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## AP®︎/College Calculus BC

# Calculus BC 2008 2d

Part 2d of the 2008 Calculus BC exam free-response section. Created by Sal Khan.

## Video transcript

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.