2011 Calculus AB free response #5b

Now ready for part b. And I've wiped all the salt off of my fingers, and I've had a glass of water. So now I'm ready for some business. Find the second derivative of W with respect to t in terms of W. Use this second derivative to determine whether your answer in part a is an underestimate or overestimate for the amount of solid waste that the landfill contains at time t equals 1/4. So let's do the first things first. Let's find the second derivative in terms of W. And we already have the first derivative over here. And in part a, I rewrote it just with slightly different notation. But I'll just use this just because I need to write down here and I can still refer to this thing. So let's just take the derivative of both sides of this differential equation with respect to t. So you take the derivative of this left-hand side with respect to t, you get the second derivative of W with respect to t as a function of t is equal to-- now you take the derivative of this with respect to t. This is the same thing as 1/25Wt minus 12. And so when you take the derivative, that constant part drops off, and you're just left with 1/25 times the derivative of Wt. Let me make that clear. This is the same thing as 1/25W as a function of t minus 12. 1/25 of 300 is 12. And you take the derivative of this, you get 1/25 times the first derivative of W and then the derivative of this with respect to t is just 0. A constant, obviously, does not change with respect to t. And so we get this right over here. Now this is the second derivative in terms of the first derivative. But the question asks us, write the second derivative in terms of W. But lucky for us, we know how to express this as a function of W. They gave that to us in the problem. This was given. This is just me rewriting it in a different notation. So this is going to be the same thing as 1/25 times the derivative of W. And the derivative of W is this. The differential equation literally tell us the derivative of W is this over here. So 1 over 25 times Wt, the function W as a function of t, minus 300. And so we can say the second derivative-- let me write it a little smaller so I don't waste space. The second derivative of W as a function of t is equal to 1/625 times W as a function of t, which is a function of t, minus 300. So we've done the first part. We've found out the second derivative of W in terms of just W. Now let's try to address the second part of their question. So we did the first part. Now use what we just figured out to determine whether your answer in part a is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time 1/4. So in part a, we found the slope. We found the slope of the tangent line at time equals 0, and we used that slope to extrapolate out to 3 months, or time equals 1/4, 1/4 of a year. Now if the function W's slope over that 1/4 was exactly the same as the slope of the tangent line, or if that slope did not change, then our extrapolation would be exactly right. If W's slope is increasing over that time, then our estimate would be an underestimate of the correct thing. And if the slope is decreasing over that time, then our estimate would be an overestimate of the actual amount in the landfill. And to figure out whether the slope is increasing or decreasing, we just have to look at the value of the second derivative. If the second derivative is positive, that means our slope is increasing, which means that our extrapolation would be an understatement. Let me just be clear here. Let me draw this. Let me make it very clear. So let me just draw a random function. So let's say that's W. So this is a case where W's slope is increasing faster, or W's slope is increasing from that starting point. So our starting point-- this was our slope. And then W's slope keeps increasing from there. And in this case, our estimate is going to be underestimate of where W actually is after 1/4 of the year. If W's slope is exactly the same as our function over the course of the first three months, it would look something like that. Maybe it diverges later on. And in this case, our approximation would be really good. It would probably be exact for W. And if W's slope, for whatever reason, goes negative after that point, and they already tells us it's an increasing function, so that is not likely. Or it could-- it doesn't have to even go negative. If W's slope decreases, it could still stay positive. And the way I drew it, it doesn't-- let me draw it like this, just to make it clear. So let's say we find out that the slope looks something like this. This is the slope of the tangent line at time equals 0. If W's slope increases from that point, then w might look something like that. And then our answer to part a would be an understatement, would be an underestimate of where W actually is after a fourth of a year. Let me draw it and make it clear to you what I'm doing. So this is my w-axis. So this is right at our initial condition of 1,400. And then this right over here is our time axis, and this is that 1/4. So in the last video, we said, hey, this is sitting right at 1,411. If W's slope increases from that point, then this is an underestimate. If W's slope stays the same, then this is actually a very good estimate because then we're going to hit that point directly with W. And if W's slope decreases-- so you can imagine maybe W looks something like this. It has that slope of the tangent line when it starts, but then the slope decreases. And it's still an increasing function, but the slope is decreasing. And in that case, we would have an overestimate. And this is a situation-- so this first situation, slope is increasing. That means W prime prime is positive. The slope is increasing. The second derivative is positive. This would be a byproduct of or this would cause the second derivative to be negative. Our slope is decreasing. And this would be our second derivative. Our second derivative is 0. If your slope isn't changing-- if your slope is constant, if your first derivative is constant, your second derivative is going to be 0. So let's just see what our second derivative is at our initial condition, and then we'll have a pretty good sense of whether we have an overestimate or an underestimate for part a. So let's just figure out what our second derivative at time 0. It's going to be equal to 1/625 times W of 0, which we know, minus 300. Well W of 0, the amount of waste we have at time 0, they told us in the problem, is 1,400 tons. 1,400 minus 1,300 is 1,100. And then 1,100 minus-- or 1,100 divided by 625 is a small number. It's 1. something. But it is a positive number. And that's the important thing here. So this thing, all of this business, it is positive. So the second derivative is positive, which means the slope is increasing. At least right at our starting point, our slope is increasing. Since our slope is increasing from that point-- and frankly, the fact that it's increasing at all and that we know the W is an increasing function-- tells us that our estimate in part a is an underestimate. So this is the case that we're dealing with. So our estimate in part a is an underestimate of the actual amount of solid waste in the landfill at time t equals 1/4.