Determining when first derivative equals average rate of change, 2025 Calculus AB free response 1b

0:00- [Narrator] We're told an invasive species of plant appears in a fruit grove at time t equals 0 and begins to spread. The function, C, defined by C(t) is equal to 7.6 times the arctangent of 0.2t models the number of acres in the fruit grove affected by the species t weeks after the species appears.

0:20It can be shown that C'(t) or the first derivative of C is equal to 38 over 25 plus t squared. And it says, "Note: Your calculator "should be in a radian mode." So this is a calculator-based free response question, or at least we can use a calculator, find the time t when the instantaneous rate of change of C equals the average rate of change of C over the time interval from t going from 0 to 4, show the setup for your calculations.

0:48So pause this video, have a go at this before we do this together. All right, so to figure out the time when our instantaneous rate of change equals average rate of change, we first have to figure out the average rate of change. And when you're doing this on an AP exam, it's really important that you show your work and you label that you're finding the average rate of change.

1:05It actually turns out that on the grading manual for this, they give two points for this, one to show that you actually calculated the average rate of change and then the other point is for getting the final answer and showing all of your work with that. So make sure to show all of your work, not just the final answer.

1:22So let's first think about what the average rate of change is going to be. Well, this is essentially we're going to find the slope between when t equals 0 and t equals 4. So one way to think about it is it's going to be the value of the function at 4 minus the value of the function at 0.

1:44All of that over 4 minus 0. Notice, this is just the slope between the (0, C(0)) and the point (4, C(4)). That's average rate of change. You actually don't need to do any calculus at this point. You're just calculating slope. So for this, the denominator is pretty straightforward.

2:08That is just 4. But the numerator, that's where we're gonna have to get our calculator out. And this is where it's useful. Let's make sure it's okay. Now I'm in degree mode, now I am in radian mode. It's good that they told us to do that. So let's see. Let's just calculate this whole thing.

2:25So first of all, we're going to have, actually I'm gonna multiply 7. Well, lemme just do it all at once. So if I were to take 0.2 times 4, times 4, that equals 0.8. And then if I were to take the inverse tangent of that, so I have tangent on this fun calculator.

2:51Now I can do inverse tangent. And if you're actually taking the AP exam, you probably want to use some type of a graphing calculator, but this is good for what I need to do right now. So I'm gonna take the inverse tangent of that and I get all of this business and then I'm gonna multiply that times 7.6, times 7.6.

3:137.6, is equal to all of this. Now from that I have to subtract C(0). So we have 0 times 0.2 is 0, an arctangent of 0 is just 0. And so the second term is all going to be 0. So this whole thing is just going to boil down to 5.128 divided by 4. So divided by 4 is equal to 1.282 all of that business.

3:46So let me just write that down here. So this is approximately equal to 1.282. And if we cared about the units here, this would be in acres per week. So then we just have to figure out when does our derivative, which is equal to 38 over 25 plus t squared, when does this equal this business?

4:111.282, so on and so forth. Well, to solve there, we just have to do some algebraic manipulation. Let's see, we can multiply both sides by 25 plus t squared and we would get 38 is equal to 1.282 times 25 plus t squared. And now we would get, let's see, I'm running outta space down here.

4:35We'd get 38 is equal to and let me get the calculator back. We have this business times 25, times, whoops. Oh, well, I said times 1. So that doesn't change it. Times 25 is equal to let's say 32.050. I'm trying to keep three decimal places here, but let's see, 32.

5:02So we get 32.050 plus 1.282 t squared. Now I can solve this if I subtract, if I subtract 32.050 from both sides, I get 1, go over here, I get 1.282 t squared is equal to 38. My brain is malfunctioning, 38 minus 32.050. And then I'd wanna divide both sides by 1.28, divide both sides by 1.28 and I'd get t squared is equal to all of that.

5:40Or I could say that t is equal to the square root and I'm just doing this all at once so I don't have to keep going to the calculator. 38 minus 32.050, all of that over, all of that over 1.28. So now let me go to my calculator and actually calculate this. So I had that 32, let me turn that.

6:04Lemme get rid of the second here. Turn that into a negative. So I'm gonna subtract that from 38. I get all of that. I'm gonna divide by 1.28, divided by 1.282. Actually, I should do 1.282, . 282 if I wanna maintain the precision. So let's just say there's a 282 there, that equals this.

6:30And then I want to take the square root of that and I get t is equal to 2.154 weeks, 2.154 weeks and I'm done. And if you're taking the actual AP exam, I encourage you to explain as much of this as possible. What I did is likely to be sufficient. I showed my average rate of change here, I showed the units, I showed that I then set my derivative to be equal to that average rate of change and I solved for t.

7:04But the more you can explain, it doesn't hurt.