculator to approximate that and find
6.31972235584...months. However, expressing the answer precisely involves something called the Lambert W function, which I'd never heard of until just now. You can see the precise expression of the value by looking at the third of the "Real solutions" on http://www.wolframalpha.com/input/?i=2%5Et+%3D+2*t%5E2 - this is a transcendental number that's not easy to even express in a formula. I'll have to read a lot more to about Lambert W now. :)(12 votes)
- How come, the quadratic graph isn't a parabola or has a parabolic shape?(2 votes)
- It is hard to see the shape because of the window settings. Rest assured, the shape of the curve is parabolic.(7 votes)
- If the graph of cars shipped to Vietnam is quadratic, then why does it look like a straight, linear line when it is graphed on the calculator?(2 votes)
- It sort of looks like a straight line because the scale had to be so large to see both of them together. Here's a much larger graph of both equations: You can clearly see the red quadratic graph is curved, but the blue exponential curve just explodes so much faster: http://tinyurl.com/ojkrdgr - let me know if that explains it.(5 votes)
- On the graphing calculator, the quadratic function looks linear. Can anyone explain this?(3 votes)
- It depends on multiple problems that can exist in your case. One, the graping plane may be zoomed in a little too much or ,two, the function may be incorrectly put in.(3 votes)
- How does the quadratic equation have a linear graphing even though it has an exponent?(2 votes)
- It is definitely curved, it's just that the exponential curves up so much faster that it doesn't look like the quadratic is curving at all. In fact (and you can see this later in calculus), the slope of an exponential increases (or decreases) exponentially, and the slope of a quadratic increases (or decreases) linearly. So the difference in how the slope changes is even more dramatic between the two functions. And a final point that is also very important in calculus, is that any smooth curve, if you zoom in far enough, looks like a straight line.(2 votes)
- the number of cars shipped to japan is modeled by function = j(t) = 2^t where 't' is time ..
how do i comprehend 2^t in real world e.g. if it was 2(t) and after 5 months i can see 2*5 = 10,
but i don't understand how i express 2^t ??
- i put 2(7)^7 and got 1647086. he mathed wrong some where. why?(1 vote)
- I think you mixed up the 2 functions. Sal's work is correct.
If you are calculating v(7), then you need 2(7)^2.
If you are calculating j(7), then you need 2^7
Neither function creates 2(7)^7.
Hope this helps.(1 vote)
- my teacher uses a common rate/ % box and i still haven't understood the subject and ive been trying for a week and have a test this Tuesday.... is there anyone who can help me understand?(1 vote)
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.