AP Calc AB FRQ: The Banana Problem

追加: 2026-05-12

[00:00:00]What's going on YouTube? My name is Mr. Almond, >> Mr. Love, >> and we are back at y'all with yet another FRQ. This one here is a calculator F FRQ dealing with rate in rate out. So, let's go ahead and give it a go. It says when a certain grocery store opens, it has 50 pounds of bananas on a display table. Okay, cool. Initial value. customers remove bananas from the display table at a rate modeled by f of t. Okay, so I'm noticing that this f of t function is the rate at which bananas are removed. And I'm also noticing that this exists from 0 to 12.

[00:00:42]>> All right, so f of t is a removal rate.

[00:00:45]All right. Um and then it says f of t is in pounds per hour. t is number of hours. Um, after the store has been open for three hours, store employees add bananas, okay, that's different, to the display table at a rate modeled by g of t. And I'm noticing that that interval is from 3 to 12, which makes sense because it happened after the store has been open for 3 hours. Okay, g of t is again in pounds per hour and t is the number of hours. Okay, so basically we got these two rates. One is the rate at which bananas are being removed and g of t is a rate at which bananas are being added. Okay. Um part a says how many pounds so a quantity of bananas are removed from the display table during the first two hours that the store is open. Okay. So if they're giving me a function that represents a rate but they're asking me about an amount that's how I know I need to integrate. Um it is just like one of those things I feel like you have to know cold that the integral of a rate represents an amount.

[00:01:55]>> Um so I'm gonna integrate the removal rate from it looks like 0 to two. So I'll write it and then Mr. Love if you want to like calculate it.

[00:02:06]>> Yep.

[00:02:06]>> Okay. So I'll say we want to calculate the integral from 0 to two of the rate at which bananas are removed which was f oftdt >> and whatever that number is should be the number of bananas that were removed.

[00:02:26]>> All right. So we're going to go ahead and go into Desmos.

[00:02:29]>> Okay.

[00:02:32]And it's good to kind of have these functions already typed in because we're just going to go ahead and do the integral from 0 to two.

[00:02:42]It's like 20.051.

[00:02:44]>> You said 051.

[00:02:46]>> On AP exam, you got to round um to three places. You can either round or truncate. That's part A.

[00:02:51]>> Yeah. All right. So, let's go ahead and look at part B. It says to find frime of seven, which should be pretty easy if we know f ofx or f of t. And then the second part says using correct units explain the meaning of frime of 7 in the context of the problem. So I'll go ahead and write what this means using units and then I guess you can calculate the the value.

[00:03:17]All right. So I'm going to go ahead and say f prime of 7 equals whatever that number is. Um, and I I'll go ahead and say is the rate at which well the way I'm thinking about it is it's it's the rate at which f is changing. And we got to remember that f represents a rate of change. So it's the rate at which a rate is changing.

[00:03:42]>> So I'll say frime of seven is the rate at which the rate of bananas being what? F is what?

[00:03:53]removed, right?

[00:03:54]>> Mhm.

[00:03:55]>> Okay. So, I'll say this is the rate at which the rate of can I say banana removal?

[00:04:04]>> Yeah.

[00:04:06]>> Okay.

[00:04:08]Is changing at 7 what? 7 t was what? Hours, right?

[00:04:14]Yeah, >> it was hours. Yeah.

[00:04:16]>> Okay. Okay.

[00:04:17]>> You want to do units?

[00:04:18]>> Oh, yeah. Units. Um because F was what?

[00:04:22]pounds per hour.

[00:04:23]>> Pounds per hour.

[00:04:24]>> So that guy's derivative would be pounds per hour per hour.

[00:04:31]>> All right. So we're going to go ahead into the Desmos and find frime of 7 appears to be 8.1195.

[00:04:39]So we can either truncate that at 119 or round to one 120.

[00:04:44]>> Okay.

[00:04:46]>> I'm not gonna lie, bro. Me knowing me, I'm the safe guy. If I'm on the test and I can truncate or round, I'm not about to mess up rounding.

[00:04:54]>> Right. Right.

[00:04:55]>> So, I'm just I'm playing it safe, bro.

[00:04:57]>> Yeah. Yeah.

[00:04:58]>> So, rate of rate of rate of change, uh, pounds per hour per hour, and then calculator for the F prime of seven part.

[00:05:07]>> Yep.

[00:05:08]>> All right. Moving right on to part C.

[00:05:12]So, this one says, "Is the number of pounds of bananas on the display table, is that amount increasing or decreasing when t is five?" And give a reason for your answer. What I'm going to do is I'm going to basically create a function that represents how many pounds of bananas are on the table based on any time t. So, what I'm going to do first is uh go back up and just double check because I believe there was an initial condition or initial value rather.

[00:05:41]>> Um, at t equals z there was 50 lbs of bananas already to begin with.

[00:05:47]>> So, I got to take that into account in my amount function.

[00:05:51]>> So, what I'm going to do is I'm going to go ahead and create a function called a of t. A stands for amount.

[00:05:58]And remember, I'm thinking quantities.

[00:06:00]So initially when t equals z there's a quantity of 50 bananas and then I remember that for the first 12 hours they were removing bananas at a rate of f oft.

[00:06:14]All right so since they were removing bananas I'm going to subtract and then I'm going to subtract the integral from zero to whatever time t we want. And that rate of removal was f.

[00:06:29]And since I can't reuse my t, I'm just going to call it f ofx dx.

[00:06:35]>> All right, so we're not done. That's just subtracting the total amount of bananas that were removed. And then I'm going to add the integral from 0 to t the integral of the rate at which bananas were being added, which I believe was what? G, right?

[00:06:53]>> G.

[00:06:53]>> Okay.

[00:06:54]>> I believe they told us that they were being added three hours after the store got open.

[00:06:57]>> Yeah. Yeah. Exactly. So the bounds the bounds I mentioned I mentioned them when I was reading them and I think I highlighted them too. Yeah.

[00:07:04]>> F and G have different bounds. Yes. So if you're not careful like I was you can miss that. F is from 0 to 12. G is from 3 to 12.

[00:07:15]>> Mhm.

[00:07:16]>> Okay. So for G, it's not going to start at zero.

[00:07:21]It's going to start at three like you said last week. That's why there's two of us, right?

[00:07:26]>> That's why there's two of us. And you know that brings up a good point too.

[00:07:29]Like if a student I'm sure students probably when they were taking this exam and did this question put zero as one of the lower bounds.

[00:07:36]>> What happened if you notice that G is a log function and you actually can't put zero into a log function. And so I'm sure that they probably run into an error on their calculator or going crazy and they're like >> so you probably would be able to hopefully course correct when you get that error and go back.

[00:07:51]>> That's true.

[00:07:52]>> Yeah.

[00:07:52]>> Sometimes they be clutching up like that man.

[00:07:55]>> Right. Right. Because without that error they would have just kept on going.

[00:07:58]>> I I know I got a wrong answer.

[00:08:00]>> Yeah. So we have a function of t that allows us to find the amount of bananas at any time. Whatever time we want, we just plug that number in for t.

[00:08:12]>> Yeah.

[00:08:13]>> So since I have my a of t function, we want to find the rate of change at five, which is a prime.

[00:08:20]Mhm.

[00:08:21]>> And then we just need to know if a rate of change is positive, then that amount is increasing and vice versa.

[00:08:28]>> Yeah.

[00:08:29]>> So the sign of a prime of five will give us our answer.

[00:08:33]>> Yep.

[00:08:33]>> Okay. B.

[00:08:34]>> And yeah, like Mr. Alman said, we can type that entire a of t function into the calculator. No, zero to zero to x, >> right? Yeah. Whatever variable you use.

[00:08:44]I use t. Well, t >> but you you defined f and g in functions of x or >> I did I did in terms of x on desmos.

[00:08:57]>> Yeah, you should be able to do t. Should be able to >> or Okay, let me let me try this. So, if I do I wonder if can you put two variables in. Okay, cool. Wait, I think this is going to be cool. Okay, so I typed in h. So 0 to t f ofx dx plus 3 to t of g of x dx.

[00:09:15]>> Mhm.

[00:09:16]>> And we want >> I'm so glad you called out that lower bound, bro.

[00:09:20]>> Yeah. Yeah, >> that was clutch.

[00:09:23]>> Um, so then we're going to do h prime of what number was it? Seven.

[00:09:28]>> Uh, five.

[00:09:30]>> Five. It was seven in part B.

[00:09:32]>> Yeah.

[00:09:32]>> Okay. So we're getting a value of -2.263.

[00:09:38]>> Okay. All right. Okay. So, I'm going to write a prime of 5= -2.263.

[00:09:44]And I'm going to say since um I'll just say da dt is negative um the amount is decreasing at t equals 5. Okay. So, we answered the question. We gave a reason. Okay. Looks good to me.

[00:09:59]>> Yep.

[00:10:00]>> All right.

[00:10:01]>> 24D.

[00:10:02]>> Okay. Nothing crazy.

[00:10:05]Now, this one says, this one is not talking about a rate of change, but it's asking for an amount. How many pounds of bananas are on the display table at time t equals 8. So, we can actually use our amount function from part C and not derive it, but just evaluate it at t equals 8.

[00:10:24]>> Yep.

[00:10:25]>> And then that should be it. And you already have your function aft still typed in?

[00:10:29]>> Yep.

[00:10:29]>> Okay, bet. So, this should be quick and easy.

[00:10:32]>> H of eight.

[00:10:34]>> Yeah. And we >> we don't need to we don't need to justify or give reasoning. We just need to give a number.

[00:10:39]>> Yep.

[00:10:41]Age of age 23.347.

[00:10:45]>> 23.347.

[00:10:49]>> Yep.

[00:10:49]>> And that is it.

[00:10:51]>> That's it.

[00:10:52]>> Okay.

[00:10:54]Right in, rate out, not too crazy. Just remember the integral of a rate represents an amount. If that rate is the rate at which something is being removed, then you have to subtract that integral. And then the rate at which something's being added, you got to add that integral. And I'm pretty sure it's safe to say that there's always going to be like some initial value. I've never seen a time where there is not.

[00:11:17]>> Yeah.

[00:11:18]>> Okay. Sweet. So, yeah, remembering to add that plus 50.

[00:11:21]>> Yep.

[00:11:23]>> That was crucial.

[00:11:24]>> Yeah. All right, man. Not too bad. Not too bad.

[00:11:29]Okay, that is yet another FRQ. Uh, if you made it this far, appreciate you for watching. Make sure to like, comment, and subscribe if you're new. All right, well, we'll catch y'all in the next one.

[00:11:38]Thanks for watching. Peace out.