2017 AP Calculus AB/BC 4b

- [Instructor] We're now going to tackle part (b) of the potato problem, and it says use the second derivative of H with respect to time to determine whether your answer in part (a) is an underestimate or an overestimate of the internal temperature of the potato at time t is equal to three. So in part (a), we found the equation of the tangent line at time equals zero and we used that to estimate what our internal temperature would be at time is equal to three. So how is the second derivative going to help us think whether that was an overestimate or an underestimate? Well, the second derivative can help us know about concavity. It'll let us know, well, is our slope increasing over this interval, or is our slope decreasing? And then we can use that to estimate whether we overestimated or not. So first, let's just find the second derivative. So we have the first derivative written up here. Let me just rewrite it, and I'll distribute the negative 1/4 'cause it'll be a little bit more straightforward then. So if I write the derivative of H with respect to time is equal to negative 1/4 times our internal temperature, which it itself is a function of time, and then negative 1/4 times negative 27, that would be plus 27 over four. Let me scroll down a little bit. So now, let me leave that graph up there 'cause I think that might be useful. What is the second derivative going to be with respect to time? So I'll write it right over here. D, the second derivative of H with respect to time is going to be equal to, well, the derivative of this first term with respect to time is going to be the derivative of this with respect to H times the derivative of H with respect to time. So this is equal to negative 1/4. That's the derivative of this term with respect to H, and then we want to multiply that times the derivative of H with respect to time. This comes just straight out of the chain rule. And then the derivative of a constant, how does that change with respect to time? It's not gonna change, that is just going to be zero. So just like that, we were able to find the second derivative H with respect to time. And now what does this tell us? Well, we talked about, in the previous video, that over the interval that we care about, for, actually, I can show you from this graph. Over the interval that we care about, for t greater than zero, it says that our internal temperature is always going to be greater than 27. And so when you look at this expression here or when you look at this expression here for dH/dt, we talked in the previous video how this is always going to be negative here. Because H is always going to be greater than 27, so that part's going to be positive. But then we're gonna multiply it by a negative 1/4, so our slope dH/dt, our derivative of our temperature with respect to time, is always going to be negative. So we could write that this or this, this is going to be negative, let me write it this way. Since H is greater than 27 for t is greater than zero, we know that dH/dt is negative, is negative. So we could say that this right over here, since dH/dt is less than zero for t is greater than zero, the second derivative of H with respect to t is going to be greater than zero for t is greater than zero. And so what does that mean? It means that we are, if your second derivative is positive, that means you're concave upward, concave upward, which means slope is increasing, slope increasing. Or you could say slope becoming less negative, slope becoming less negative. Now what does that mean? And you could see it intuitively. If the slope is becoming less and less negative, then that means when we approximated what our temperature is at t equals three, we used a really negative slope. When, in reality, our slope is getting less and less and less negative. So what we would've done is we would've over-decreased our temperature from t equals zero to t equals three. So that would mean that we would have underestimated it. So let me write that down. And I'm running out of a little bit of space, but let me write it right over here. So that implies, so this implies that underestimate, underestimate in part (a). And if I were taking this on the AP exam, I would flesh out my language a little bit here to make it clear, but hopefully that makes sense.