MARK KIRALY: Hi Virge.
VIRGE CORNELIUS: Hi Mark.
How are you doing?
MARK KIRALY: All right, we're back
for another AP Daily Practice session with AP Calculus.
And in this video, we're going to go over a free response
question where we get to use the calculator quite a bit
actually.
VIRGE CORNELIUS: Yes.
So if you have not yet downloaded the question,
now is definitely a good time to do that.
And also make sure your calculator is ready to go.
I'm Virge Cornelius.
I teach at Lafayette High School in Oxford, Mississippi and--
MARK KIRALY: I'm Mark Kiraly and I
teach at Ryan High School in Denton, Texas.
So here's our opening.
An online bookseller receives orders at a rate of B of t
equals the square root of 480t minus 40 t squared
in orders per hour since t equals 0 after opening.
The bookseller begins shipping orders at t
equals 1 hour after opening at a rate of S of t equals 9 plus 3
to the 0.035 t squared orders per hour.
When the bookseller opens, 0 hours after opening,
there are 57 orders in the system waiting to be shipped.
And we're going to take this thing one piece at a time.
And piece a is find B prime at 2.
Using correct units interpret the meaning of your answer
in the context of this problem.
I've got my B of t and my S of t but we are clearly
using B down here.
Now Virge and I we're talking about this
when we were doing our setup yesterday or a couple of days
ago that we don't always store their functions
in the calculator when it's multiple choice.
But, for free response, it makes sense.
It's multiple parts, so I'm going to probably refer back
to these.
So you'll see, I've put my B of t
and my y1 which is the square root of 480x minus 40
x squared.
I've changed the spelling of t to x.
And I've got my S of t in there.
I'm not using S of t in this part,
but I anticipate I'm going to need it,
so I could enter it later if I wish to.
So I've already re copied my B prime of 2
here because I do want to just recopy what I'm doing.
I'm going to go out to my home screen.
And you have to be familiar with your own machine.
I know Virge talked about her yellow sheet in the last video.
I think you should practice with whatever calculator
you're going to use on the exam and know where this stuff is.
So for me, I go to the math venue.
You can also go to F2 on this machine,
but different machines have it but math 8
is what I have memorized.
And we do math 8.
I want to put my x there because I'm
taking the derivative with respect to x.
And I can type it in right here.
Great Virge.
I can just put second square route and so on, but I've got
it stored in white so I'm going to do--
and I get 5.6568 and a bunch of other stuff.
On the AP exam, they want it to three decimal places accuracy.
That means you can write as many as you want.
So what I wrote is fine.
I can write down everything that's on the calculator.
I could just stop at 6 or I could write it is 5.657.
I can round that or truncate.
This is called rounding and of course, this
is called truncating.
And either one's fine whatever you're comfortable with.
Honestly, I think the safest thing is just
to write a bunch of numbers but that's B prime of two.
VIRGE CORNELIUS: I would say that that's also a good idea
because with intermediate values, which this isn't.
But if you have an intermediate value like an intercept
or something, you need that to a higher precision.
So it's always a good habit to have
more than three decimal places.
MARK KIRALY: Yes.
Just for storing purposes.
If you didn't, you can at least type in a large number
of accurate decimals later.
Now it says use correct units interpret the meaning of this.
Well, B of t was a rate.
It's the rate in which orders are received.
So B prime of two, so the rate at which orders are received.
That's what B is, right?
VIRGE CORNELIUS: Mm-hmm.
MARK KIRALY: Rate at which orders I received.
And is that a positive?
It indicates what's happening, Virge.
VIRGE CORNELIUS: Increasing.
MARK KIRALY: Is increasing at a rate of 5.656.
Well, B was measured in orders per hour.
And I took the derivative with respect to what, Virge?
What was my independent variable?
VIRGE CORNELIUS: Time.
MARK KIRALY: Time.
So per hour again.
VIRGE CORNELIUS: Essentially dividing by time,
you're finding that slope.
MARK KIRALY: Im doing d/dt of B of t.
So B of t's units per time unit.
And I could, of course, write that as orders per hour
squared.
I'm not done interpreting because this
is specific to at t equals 2 hours after opening
because of that, there's a parenthetical.
So it's the rate at which orders are received
is increasing at a rate of 5.656 orders per hour per hour
at time equals 2 hours after opening.
And of course, prepositional phrases can be rearranged.
I could start with at 2 hours after opening
and then talk about the rate increasing
but hope that makes sense.
VIRGE CORNELIUS: It's good and I'm
glad you did it because it's a very awkward interpretation,
I think but if you follow the recipe on that interpretation--
MARK KIRALY: Yes, and like I said,
the recipe is units of what you started
with per units of what you took the derivative with respect to.
All right so part B. Very straightforward.
Find the integral of 1/4 of the integral from 1 to 5
of S of t dt.
And again, using correct units, interpret
the meaning in the context.
A little more writing.
More writing than perhaps we're used to in a math class
but that's all right.
S of t.
That's the other function, isn't it, Virge?
VIRGE CORNELIUS: Guess where it is.
MARK KIRALY: So there it is and it's in my white too.
VIRGE CORNELIUS: You got it.
[LAUGHTER]
MARK KIRALY: How do you use it?
So again, I'm going to grab my very convenient machine here.
This is under math 9 on this machine.
But again, you should know where your definite integral is.
I'm going to go 1 to 5.
I did leave that 1/4 off so remind me to divide
by 4 here at the end, Virge.
VIRGE CORNELIUS: You got it.
It won't be very long from now.
MARK KIRALY: So let's see.
So I am doing y2 this time divided by 4
or multiplied by 1/4--
VIRGE CORNELIUS: Oh, I thought you were going to hit Enter
and then divide it by 4.
But yeah, that's OK too.
MARK KIRALY: I don't mind doing that.
I thought about doing that just to discuss something
and then divide by 4.
So we get-- and again, I'm not going
to write it up here in this time only
because that's not the way it would be printed
on an actual test book, so I'm going to go ahead and write
the whole thing again.
And it's 10.545 this time whether I
round it or truncate it.
But again, if I wanted to write that 07859, I could.
So it says interpret the meaning in the context of your problem.
Well, what was the units on S of t?
VIRGE CORNELIUS: Same as B of t.
MARK KIRALY: Orders per hour.
So when I integrate that with respect to hours,
what does 42.180 represent, Virge?
VIRGE CORNELIUS: Orders.
MARK KIRALY: That was the number of orders taken
from time and end of time.
VIRGE CORNELIUS: In that time period.
Yeah.
MARK KIRALY: And then I divide it by the four hours.
So this is orders per hour again.
VIRGE CORNELIUS: Yeah, it's another rate.
MARK KIRALY: Because we've taken that time unit off there
but then put it back in with this 4
because that 4 is really disguised
as 1 over 5 minutes 1.
In talking about that, I just started shooting into my answer
here that this matches this form.
There's the famous thought bubble.
VIRGE CORNELIUS: Ooh, that goes on your yellow sheet
from last video.
MARK KIRALY: Yeah.
That is the average value of f of x on the interval.
a, b.
Box it if you're assigned your yellow sheet, right?
VIRGE CORNELIUS: Yep.
MARK KIRALY: So interpreting that.
The average in--
S of t again is a rate.
It says right here at a rate of.
So the average rate of orders in essence
were shipped from time t equals 1 to t
equals 5 hours after opening.
I was really getting some messy handwriting,
is 10.545 orders per our.
OK.
Part C. Write but do not evaluate an integral expression
that gives the total number of orders waiting
to be shipped at time t equals 3 hours
after the bookseller opens.
That's a lot to unwrap.
So total number of orders that are waiting to be shipped.
Well, we started with a certain amount, Right
VIRGE CORNELIUS: Yes.
MARK KIRALY: We're only using this 57.
We started the day with 57 orders waiting to be shipped.
And they immediately started taking orders.
They started taking orders B of t here from time equals 0.
And work here is about time equals 3.
So these orders, this is that rate n.
Virge mentioned rate n's.
Orders are coming in from 0 to 3 at a rate of 480t
minus 40 t squared.
Of course, if I didn't want to write that expression,
what could I write there, Virge?
VIRGE CORNELIUS: B of t.
In fact, I think we'll probably do that on our area volume
problem later.
MARK KIRALY: I highly recommend writing.
The great thing about that is you
have to worry about, if I miss copy this,
I'm not going to earn full points with a missed copy so
much easier to write B of t than that square root function
and risk missing it.
So this is the rate at which orders are coming in.
But then we do start shipping.
When do we start shipping?
VIRGE CORNELIUS: After they finish drinking their coffee.
MARK KIRALY: So one hour after they get their coffee.
Maybe check the morning paper or the internet
since no one reads papers anymore.
And I'm just going to write S of t
this time because that is the easier way that--
VIRGE CORNELIUS: And you can mix and match.
You could have an expression that's got to B of t
and an S of t or whatever.
MARK KIRALY: Now this is the whole rate in, rate out.
Orders are coming in but they're shipping out.
So we're going to subtract.
That is [INAUDIBLE] 1 to 3.
We don't have to box.
Our final answers our final answer
but I'm going to box that.
VIRGE CORNELIUS: But either of those two lines
would have been fine.
MARK KIRALY: Absolutely.
VIRGE CORNELIUS: You don't have to give both of them.
MARK KIRALY: So this is our beginning.
This is our in and that's our out.
So that's how many our total are going
to be waiting for us at a time 3 hours after they open.
So pretty neat.
Works out well.
Didn't have to evaluate it so I can walk away
from it at that point, right?
I did want to say, if these bounds matched,
you could combine them as a single interval.
You can do your rate in minus your rate out
if the times match.
But since the time intervals don't match,
we're not going to be able to consolidate this at all.
VIRGE CORNELIUS: Mm-hmm.
MARK KIRALY: And on to part d.
Now is the number of orders waiting
to be shipped increasing or decreasing at t
equals 10 hours after the bookseller opens?
Virge, you said you worked these as well, correct?
VIRGE CORNELIUS: Mm-hmm.
MARK KIRALY: Did you write a function
for the number of orders waiting or did you just
jump into the rate of change?
VIRGE CORNELIUS: OK.
So I wrote a function n of t, like number of orders
or something like that.
OK that's good.
MARK KIRALY: This is what I did and I don't think
this is natural necessarily.
It's a really good practice.
And I'm using x as my dummy variable and t is my--
so this would be W of t is how many are waiting which that
should look familiar because we built one for 3:00 here,
How many were waiting at 3:00?
So I just replaced the 3's with t's
VIRGE CORNELIUS: Well, and the reason
that this is such a good idea to do because a lot of times
when students read d, they think we'll
have to take the derivative of something.
I'm being asked increasing or decreasing.
Well, I would want to think about the derivative
and see if the derivative is positive or negative at t
equals 10.
For me anyway, when I read a question and I'm being asked
is it increasing or decreasing?
I immediately go to oh, I have to take the derivative
and analyze it.
But I wasn't given anything like that.
I was given derivatives.
So I actually was given something
that I could right off the bat analyze.
Yeah.
MARK KIRALY: And here it was in the words, right?
We were given rates.
So if I just take the rate at which orders are coming in
minus the rate in which they're shipping,
that'll tell me if we're getting ahead or behind
at our bookshop.
VIRGE CORNELIUS: At time t equals 10 hours, yeah.
MARK KIRALY: At 10.
So it was really B of 10 minus S of 10.
And whether you built a function like this or you just reason
through it as orders coming in at 10 minus orders
going out at 10 will tell me if this is increasing or not.
And you we're going to use that alpha y 1.
And you can do function notation on the calculator.
That's so fabulous
VIRGE CORNELIUS: I love that.
MARK KIRALY: So grab my functions again.
Please, reach out to your teachers,
your classmates, your internet if I'm
doing things on the calculator that you be like wow,
I didn't know it did that.
Go find those resources.
They're out there.
VIRGE CORNELIUS: Yeah, watch the videos.
Not our videos.
Somebody else's videos.
MARK KIRALY: TI, HB, I am sure they all have their own, right?
VIRGE CORNELIUS: Mm-hmm.
MARK KIRALY: Negative 27.481.
And then they just asked me to interpret it.
Now that is orders per hour but more importantly the sign.
We mentioned the sign earlier.
That is negative so we are decreasing.
And they said give a reason.
Because B of 10 minus S of 10 was less than 0.
VIRGE CORNELIUS: Yep.
MARK KIRALY: And I think that's enough.
That's a reason.
I don't have to get into a justification
where it says give a reason.
VIRGE CORNELIUS: Yeah, and honestly, I
don't even think you need that inequality statement.
I think you can just have B of 10 minus S
of 10 and then that negative 27 right there, which is correct.
And then the declaration of decreasing is good.
Yeah.
MARK KIRALY: So what are our takeaways?
Don't forget you have a calculator.
We didn't talk about this within the video but a lot of times
we'll see students start taking elaborate derivatives
or integrals they can't even find because
of the limitations of what we've done in Cal 1
and Cal 1 and then realize, oh, I have a calculator.
So when you're asked to do a derivative
or find an integral with a calculator,
you're allowed to do those.
You're allowed to solve an equation.
So remember that you have that tool when it's allowed.
Virge, I want you to say this one.
VIRGE CORNELIUS: This is my favorite.
If you don't know what to do, particularly on these big word
problems, just do what you know.
And you really two things.
You know how to differentiate and you know how to integrate,
so start taking derivatives or start integrating.
That goes in to keep your pencil moving because that just--
Just do something even though you feel like you don't even
know where to start.
MARK KIRALY: Yeah, keep your pencil moving and putting
things into the calculator falls under that as well, right?
VIRGE CORNELIUS: Mm-hmm.
MARK KIRALY: All right.
Thank you.
We'll see you in the next video for another free response
this time without a calculator.
VIRGE CORNELIUS: That's right, and I'll be working that one.
So see everybody later.
MARK KIRALY: By Virge.
VIRGE CORNELIUS: Bye.