Calculus BC 2008 2d

Calculus

Calculus BC sample questions

AP Calculus BC Exams: 2008 1 a
AP Calculus BC Exams: 2008 1 b&c
AP Calculus BC Exams: 2008 1 c&d
AP Calculus BC Exams: 2008 1 d
Calculus BC 2008 2 a
Calculus BC 2008 2 b &c
Calculus BC 2008 2d
2011 Calculus BC Free Response #1a
2011 Calculus BC Free Response #1 (b & c)
2011 Calculus BC Free Response #1d
2011 Calculus BC Free Response #3a
2011 Calculus BC Free Response #3 (b & c)
2011 Calculus BC Free Response #6a
2011 Calculus BC Free Response #6b
2011 Calculus BC Free Response #6c
2011 Calculus BC Free Response #6d

Calculus BC 2008 2d Part 2d of the 2008 Calculus BC exam free-response section

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Calculus BC 2008 2d

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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.

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