Part 2d of the 2008 Calculus BC exam free-response section. Created by Sal Khan.
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- How to solve the last integral expression without using a calculator?(3 votes)
Welcome back. We're ready to do part D, and let me copy and paste that in as well. See, I don't think that's going to need this graph, so let me just remove that with a color other than yellow. It's copy and pasted. I don't know if you can read it, but it's helpful for me to review the problem on our clipboard. OK. The rate at which tickets were sold for t -- for over this range is modeled by r of t-- let me write that in case you can't see it-- is the rate at which tickets were sold. r of t is equal to 550 te to the minus t over 2 tickets per hour. Based on the model, how many tickets were sold by 3:00 PM? So my t equals 3-- to the nearest whole number? So that's sometimes important. You don't want to give a decimal answer. So this is the rate at which tickets are sold. So this is the derivative of the total tickets sold function. Or another way that we could write it is the total tickets sold-- so let's call that, I don't know, capital T sub-- well let me-- I don't want to do T of t, that's [UNINTELLIGIBLE] So let's say the tickets sold as a function of time is going to be equal to the definite integral-- well, we could say is at any time t, the tickets sold-- and this is the fundamental theorems calculus, I think it might be one of its correlaries or actually sometimes it is the fundamental theorem of calculus, I always forget my definitions. Between time equals 0 and t-- or if we want to know the tickets sold, between time equals 0 and t is equal to the integral of the rate at which the tickets sold was changing. So that's equal to 550te to the minus t over 2dt. Right? That's it. And so if we want to know how many tickets were sold at time equals 3, that's just equal to the definite integral from 0 to 3, or we could also view it as the area under this curve, from time equals 0 to time equal to 3 of 550te to the minus t over 2dt. Now this integral right here, you can solve it analytically using integration of parts, which I just called the reverse product rule, but you only have 45 minutes to do all three of these problems, and they'll let you use your graphing calculator, and your graphic calculator is excellent at doing definite integrals, and they just want the number, right? So let's use our graphic calculators to get that number. Let's see. I don't want to copy, so how do we do that? We just do second the division button but that's calc-- definite, let me use the definite integral, and like what was, let's see, let me make sure I have that-- 550 let's just use x. 550 times x times second e to the minus x divided by 2. I think that's the whole function. And let's see. My independent variable is x, I've just swapped t for x there, and I'm taking the integral from 0 to 3. Click enter, let the calculator do the work. This would have taken you quite a while if you had to actually do the integral yourself. 972.78, and they want us to round to the nearest whole number. So the nearest whole number is 973. So we say 973 tickets sold by 3:00 PM. And we're done. That only took us four minutes. And it would have taken us even less if we didn't have to explain it. Anyway. I will see you in problem number three.